# A Critical Study of the Finite Difference Time Dependent Schrödinger Equation

A Critical Study of the Finite Difference and Finite Element Methods for the Time Dependent Schrödinger Equation Simen Kvaal Thesis Submitted for the Degree of Candidatus Scientiarum Department of Physics University of Oslo March 2004 Preface This thesis is perhaps a bit lengthy compared to the standards of a cand. scient. degree (or Master Degree as it will be called in the future in order to comply with international standards). It is however aimed at a broad audience; from my fellow physics students to mathematicians and other non-physicists that may have interests in the area. I have tried to cut away material in order to make it a little bit shorter, and some less central material is moved to appendices for the specially interested readers. The title of the thesis pretty much describes the aim of this cand. scient. project. The ﬁnite diﬀerence and ﬁnite element methods are two widely used approaches to solving partial diﬀerential equations. Traditionally the ﬁnite element method has been reserved for engineering projects and not that much in basic research ﬁelds such as atomic physics where the ﬁnite diﬀerence method is the main method. One reason may be that ﬁnite element methods are very complicated to implement and that they utilize a wide range of complicated numerical tools, such as sparse matrices and iterative solvers for linear equations. For this reason ﬁnite element solvers are usually expensive commercial products (such as FemLab, see Ref. [1]) whose operation hides the numerical details for the user, an approach that in a way makes scientists feel that they lose control of the situation. Some work has been done on the Schrödinger equation with ﬁnite element methods early in the eighties and nineties, see for example Refs. [2–4], but for some reason the development has seemed to stagnate. One reason might be the above mentioned complexity in implementation. There is a huge threshold to climb if one wants to generalize a simple formulation which can be coded in a few hundred lines, experiment with diﬀerent element types and so on. Fortunately, the programming library Diﬀpack is tailored for scientiﬁc problems, making available very powerful and ﬂexible class hierarchies, interfaces and visualization support. The newly initiated Centre for Mathematics with Applications (CMA) tries to join the forces of mathematics, physics and informatics among others, and a thorough yet basic exposition into quantum mechanics might be in the spirit of such collaborative work. Chapters 1 and 2 contain the fundamentals of quantum mechanics, and physicists may skip these chapters and go directly to chapter 3, in which I discuss more speciﬁc physical problems. Others may ﬁnd the preceding chapters illuminating and interesting. A lot of work has been put into making an understandable discussion aimed at practitioners of natural sciences with a foundation in mathematics. It places itself in the middle of an intermediate course of quantum mechanics and a guide for mathematically trained people with an interest in physics. Chapter 4 deals with ordinary and partial diﬀerential equations and numerical methods. A recipe-based approach is taken, retaining a touch of rigor and analysis along the way. Chapter 5 is a short review of numerical methods for problems from linear algebra; more speciﬁcally square linear systems of equations and eigenvalue problems. We also discuss the implementations of these methods in Diﬀpack, at the same time giving an introduction to this powerful programming library for partial diﬀerential equations and ﬁnite element methods. Chapter 6 turns to quantum mechanical eigenvalue problems and discusses their i Preface importance and their applications in time dependent problems, which are presented in chapter 7. Some numerical experiments are performed and analyzed, yielding an insight into the behavior of both the ﬁnite diﬀerence and ﬁnite element discretizations of the eigenvalue problem. Chapter 7 is the pinnacle of the thesis, around which all other material build up. We discuss the solution of the time dependent Schrödinger equation for a two-dimensional hydrogen atom with a laser ﬁeld perturbation; a system which is both numerically, physically and conceptually interesting. The ﬁnite element implementation is ﬂexible and allows for generalizations such as diﬀerent gauges, diﬀerent geometries, new time integration methods and so on. Chapter 8 concludes and sums up the contents of the thesis, discussing important results and ideas for further work. There are a few appendices; the ﬁrst one picks up some technical details of calculations and formalism, while appendix B contains complete program listings for every program written for the thesis. As is often the case with the work on a cand. scient.-degree the creative process accelerates enormously towards the ﬁnal days before deadline. Ideas pop up, intensive research is being done and many good (and some bad) ideas emerge. Alas, one cannot follow each and every idea. Rather late in the work with the thesis I realized that the mass matrix of ﬁnite element discretizations is in fact (obviously!) positive deﬁnite. This allows for converting every generalized eigenvalue problem arising from ﬁnite element discretizations into a standard eigenvalue problem by means of the Cholesky decomposition. Furthermore, incorporating this into the HydroEigen class deﬁnition should only amount to another hundred lines of code, but with proper testing rituals and debugging sessions it would require another week of work while not adding to the ﬂexibility of the program. On the other hand, this optimization would allow use of much faster eigenvalue searching, which in turn would mean more numerical experiments and more detailed results. There are several computer programs written for this thesis in addition to essential downloadable documentation and texts. For this reason I have created a web page, see Ref. [5], giving access to all source code, reports, animations, this text and more. I would like to thank my supervisors Prof. Morten Hjorth-Jensen and Prof. Hans Petter Langtangen for invaluable support and guidance during the work with the thesis. Morten has spent quite a few hours of valuable time on interesting discussions, providing constructive criticism and proofreading of the text, and making available his deep and thorough insight into both numerical methods and quantum mechanics. Hans Petter was co-supervisor and is responsible for this thesis being possible at all with his thorough knowledge of Diﬀpack, his oﬀering summer holiday work relevant to my thesis and his constant encouragement and conﬁdence in my abilities as a student and programmer. I would also like to thank Prof. Ragnar Winther at CMA for valuable help with section 4.6 on stability analysis of ﬁnite element schemes. In addition, 1st Am. Per Christian Moan (also at CMA) has provided interesting discussions on gauge invariance and time integration. During moments of frustration, slight despair and writer’s block my girlfriend, my family and my friends have (as always!) put up with me. They have provided immense encouragement and support for which I am truly thankful. I dedicate this work to all of you, for as they say, no man is an island. Oslo, March 2004 Simen Kvaal ii Contents 1 A Brief Introduction to Quantum Mechanics 1.1 A Quick Look at Classical Mechanics . . . . . . . . . . . . . . . . 1.2 Why Quantum Mechanics? . . . . . . . . . . . . . . . . . . . . . 1.3 The Postulates of Quantum Mechanics . . . . . . . . . . . . . . . 1.4 More on the Solution of the Schrödinger Equation . . . . . . . . 1.5 Important Consequences of the Postulates . . . . . . . . . . . . . 1.5.1 Quantum Measurements . . . . . . . . . . . . . . . . . . . 1.5.2 Sharpness and Commuting Observables . . . . . . . . . . 1.5.3 Uncertainty Relations: Heisenberg’s Uncertainty Principle 1.5.4 Ehrenfest’s Theorem . . . . . . . . . . . . . . . . . . . . . 1.5.5 Angular Momentum . . . . . . . . . . . . . . . . . . . . . 1.5.6 Eigenspin of the Electron . . . . . . . . . . . . . . . . . . 1.5.7 Picture Transformations . . . . . . . . . . . . . . . . . . . 1.5.8 Many Particle Theory . . . . . . . . . . . . . . . . . . . . 1.5.9 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Electromagnetism and Quantum Physics . . . . . . . . . . . . . . 1.6.1 Classical Electrodynamics . . . . . . . . . . . . . . . . . . 1.6.2 Semiclassical Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 4 6 11 12 12 13 15 16 17 21 22 24 26 27 28 30 2 Simple Quantum Mechanical Systems 2.1 The Free Particle . . . . . . . . . . . . 2.1.1 The Classical Particle . . . . . 2.1.2 The Quantum Particle . . . . . 2.1.3 The Gaussian Wave Packet . . 2.2 The Harmonic Oscillator . . . . . . . . 2.2.1 The Classical Particle . . . . . 2.2.2 The Quantum Particle . . . . . 2.3 The Hydrogen Atom . . . . . . . . . . 2.3.1 The Classical System . . . . . . 2.3.2 The Quantum System . . . . . 2.4 The Correspondence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 35 35 36 37 39 39 41 45 47 48 52 3 The Time Dependent Schrödinger Equation 3.1 The General One-Particle Problem . . . . . . . 3.1.1 Uni-Directional Magnetic Field . . . . . 3.1.2 A Particle Conﬁned to a Small Volume 3.1.3 The Dipole-Approximation . . . . . . . 3.2 Physical Problems . . . . . . . . . . . . . . . . 3.2.1 Two-Dimensional Models of Solids . . . 3.2.2 Two-Dimensional Hydrogenic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 53 54 56 56 56 57 58 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Contents iv 4 Numerical Methods for Partial Diﬀerential Equations 4.1 Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Ordinary Diﬀerential Equations . . . . . . . . . . . . . . . . . 4.1.2 Partial Diﬀerential Equations . . . . . . . . . . . . . . . . . . 4.2 Finite Diﬀerence Methods . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 The Grid and the Discrete Functions . . . . . . . . . . . . . . 4.2.2 Finite Diﬀerences . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Simple Examples . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Incorporating Boundary and Initial Conditions . . . . . . . . 4.3 The Spectral Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 The Discrete Fourier Transform . . . . . . . . . . . . . . . . . 4.3.2 A Simple Implementation in Matlab . . . . . . . . . . . . . . 4.4 Finite Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 The Weighted Residual Method . . . . . . . . . . . . . . . . . 4.4.2 A One-Dimensional Example . . . . . . . . . . . . . . . . . . 4.4.3 More on Elements and the Element-By-Element Formulation 4.5 Time Integration Methods . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 The Theta-Rule . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 The Leap-Frog Scheme . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Stability Analysis of the Theta-Rule . . . . . . . . . . . . . . 4.5.4 Stability Analysis of the Leap-Frog Scheme . . . . . . . . . . 4.5.5 Properties of the ODE Arising From Space Discretizations . . 4.5.6 Equivalence With Hamilton’s Equations of Motion . . . . . . 4.6 Basic Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Stationary Problems . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Time Dependent Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 62 62 63 64 65 66 67 70 71 72 75 77 78 80 83 85 86 88 89 91 94 95 95 96 99 5 Numerical Methods for Linear Algebra 5.1 Introduction to Diﬀpack . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Finite Elements in Diﬀpack . . . . . . . . . . . . . . . . 5.1.2 Grid Generation . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Linear Algebra in Diﬀpack . . . . . . . . . . . . . . . . . 5.2 Review of Methods for Linear Systems of Equations . . . . . . 5.2.1 Gaussian Elimination and Its Special Cases . . . . . . . 5.2.2 Classical Iterative Methods . . . . . . . . . . . . . . . . 5.2.3 Krylov Iteration Methods . . . . . . . . . . . . . . . . . 5.3 Review of Methods for Eigenvalue Problems . . . . . . . . . . . 5.3.1 Methods For Standard Hermitian Eigenvalue Problems . 5.3.2 Iterative Methods for Large Sparse Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 102 102 103 104 107 107 109 111 113 113 115 6 Quantum Mechanical Eigenvalue Problems 6.1 Model Problems . . . . . . . . . . . . . . . . . . 6.1.1 Particle-In-Box . . . . . . . . . . . . . . . 6.1.2 Harmonic Oscillator . . . . . . . . . . . . 6.1.3 Two-Dimensional Hydrogen Atom . . . . 6.2 The Finite Element Formulation . . . . . . . . . 6.3 Reformulation of the Generalized Problem . . . . 6.4 An Analysis of Particle-In-Box in One Dimension 6.5 The Implementation . . . . . . . . . . . . . . . . 6.5.1 Class methods of class HydroEigen . . . 6.5.2 Comments . . . . . . . . . . . . . . . . . . 6.6 Numerical Experiments . . . . . . . . . . . . . . 6.6.1 Particle-In-Box . . . . . . . . . . . . . . . 6.6.2 Two-Dimensional Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 120 120 122 123 124 126 128 131 132 135 136 136 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents 6.7 6.8 6.9 Strong Field Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Intermediate Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . 150 Discussion and Further Applications . . . . . . . . . . . . . . . . . . . . 151 7 Solving the Time Dependent Schrödinger Equation 7.1 Physical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Time Stepping . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Description of the Member Functions . . . . . . . . . . . . . 7.2.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Building and Solving Linear Systems . . . . . . . . . . . . . . 7.3.2 Comparing the Crank-Nicholson and the Leap-Frog Schemes 7.3.3 Comparing Linear and Quadratic Elements. . . . . . . . . . . 7.3.4 A Simulation of the Full Problem . . . . . . . . . . . . . . . . 7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Conclusion 155 155 156 156 158 160 161 162 164 167 170 173 175 A Mathematical Topics A.1 A Note on Distributions . . . . . . . . . . . . . . . . . . . . . . A.2 A Note on Inﬁnite Dimensional Spaces in Quantum Mechanics A.3 Diagonalization of Hermitian Operators . . . . . . . . . . . . . A.4 The Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . A.5 Time Evolution for Time Dependent Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 179 179 180 181 182 B Program Listings B.1 DFT Solver For One-Dimensional B.1.1 fft schroed.m . . . . . . B.2 The HydroEigen class . . . . . . B.2.1 HydroEigen.h . . . . . . B.2.2 HydroEigen.cpp . . . . . B.3 The TimeSolver class . . . . . . B.3.1 TimeSolver.h . . . . . . B.3.2 TimeSolver.cpp . . . . . B.3.3 main.cpp . . . . . . . . . B.4 The EigenSolver class . . . . . . B.4.1 EigenSolver.h . . . . . . B.4.2 EigenSolver.cpp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 185 185 186 186 188 198 198 199 207 207 207 208 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Chapter 1 A Brief Introduction to Quantum Mechanics We will present an introduction to the quantum mechanics of a single particle in this and the subsequent two chapters; its background, formalism and application on simple systems and examples. It will serve as an introduction to the real topic of this thesis, which is the numerical solution of the time dependent Schrödinger equation (which we will introduce later). In this chapter, we will start with some basic classical mechanics to provide means to compare quantum physical concepts to “ordinary” Newtonian concepts, such as measurements and observables, equations of motion and probabilistic features. Then we proceed with the postulates of quantum mechanics. By “postulates” we shall mean concepts that are not obtainable by means of mathematics or deduction from other areas of physics, but on the other hand are necessary and suﬃcient for developing all quantum mechanical results. 1.1 A Quick Look at Classical Mechanics A particle described by classical mechanics behaves in a perfectly ordinary way. That is, it obeys the everyday mechanics of golf balls, planets and coﬀee-cups, also known as Newtonian mechanics. The central force behind Newtonian mechanics is Newton’s second law. This is the well-known diﬀerential equation stating the relation between a particle’s acceleration and the force exerted on it by its surroundings:1 mẍ = F . (1.1) Although Newtonian mechanics is seemingly a well-established theory in this form, mathematicians and physicists have developed other equivalent and in some respects more complicated ways of formulating it. On the other hand, the reformulations yield deep insight into classical mechanics. In fact, the mathematical framework of classical mechanics is much deeper than we will ever be able to present in even a lengthy exposition, and it represents a huge area of mathematics. For an excellent account of classical mechanics, see Ref. [6]. We will give a brief introduction to the Hamiltonian formulation of classical mechanics. Quantum mechanics may be built on this formalism, and the similarities and 1 Note the dot-notation for the time derivative: n dots means n diﬀerentiations with respect to time. 1 A Brief Introduction to Quantum Mechanics z m r(q(t)) q y x Figure 1.1: Particle in three dimensions with constraints parallels are quite instructive. (There is also another and equivalent way of introducing quantum mechanics through the Lagrangian formulation of classical mechanics, see for example Ref. [7].) Let us ﬁrst introduce the material system: A system consisting of n point-shaped particles moving in D-dimensional space. By point-shaped we mean that its conﬁguration is completely determined by a single position in D-space, and that each particle has a non-zero mass mi . The conﬁguration of a material system is then speciﬁed by nD real numbers q = (q1 , q2 , . . . , qnD ), the set of which constitutes the so-called conﬁguration space. To specify the state of the material system however, we need to specify the motion of the system at an instant, and this is done through the momenta p = (p1 , p2 , . . . , pnD ). To put it simple, the momenta represents the velocity of each particle in each direction in D-space. The conﬁguration and momentum of the system are not necessarily the usual cartesian coordinates (xi , yi and zi , i = 1 . . . n) and momenta (mi vx,i , mi vy,i and mi vz,i ), but rather so-called generalized coordinates and momenta. They are also called canonical variables and p and q are called canonical conjugates of each other. For a single particle moving in three-dimensional space, there are initially three coordinates x, y and z in addition to the momenta px , py and pz , but if we impose constraints on the motion, such as forcing the particle to move on an arbitrary shaped wire like in Fig. 1.1, the number may be reduced. In this case we have only one generalized coordinate q. The cartesian coordinates are then given as a function of q. We will not state the general deﬁnition of generalized momenta. For the purpose of the applications in this thesis it suﬃces to think of the pi as the velocity of qi . However, the momenta are deﬁned in such a way that the equations of motion in Hamiltonian mechanics equivalent to Newton’s second law are given by Hamilton’s equations of motion: ∂H ∂pi ∂H ṗi = − ∂qi q̇i = (1.2) (1.2 ) The so-called Hamiltonian is a function of the individual coordinates and momenta and possibly explicitly of time, viz., H = H(q, p, t). The Hamiltonian may usually be thought of as the system’s energy function, i.e., its total energy. In some systems we consider this is however not the case. We will 2 1.1 – A Quick Look at Classical Mechanics x=(0.4,0.3,0.7) p=(0.1,0.3,-0.1) x=(0.4,0.3,0.7) p=(0.1,0.3,-0.1) Figure 1.2: A classical particle and measuring its state introduce electromagnetic forces to the system, and this alters the Hamiltonian and the deﬁnition of the canonical momenta pi . If H is the total energy of the system, and if it is explicitly independent of time, we may easily derive an important conservation law: ∂H ∂H ∂H dH ∂H ∂H = + + . q̇i + ṗi = (−ṗi q̇i + q̇i ṗi ) = dt ∂t ∂qi ∂pi ∂t ∂t i i i Thus, the energy is conserved for a material system if the Hamiltonian does not explicitly depend on time. We shall see examples of this in chapter 2 when we discuss simple quantum mechanical systems and their classical analogies. An important special case of Hamiltonian systems are systems composed of a single particle with mass m moving in three-dimensional space and under inﬂuence of an external potential V (x ), where x is the cartesian coordinates of the particle, viz., H(x , p, t) = T + V = p2 + V (x , t). 2m Here, T is the total kinetic energy. Writing out Hamilton’s equations of motion yields ẋ and ṗ p m = −∇V (x , t), = which is exactly Newton’s second law. (Recall that in a potential ﬁeld the force is given by −∇V .) We will not elaborate any further on Hamilton’s equations until chapter 2, but only note that since they are a set of ordinary diﬀerential equations governing the time evolution of a classical system, classical dynamics is perfectly deterministic, in the sense that given initial conditions x (0) and p(0) for the location and momentum of the particle, one may (at least in principle) calculate its everlasting trajectory through conﬁguration space by solving these equations. Furthermore, the concept of ideal measurements in classical mechanics is rather simple. If we have some dynamical quantity ω(x , p) it may be calculated at all times using the coordinates and momenta at that time. There is no interference with the actual system. We can picture the particle as if it had a display attached, with x and p continuously updated and displayed for us to read, as in Fig. 1.2. Functions ω of the canonical coordinates and momenta are called classical observables. A measurable quantity can only be deﬁned in terms of the state of the system, hence it must be such a function. On the other hand, x and p are measurable so that any function of these also must be an observable. Newtonian mechanics is in most respects very intuitive and agrees with our ordinary way of viewing things around us. Things exist in a deﬁnite place, moving along deﬁnite paths with deﬁnite velocities. Indeed, this mechanistic viewpoint was securely founded 3 A Brief Introduction to Quantum Mechanics ψ1 ψI S1 S1 I1 I 1+2 S2 ψ2 Figure 1.3: Schematic view of double-slit experiment (left) and discovery of photons (right) already in the 18th century, when Pierre Simon De Laplace (1749–1827) proposed his deterministic world view, see Ref. [8]: “We may regard the present state of the universe as the eﬀect of its past and the cause of its future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.” This view was however profoundly shaken through the advent of quantum mechanics. One simply cannot measure the positions and velocities with inﬁnite accuracy. In addition, the new science of chaos makes the Laplacian determinism somewhat hollow. If we consider Hamilton’s equations of motion that govern many physical systems, they turn out to exhibit chaotic behavior, i.e., small variations in initial conditions increase exponentially with time, making long-term predictions of the system’s conﬁguration impossible, see Ref. [9]. 1.2 Why Quantum Mechanics? We will not go into the historical reasons of how and why quantum mechanics came to be, although this is very interesting in its own respect. Quantum mechanics arose in the ﬁrst decades of the 20th century when modern experiments and measurements contradicting the classical theories created a crisis in the physical communities. In Refs. [10, 11] excellent and entertaining accounts of these matters are given. We will instead make an illustration that is quite popular, explaining why we need quantum mechanics and at the same time introducing some of the concepts, making it easier to follow the introduction of the postulates of quantum mechanics in the next section. Imagine that we direct a beam of monochromatic light described by a wave ψI onto a plate with two slits S1 and S2 . At some distance behind the plate we place a photosensitive screen. The experimental setup of this double-slit experiment is shown in Fig. 1.3. Some of the light will pass through the plate. The outgoing waves ψ1 and ψ2 coming from S1 and S2 , respectively, will interfere with each other, producing an interference pattern I1+2 at the screen. (The subscript indicates what the slits are open.) This is very well understood if we assume the wave-nature of electromagnetic ﬁelds; an assumption that has been made for more than a century, after Maxwell proposed 4 1.2 – Why Quantum Mechanics? S1 I1 S2 I2 I 1+ I 2 Figure 1.4: Classical particles scattered by double-slits, and their combined arrival distributions his famous equations that show that light propagates as waves in a way very similar to waves on the surface of water.2 In fact we may imagine an analogous experiment with water-waves and observe the results above. Suppose now that we close S2 and lower the intensity of the beam ψI . Then we will notice that the light no longer arrives in a continuous fashion at the screen; we observe small bursts of light. We have made the monumental discovery that light actually comes in small bundles – they have a particle nature. If we observe a large number of these so-called photons and make a histogram of their arrival-coordinate x, we will of course produce the pattern I1 , see Fig. 1.3. The fact that light turned out to be particles was surprising, but not in direct opposition to classical physics. Physicists are used to the idea that phenomena appearing continuous in reality are composed of small and discrete components. A drop of water is for example composed of tiny molecules. But classical physics was anyway in for a surprise. Experimentally we may ﬁnd that each photon carries the same energy E = hν, where ν is the frequency of the light waves. Furthermore, we may ﬁnd that they all share the same momentum p = h/λ, where λ is the wavelength, related to the frequency by λν = c, the speed of light. The number h is called Planck’s constant, and h ≈ 6.626 068 076 · 10−34 Js, a very small number.3 What happens when we open S2 again? If the photons are particles, they will pass either S1 or S2 , creating the pattern I1 + I2 . The photons passing S1 will contribute to I1 , and the photons passing S2 will contribute to I2 . This picture of the photons as classical particles is shown in Fig. 1.4. Instead, we observe that the pattern is indeed very diﬀerent from I1 + I2 , even with the very low intensity, which can only mean that the photons cannot pass through one single slit in a well-deﬁned way. In some strange way they must pass both! It turns out that the classical particle picture has some ﬂaws after all. The same experiment may be used with electrons; physical objects that until the 1920s were solely regarded as particles in the classical sense. But as the joint diﬀraction patterns I1 + I2 fails to reproduce I1+2 in this case as well, we must abandon the Newtonian particle nature of the electrons; at least when they pass the slits S1 and S2 . In fact, electron diﬀraction experiments have been carried out with an electron intensity as low as one electron per minute, see Ref. [13]. In 1926 De Broglie proposed a daring hypothesis (see Refs. [10, 14]), stating that 2 See 3 The section 1.6. uncertainty of the cited h lies in the two last digits. Cited from Ref. [12]. 5 A Brief Introduction to Quantum Mechanics any physical object can be ascribed a wavelength λ given by the relation λ= h , p (1.3) which is the same expression as for the wavelength of photons, i.e., λ= c hc hc hc h = = = = . ν hν E pc p Indeed, with De Broglie’s hypothesis the results from the double-slit experiment make very much sense when we assume that a wave – not a particle – passes the slit, but that a particle is being found at the screen. Before the ground-shaking surprise the wave ψ could be ascribed because the electrons and photons were so large in number. Quantum mechanics says that each particle is also a wave, in the sense that it is ascribed a wave ψ(x), whose square amplitude |ψ(x)|2 is the probability density that the particle will be found at the position x. Each particle is of course observed at some speciﬁc place x along the screen when the wave arrives. However, we cannot tell what the wave itself looks like. This is because the wave is only a probability density for one single particle’s position. Only if a large ensemble of particles starting out the same way hit the screen we may deduce from a histogram what |ψ(x)|2 looks like. On the other hand an intense electron beam will completely obscure the particle-nature, and we will observe a continuous wave-front of particles, very much like as in the case with light. We see that the probabilistic nature of measurements is fundamental to quantum mechanics. So why do we have Newtonian physics in the ﬁrst place? The answer lies in the number h, which is a very small number. If we calculate the wavelength of, say, a 1 kg basket ball moving with 1 m/s, we will get λ ≈ 6 × 10−27 m, a number which is extremely small compared to the size of the basketball. To observe diﬀraction patterns this wavelength must be comparable to the size of the slits which is absurd. It is very diﬃcult to come up with an experiment showing the wavelike nature of basketballs. In summary we have experimentally turned down the concept of microscopic particles moving along deﬁnite trajectories. Instead we say that they have a wave-like nature and that their motion is associated with some wave ψ(x). Classical Newtonian physics cannot be used, at least in cases where the De Broglie wavelength h/p is comparable to the size of the system considered. 1.3 The Postulates of Quantum Mechanics In this section we will introduce the postulates of non-relativistic quantum mechanics. Even though we until now have been talking about waves distributed in space we shall make a formulation in more abstract mathematical terms. The connection between these abstract terms and the concrete example of the double-slit experiment will become clear as the discussion progresses. The language of quantum mechanics is linear algebra, with all its implications. This connection is clearly seen through the ﬁrst three postulates. Postulate 1 The state of a quantum mechanical particle is described by a vector Ψ in a complex Hilbert space H. On the other hand, all the possible states of the particle are exactly H minus the zero vector. Not every vector of H corresponds to distinct states. Every vector Ψ = αΨ 6 1.3 – The Postulates of Quantum Mechanics with α a (complex) scalar, corresponds to the same state. The diﬀerent “rays” or directions in Hilbert space thus corresponds to distinct physical states. We will show this explicitly later, when discussing observables and measurements. Particles exist in physical space, and this is reﬂected through the fact that our Hilbert space almost always contains L2 (R3 ), the set of square-integrable complexvalued functions on R3 . Indeed, the wave-shape of the double-slit experiment is just the vector Ψ represented as an L2 -function. (See Refs. [15–17] for a discussion of L2 .) But unlike in classical physics a particle in quantum mechanics has not only spatial degrees of freedom. It may also have so-called spin, which is an additional intrinsic degree of freedom found in many elementary particles. In that case our Hilbert space looks like H = L2 (R3 ) ⊗ Cn , where n depends on the fundamental particle in consideration. In other words, the function Ψ(x ) has n complex components Ψ(σ) , σ = 1, 2, . . . , n. For so-called spin1 2 particles we have n = 2. The most fundamental examples of such particles are electrons, protons and neutrons, the constituents of all atoms and also most of the known universe. Photons, by the way, are spin-1 particles, with n = 3, but they do not obey non-relativistic quantum mechanics as they have zero rest mass. Sometimes we reduce our Hilbert space (in some appropriate way) and consider particles in one or two dimensions. We may also neglect the spin degrees of freedom as in the double-slit illustration, or even neglect L2 when our particle is conﬁned to a very small volume in space.4 In addition, other ways of approximating H by neglecting some parts of it are used. Indeed, the ﬁnite element methods to be discussed can be viewed in this way. There are obviously striking diﬀerences between the classical way of describing a particle’s state and the quantum mechanical way. The classical state is fully speciﬁed with momentum x and position p; six degrees of freedom in total. But in quantum mechanics the particle’s state is only completely speciﬁed with an inﬁnity of complex numbers, as L2 is an inﬁnite-dimensional vector space. This diﬀerence stems from the probabilistic interpretation of the wave function hinted at in the previous section; we need a probability density for all points in space. The second postulate concerns physically observable quantities. Postulate 2 Every measurable quantity (i.e., observable) is represented by an Hermitian linear operator in H. For every classical dynamical variable ω(x , p) there corresponds an operator Ω obtained by operator substitution of the fundamental position and momentum operators X and P, respectively: Ω = ω(x → X , p → P). The components of X and P are operators deﬁned through the fundamental quantum commutation relation [Xi , Pj ] = iδij . When Ψ is represented as an L2 -function, the position operator Xi is just a multiplication of Ψ(x ) with xi . The momentum operator then becomes Pi = −i ∂ , ∂xi Note however that this choice is not unique, as shown in Ref. [7]. Furthermore, as it stands now it really makes an assumption on the diﬀerentiability of Ψ. 4 See chapter 3 for a further discussion. 7 A Brief Introduction to Quantum Mechanics We have simpliﬁed things somewhat. Given ω, the operator Ω may be ambiguous. As an example consider the dynamical quantity ω = xp for a particle in one dimension. Since X and P do not commute we cannot tell what operator XP or P X to prescribe, and the physical results are not equivalent when using the diﬀerent operators. In addition, neither operator is Hermitian. The remedy is most often to use the so-called Weyl product 1 Ω = (XP + P X) 2 instead, but there is no universal recipe in more general cases where this fails. We will not elaborate anymore on this, but for a more thorough discussion, see Ref. [18]. The observables Ω described above have a classical counterpart. But quantum particles have degrees of freedom that have no classical meaning. In this part of Hilbert space we also have linear Hermitian operators, and these cannot be interpreted classically. Examples of such observables are the spin operators that are used to measure the direction of the intrinsic spin of spin- 12 particles. The third postulate concerns (ideal) quantum mechanical measurements of observables. The meaning of ideal measurements are delicate, see page 122 of Ref. [7], and we will only consider them in a mathematical and in an operational way, without discussing the deeper physical meanings of them, such as if ideal measurements are possible at all. Postulate 3 The only possible values obtainable in an (ideal) measurement of Ω are its eigenvalues ωn . They have a probability (1.4) P (ωn ) ∝ (Ψ, Πn Ψ) of occurring, with Πn the projection operator onto the eigenspace of ωn . Immediately after a measurement of Ω, the quantum state collapses into Πn Ψ. This postulate contains a great deal of information, vital in order to understand how quantum mechanics work. We assume for simplicity that the spectrum ωn of Ω is discrete. First of all we have the collapse of the state vector. When we have obtained a value ωn for the observable Ω the state of the particle changes into Πn Ψ, the projection of Ψ onto the eigenspace of ωn . Contrary to a classical system, measuring an observable quantity interferes with the system, and it is inevitable as well. This is perhaps the most fundamental diﬀerence between classical physics and quantum physics. An implication of this in terms of the double-slit experiment, is that we cannot follow the particle’s trajectory through the slits. Indeed, there is no such thing as a trajectory. When we measure the position we destroy the wave-pattern that the interaction with the slits created, and the particle will never reach the screen and contribute to the distribution dictated by the same wave-pattern. Thus, we cannot observe both the particle nature and the wave nature at the same time – an important fact. But what about the proportionality factor in Eqn. (1.4)? This is where we ﬁnd out about which vectors in H correspond to distinct physical states. Since the only obtainable values for Ω are the ωn s, we must have P (ωn ) = 1. n That is, the total probability of getting one of the eigenvalues when performing a measurement is unity. Since Ω is an Hermitian operator, it has a complete set of orthonormal eigenvectors Φn so that we may expand Ψ in this basis, viz., (Φn , Ψ)Φn . Ψ= n 8 1.3 – The Postulates of Quantum Mechanics For simplicity we assume that we have no degeneracy of the eigenvalues ωn , that is they are all distinct. Since Φn are orthonormal, we get Πn Ψ = (Φn , Ψ)Φn , and 1= n P (ωn ) = C n (Ψ, Πn Ψ) = C Ψ, (Φn , Ψ)Φn = C(Ψ, Ψ). n For this to hold, we must choose C = Ψ−2 . In that case, we have an explicit formula for P (ωn ): (Ψ, Πn Ψ) P (ωn ) = . (Ψ, Ψ) Scaling Ψ with some arbitrary nonzero complex number α, we see that the probabilities are all conserved. This must hold for every observable, and thus the probability distribution of every possible observable is conserved when we scale Ψ. This is why each direction in Hilbert space corresponds to distinct physical states, rather than each individual vector.5 We simply cannot observe any diﬀerence among the diﬀerent vectors along the ray. We only have a probabilistic way of forecasting what value an observable Ω will have if we measure it. The double-slit experiment demonstrated this clearly when we measured the position of each particle. We have assumed here that the spectrum of Ω is discrete. This is not always the case, and in the case of a continuous spectrum, we have to change our formulation a little bit: The only obtainable values becomes ω(x), where x is a parameter in some interval (instead of a discrete index), and P (ω(x)) becomes a probability density instead of a probability. So far we have discussed the state Ψ of a particle and the description of observable quantities. What is left is the time development of the system, and this is exactly what is given in the fourth postulate. Postulate 4 The time development of the quantum state Ψ is given by time dependent Schrödinger equation, viz., ∂ (1.5) i Ψ(t) = HΨ(t). ∂t We may interpret this postulate for the double-slit example. Eqn. (1.5) tells us in what way the particle-wave will move through the slits. Compare the Schrödinger equation with Hamilton’s equations of motion and Newton’s second law, which are two equivalent classical counterparts of the fourth postulate. They dictate the unique evolution in time of the state of a material system. Since Eqn. (1.5) is a ﬁrst order diﬀerential equation, knowledge of the wave Ψ at one instant makes us able to forecast the evolution at all times, past and future. We may demonstrate this explicitly. Note that when we describe the state Ψ as a function of spin and the spatial coordinates, Eqn. (1.5) becomes a coupled set partial diﬀerential equations of second order in space. We will investigate (partial and ordinary) diﬀerential equations later in chapter 3 and 4. We will assume that H is independent of time. Given Ψ(t1 ), suppose we want to ﬁnd Ψ(t2 ). Even though Ψ(t) is a vector, we may create a Taylor expansion of Ψ(t2 ) 5 There is also a reason connected to the fourth postulate, on time development of Ψ. Does αΨ develop diﬀerently in time compared to Ψ? The answer is negative due to the Schrödinger equation being linear. 9 A Brief Introduction to Quantum Mechanics around t = t1 :6 ∞ ∆tn ∂ n Ψ Ψ(t2 ) = , n! ∂tn t=t1 n=0 where ∆t = t2 − t1 . But according to Eqn. (1.5), ∂Ψ(t) i = − HΨ(t1 ), ∂t t=t1 so that our Taylor expansion becomes: n ∞ 1 i i Ψ(t2 ) = Ψ(t1 ) = exp −∆t H Ψ(t1 ), −∆t H n! n=0 where we have used the deﬁnition of the exponential of an operator. Deﬁning the so called propagator, or time evolution operator, i U(t2 , t1 ) = exp −(t2 − t1 ) H , (1.6) we may write Ψ(t2 ) = U(t2 , t1 )Ψ(t1 ). In other words we have found a linear operator U that solves the time dependent Schrödinger equation in the sense that it takes a quantum state Ψ(t1 ) into itself at time t2 , at least formally. Note that since H is Hermitian, that is H † = H, we have i U † (t2 , t1 ) = exp (t2 − t1 ) H = U(t1 , t2 ) = U(t2 , t1 )−1 , so that U is a unitary operator. Alternatively, we may note that the adjoint is simply a Taylor series backwards in time, and thus unitarity follows immediately. For time dependent Hamiltonians things are a little bit more complicated, and we defer its discussion until appendix A. The operator U is still a unitary operator from the Taylor expansion argument. An important implication of the unitarity of the propagator, is that the norm of Ψ(t) is conserved in time, viz., Ψ(t2 ) = U(t2 , t1 )Ψ(t1 ) = (Ψ(t1 ), U † (t2 , t1 )U(t2 , t1 )Ψ(t1 )) = Ψ(t1 ) . 2 2 2 Conservation of norm is equivalent to conservation of probability. Recall that we may decompose a state Ψ into its components along an orthonormal basis, viz., Ψ= cn Φn , n where the Φn may be precisely the eigenvectors of some observable Ω, again assumed to be non-degenerate and discrete for simplicity. Then, the probability of obtaining the eigenvalue ωn from a measurement of Ω is P (ωn ) = (Ψ, Πn Ψ) 2 Ψ = |cn |2 2 Ψ = |(Φn , Ψ)|2 Ψ 2 , 6 This requires, of course, that the solution to the diﬀerential equation exists, which we silently assume here. 10 1.4 – More on the Solution of the Schrödinger Equation and all these add up to unity, viz., P (ωn ) = n |cn |2 n Ψ 2 = Ψ = Ψ−2 (Ψ, −2 |cn |2 = Ψ −2 n n Πn (Ψ, Φn )(Φn , Ψ) Ψ) = Ψ−2 Ψ2 = 1. n We have used that the eigenvectors of Ω constitute a basis, that is the sum of all the projection operators on the eigenspaces is the identity, viz., Πn = 1. n Thus, when Ψ is conserved, so is the total probability for obtaining some of the eigenvalues ωn when measuring Ω. In short, unitarity of U(t2 , t1 ) allows for probabilistic interpretation of Ψ. 1.4 More on the Solution of the Schrödinger Equation In this rather compact section we will take a closer look at the solution of the Schrödinger equation (1.5) for time independent Hamiltonians, i.e., where the Hamiltonian does not contain an explicit dependence on time, viz., ∂ H ≡ 0. ∂t Since H is Hermitian, we may ﬁnd a basis H of eigenvectors of H, viz., HΦn = En Φn , (1.7) and we assume for simplicity that the energy spectrum (i.e., the eigenvalues of H) are discrete and non-degenerate. Eqn. (1.7) is referred to as the time independent Schrödinger equation. We expand the wave function Ψ(t) in this basis, viz., cn (t)Φn . Ψ(t) = n Inserting this expansion into the time independent Schrödinger equation yields i dcn n dt Φn = En cn (t)Φn . n Both sides represent a non-zero element in H in terms of a basis. Hence, the terms must be equal as well, i.e., dcn = En cn (t) i dt which implies cn (t) = cn (0)e−iEn t/ . The solution to the time dependent Schrödinger equation then reads Ψ(t) = cn (0)e−iEn t/ Φn , (1.8) n where Ψ(0) = n cn (0)Φn . 11 A Brief Introduction to Quantum Mechanics The coeﬃcients cn along the energy basis rotate with angular frequency En /. The magnitude |cn | is easily seen to be constant, and hence the probability of ﬁnding the system in the state Φn remains constant. It is then easy to see why Φn is called a stationary state. If Ψ = Φn for some n the system remains in the eigenstate at all t. Notice that the solution (1.8) is simply the application of the time evolution operator from Eqn. (1.6) when expressed in a basis of eigenvectors for H, i.e., when H is diagonal, viz., i Ψ(t) = e− tH Ψ(0). A consequence of Eqn. (1.8) is that solving the time dependent Schrödinger equation for time independent Hamiltonians is equivalent to diagonalizing H; that is equivalent to ﬁnding its eigenvalues and eigenvectors. This may be looked upon as a simpler problem than solving the original Schrödinger equation (1.5), and indeed in light of Eqn. (1.8) it is not surprising that ﬁnding an algorithm for solving the Schrödinger equation numerically for time dependent Hamiltonians is more diﬃcult than for time independent ones. 1.5 Important Consequences of the Postulates The postulates together with the facts of linear algebra in Hilbert spaces lay the foundation for quantum mechanics, and now is the time for gaining some physical insight based on the postulates. The goal of this section is to present some important results whose contents are vital for understanding and working with quantum mechanics. 1.5.1 Quantum Measurements The second and third postulate tell us about observables in quantum mechanics. When measuring an observable ideally, in a sense that we shall not deﬁne, the quantum state collapses onto the eigenvector (or rather onto the eigenspace) corresponding to the eigenvalue found. Thus, we destroy our perhaps painstakingly constructed quantum system in the process. As we have noted, quantum mechanics is probabilistic in nature. Some things we do not know for certain, only with a certain probability. Measuring a quantum state destroys it, making further investigations useless. It is clear that statistical methods will become in handy. In statistics, if we have some quantity A taking on the values An , where n is some index, with some probability distribution Pn , then the expectation value A is deﬁned as: Pn An . A := n When performing a very large sequence of experiments, each obtaining one of the values An , we will on average get the value A. Applying this on a quantum observable Ω and its eigenvalues with their probabilities, we get, when we assume Ψ = 1: Ω = n P (ωn )ωn = n (Ψ, Φn )(Φn , Ψ)ωn = (Ψ, ΩΦn )(Φn , Ψ) = (Ψ, ΩΨ), n where we have used that the eigenvectors Φn constitute an orthonormal basis for our Hilbert space. If Ψ is not normed to unity, then we must introduce an additional factor −2 Ψ as is easily seen. 12 1.5 – Important Consequences of the Postulates Theorem 1 The expectation value of some observable A in the state Ψ is given as: A = (Ψ, AΨ) . (Ψ, Ψ) Although we have only shown this for a discrete, non-degenerate spectrum, it also holds in general for continuous and degenerate spectra, the key reason being the orthonormality of the eigenvectors and the fact that they span the whole Hilbert space H. Finding the average value of an observable thus amounts to applying the observable (i.e., its operator representation) to the state, and projecting the result back onto the state. This is the best prediction we can make of the outcome of a measurement. 1.5.2 Sharpness and Commuting Observables When we measure an observable Ω the state collapses onto the eigenspace of the obtained value ωn . If we try to measure the same observable immediately after the previous measurements, we will with certainty get ωn as result, because by the third postulate we know that Ψ = Φn , the eigenstate corresponding to ωn . We say that ωn is a sharply determined value of Ω in the state Ψ. In other words, the probability is 1 that we ﬁnd ωn when measuring the observable for the state Ψ. If the system in consideration is in an eigenstate of an observable Ω, then we know with certainty that a measurement will yield the corresponding eigenvalue. Conversely, if we want P (ωn ) = 1, then (Φm , Ψ) = 0 for m = n. Thus, having a sharp eigenvalue of some observable for a system is equivalent to the system being in an eigenstate of the observable. An important question arises, and indeed it is one of the most important: What conditions must be fulﬁlled for two observables to have sharply determined values at the same time? Theorem 2 A suﬃcient and necessary condition for two observables A and B to have a common set of orthonormal eigenvectors is that A and B commute, i.e., if [A, B] := AB − BA = 0. Proof: Again we show this for a discrete spectrum, but this time degeneracy is also included. An immediate requirement for two observables to have sharp values in the same state at the same time is that the observables must have common eigenvectors, because a sharp value is only obtained in the case of the system being in an eigenstate. Assume that an and bn are the eigenvalues of A and B, respectively, and that Φn are the corresponding eigenvectors of both operators: AΦn BΦn = = a n Φn bn Φn Operating on the ﬁrst relation with B and the second with A, and subtracting yields (AB − BA)Φn = (an bn − bn an )Φn = 0, and since Φn is not the zero vector, we get [A, B] = 0. 13 A Brief Introduction to Quantum Mechanics For the converse, assume that [A, B] = 0, and assume that we are given the eigenvalues and eigenvectors of A, viz., AΦn = an Φn . Using commutability we get A(BΦn ) = B(AΦn ) = an (BΦn ), implying that BΦn is an eigenvector of A with the eigenvalue an . Assume that an is a non-degenerate eigenvalue. Then BΦn must be some scalar multiple of Φn , because the eigenspace has only dimension one. Let us call this scalar bn , viz., BΦn = bn Φn , and we are done. Assume then that an is degenerate, and that the eigenspace has (inﬁnite or ﬁnite) dimension g. Assume that Φm n , m = 1 . . . g is a basis for the eigenspace. BΦn must be in the eigenspace and thus be a linear combination of the basis vectors, viz., BΦm Mmi Φin . n = i This vector is not necessarily an eigenvector of B. But a linear combination Ψ might be, viz., Ψ= ci Φin . (1.9) i We assume that Ψ is an eigenvector for B with eigenvalue b, viz., BΨ = bΨ, and writing out the left hand side, we get ci Φin = ci Mij Φjn . BΨ = B i i j On the right hand side we have bΨ = b cj Φjn . j Since the vectors Φm n are linearly independent, the right and left hand side must be equal term by term also, viz., Mij ci = bcj , i which is nothing more than the jth component of the matrix equation M c = bc. This is a new eigenvalue problem of dimension g. But since B is Hermitian, so is the g × g matrix M , and thus we may ﬁnd g eigenvalues bj and eigenvectors c. This gives coeﬃcients ci for Eqn. (1.9), and thus we have found g (orthonormal) eigenvectors of B which also are eigenvectors of A. We have found that A and B have a common set of eigenvectors, and we are ﬁnished. Note that even though Φm n are eigenvectors corresponding to the same eigenvalue an for A, the eigenvalues bj found for B may be diﬀerent. 14 1.5 – Important Consequences of the Postulates 1.5.3 Uncertainty Relations: Heisenberg’s Uncertainty Principle The converse of sharpness is that Ψ is not an eigenstate for our observable A, and thus there is some probability that we will get another value than an when measuring the observable. As a measure of the degree of sharpness we may use the standard deviation ∆A, viz., (1.10) ∆A := A2 − A2 . For an eigenstate Φn of A, we trivially get ∆A2 = (Φn , A2 Φn ) − (Φn , AΦn )2 = 0, so we have zero uncertainty in the case of a sharp value for the observable A. If A has standard deviation ∆A (in a given state), what is the optimal standard deviation ∆B for another observable B? The question has fundamental importance, and the answer is one of the most striking facts of quantum mechanics. Theorem 3 Given two observables A and B, we have the uncertainty relation ∆A∆B ≥ 1 |[A, B]| . 2 (1.11) Proof: The standard text-book proof is a rather elegant one. Deﬁne two new operators Â and B̂ by: Â = A − A1, and B̂ = B − B1. Of course, Â and B̂ are also Hermitian. Note that [Â, B̂] = [A, B]. Let Ψ be an arbitrary vector in H, and deﬁne H = Â + iαB̂, where α ∈ R is arbitrary. We must have HΨ2 = (Ψ, H † HΨ) ≥ 0, by deﬁnition of the norm. Calculating the product H † H yields H † H = Â2 + iα[Â, B̂] + α2 B̂ 2 , and thus HΨ2 = Â2 + iα[Â, B̂] + α2 B̂ 2 . Note that we must have i[Â, B̂] ∈ R since the norm is a real number.7 We then choose i[Â, B̂] . α=− 2B̂ 2 Insertion into the norm gives us Â2 ≥ − [Â, B̂]2 4B̂ 2 . When noting that Â2 = ∆A2 and similarly for ∆B, we immediately get the desired result (1.11). A very important special case is the Heisenberg uncertainty principle: ∆Xi ∆Pj ≥ 7 It δij . 2 (1.12) is easy to show that [A, B]† = −[A, B], and thus i[A, B] is Hermitian. 15 A Brief Introduction to Quantum Mechanics This relation states that if the particle is localized to degree ∆Xi in the ith spatial direction, then the corresponding momentum is sharp to a degree limited by the Heisenberg uncertainty principle. Contrast this result to the classical principle depicted in Fig. 1.2. The classical particle oﬀers complete information of its state at all times through the coordinates and momenta. It has ∆Xi · ∆Pi = 0 in all situations. The quantum particle behaves however in a more complex way. A quantum particle localized perfectly like this will have inﬁnite ∆Pi ! 1.5.4 Ehrenfest’s Theorem We will state and prove the so-called Ehrenfest’s theorem, which in some sense connects the time development of the expectation values of Xi and Pi to Hamilton’s equations and the movement of a classical particle. But ﬁrst we need a more general result concerning time dependencies of expectation values. Theorem 4 Given an observable A, its expectation value develops in time according to the diﬀerential equation i ∂A d A = [H, A] + . (1.13) dt ∂t The last term accounts for the explicit dependence on time of A. Proof: We begin by noting that with an expression such as (Ψ, AΨ), we may use the familiar product rule for diﬀerentiation (with respect to t). This can be done since we may expand both operators and states in Taylor series in time,8 viz., ∂Ψ(t) + O(∆t2 ) ∂t ∂A(t) + O(∆t2 ). and A(t + ∆t) = A(t) + ∆t ∂t Using these expansions when diﬀerentiating for example AΨ we get Ψ(t + ∆t) = Ψ(t) + ∆t ∂ 1 (A(t)Ψ(t)) = lim (A(t + ∆t)Ψ(t + ∆t) − A(t)Ψ(t)) ∆t→0 ∆t ∂t ∂Ψ 1 ∂A = lim )(Ψ(t) + ∆t ) − A(t)Ψ(t) (A(t) + ∆t ∆t→0 ∆t ∂t ∂t ∂A(t) ∂Ψ(t) = Ψ(t) + A(t) . ∂t ∂t If we are able to expand some time dependent quantity in Taylor series, this argument will hold. Thus, we get ∂A ∂ ∂Ψ ∂A ∂Ψ = (Ψ, AΨ) = ( , AΨ) + (Ψ, Ψ) + (Ψ, A ) ∂t ∂t ∂t ∂t ∂t i ∂A i = (− HΨ, AΨ) + (Ψ, − AHΨ) + ∂t i ∂A i = (Ψ, HAΨ) + (Ψ, − AHΨ) + ∂t i ∂A = [H, A] + . ∂t 8 Again 16 we are assuming that the solution Ψ(t) exists. 1.5 – Important Consequences of the Postulates This completes the proof. The important special case named Ehrenfest’s theorem concerns the operators Xi and Pi in particular. Theorem 5 Assume H = L2 (R3 ), that is we are working with square integrable functions and without spin degrees of freedom. Assume that ∇2 Ψ ∈ H. Assume that the Hamiltonian is given by P2 + V (X ), H= 2m where V is a diﬀerentiable function of X . Then, ∂ Xi ∂t ∂ and Pi ∂t 1 Pi m ∂V = − . ∂Xi = (1.14) (1.15) Proof: We begin by computing a commutator: P2 P2 1 1 Xi + V Xi − Xi − Xi V = [P 2 , Xi ] = [P 2 , Xi ] 2m 2m 2m 2m i 1 (Pi [Pi , Xi ] + [Pi , Xi ]Pi ) = Pi . = 2m 2im [H, Xi ] = This gives us Eqn. (1.14) upon insertion into Eqn. (1.13). Next, we compute another commutator: [H, Pi ] = [V, Pi ] = −i[V, ∂ ∂V ] = i , ∂Xi ∂Xi when we regard the operators as operating on functions. This immediately gives Eqn. (1.15), when we insert the commutator into Eqn. (1.13). In other words: Hamilton’s equations of motion (1.2) are satisﬁed for the expectation values. This is not a strong link to classical theory as one might think. Consider a particle with a high degree of localization. Then we may take ∆Xi ≈ 0. In addition, if the particle has low speed (or high mass) we may take ∆Pi ≈ 0. The distribution for position and velocity exhibit the features of the state of a classical particle. Furthermore, the expectation value on the right hand side of Eqn. (1.15) become simply a sampling of the gradient of V at the particle’s now well-deﬁned position. For a general quantum particle one must sample the gradient of the potential in whole space, and not locally as in Newtonian mechanics. This is a very big diﬀerence. Similar results exist regarding semiclassical electrodynamics, as considered in section 1.6. The expectation values do not only sample the electromagnetic forces locally through a gradient or a curl, but through an integration of these over whole space. 1.5.5 Angular Momentum An important class of observables comprises the angular momentum operators. They are deﬁned as three Hermitian operators J1 , J2 and J3 constituting a vector operator J obeying the following commutation relations: [J1 , J2 ] = iJ3 [J2 , J3 ] = iJ1 [J3 , J1 ] = iJ2 (1.16) 17 A Brief Introduction to Quantum Mechanics In short, the three operators Ji obey [Ji , Jj ] = iijk Jk , where we have assumed Einstein’s summation convention (summation over repeated indices) and the Levi-Civita symbol ijk which is equal to the sign of the permutation of the numbers 1, 2 and 3, if ijk is a permutation of these, and zero otherwise. The angular momentum operators get their names from the fact that the observables associated with each of the three components of the classical orbital momentum obeys the relations (1.16). The orbital momentum is given by i j k L := r × p = x y z px py pz = (ypz − zpy )i + (zpx − xpz )j + (xpy − ypx )k . (1.17) We will return to this later, when discussing the solution of the hydrogen atom. Let us deﬁne J 2 as the operator given by J 2 := Ji Ji = J12 + J22 + J32 . We call J 2 the square length (or square norm) of the angular momentum. The components of the quantum mechanical angular momentum cannot be measured sharply at the same time, as we know from Theorem 3. For example, ∆J1 ∆J2 = |J3 |. 2 However, is a very small number in macroscopic terms, actually essentially zero according to the correspondence principle. Thus, for the classical orbital angular momentum of Newtonian mechanics we may measure each component Li simultaneously with in reality perfect accuracy. However, there are angular momentum operators that are not derived from classical quantities yielding important quantum mechanical effects. The intrinsic spin of particles such as electrons will display this, as we will see later. It is rather straightforward to show that J 2 commutes with all three Ji . Theorem 6 Given three angular momentum operators Ji , i = 1, 2, 3, the square norm J 2 = Ji Ji obeys the following commutation relation: [J 2 , Ji ] = 0, i = 1, 2, 3. Proof: We begin by writing out the ﬁrst argument of the commutator, viz., [J 2 , Ji ] = [Jj Jj , Ji ] = Jj Jj Ji − Ji Jj Jj = Jj Jj Ji − ([Ji , Jj ] + Jj Ji )Jj = Jj [Jj , Ji ] − iijk Jk Jj = ijik Jj Jk − iijk Jk Jj In the last expression, we are summing over both j and k. Swapping their names in one of the terms makes the expression vanish identically, viz., ijik Jj Jk = ikij Jk Jj = iijk Jk Jj . We have used that ijk = kij since cycling the symbols does not change the sign of the permutation. According to Theorem 2, J 2 has a common set of eigenvectors with each of the components Ji . However, since the components by deﬁnition do not commute among 18 1.5 – Important Consequences of the Postulates themselves, we cannot have a common set of eigenvectors for J 2 and more than one component at the same time. We may then have a sharp value of J 2 and for example J3 , but J1 and J2 will not be sharply determined. Comparing this to the classical orbital momentum, this is like saying that we may know the absolute value of the angular momentum and the z-component, but not the x and y component. We will now state a very important and striking theorem concerning the angular momentum operators. The theorem puts restrictions on the possible eigenvalues of Ji and J 2 . Its proof rests solely on the deﬁning relations (1.16). In the proof, we use a ladder operator technique very similar to that of the solution of the harmonic oscillator in section 2.2 later. Thus it might be of interest to read these sections in conjunction. The ladder operators are deﬁned as J+ and J− := J1 + iJ2 , := J1 − iJ2 . † ; they are Hermitian adjoints of each other. Note that J− = J+ Theorem 7 The square of the norm of the angular momentum, J 2 , has a common set of orthonormal eigenvectors Φjm with J3 . The eigenvalues of J 2 and J3 may only be given by J 2 Φjm and J3 Φjm = j(j + 1)2 Φjm , = mΦjm , where j ≥ 0 is integral or half-integral, and where m takes the values m = −j, −j + 1, . . . , j − 1, j. Thus, m is also integral or half-integral, and the eigenvalues of J 2 are degenerate with degeneracy (at least) 2j + 1. Proof: At this stage, m and l are arbitrary, and we will gradually reduce their freedom to the restrictions stated in the theorem. First of all j and m are dimensionless numbers, since has the same dimension as Li , as can be seen from Eqn. (1.16). Furthermore, we see that j(j + 1) ≥ 0, since L2 is positive deﬁnite, and we may deﬁne j ≥ 0 since the equation j(j + 1) = x has a non-negative solution j for all x ≥ 0. We claim that −j ≤ m ≤ j. We prove this by considering the norm of the vectors Ψ− and Ψ+ , which we deﬁne as Ψ± := J± Φjm . The norm of Ψ± is easily calculated: 2 Ψ± = (Φjm , J± J∓ Φjm ) = (Φjm , (J1 ± iJ2 )(J1 ∓ iJ2 )Φjm ) = (Φjm , (J12 + J22 ± i[J2 , J1 ])Φjm ) = (Φjm , (J − J32 ± J3 )Φjm ) 2 = 2 (j(j + 1) − m(m ∓ 1)) Φjm . Since the norm must be non-negative, we get −j ≤ m ≤ j + 1, and − j − 1 ≤ m ≤ j, 19 A Brief Introduction to Quantum Mechanics for the plus and the minus sign, respectively. Hence, −j ≤ m ≤ j, as were claimed. Next, we claim that m can only diﬀer from j by an integer, and that j is either integral or half-integral. Equivalently, j is integral or half-integral, and so is m. To prove this, we begin by noting that J 2 commutes with both J− and J+ , so J 2 Ψ± = 2 j(j + 1)Ψ± , and Ψ± is also an eigenvector for J 2 with the same eigenvalue as Φjm . In other words, Φ± lies in the same eigenspace for the J 2 operator’s eigenvalue as Φjm . We then operate with J3 on Ψ± : J3 Ψ± = J3 (J1 ± iJ2 )Φjm = ([J3 , J1 ] + J1 J3 ± i[J3 , J2 ] ± iJ2 J3 )Φjm = (iJ2 ± iJ2 J3 ∓ J1 + J1 J3 )Φjm = (iJ2 ± imJ2 ∓ J1 + mJ1 )Φjm = (m ± 1)(J1 ± iJ2 )Φjm = (m ± 1)Ψ± . That is, Ψ± is an eigenvector for J3 with eigenvalue (m ± 1), that is Ψ± = J± Φjm = CΦj(m±1) . By operating successively on one single Φjm , we get a ladder of increasing and decreasing eigenvalues and corresponding eigenvectors, but these cannot have eigenvalues outside the range −j . . . j. Observe that |C| = j(j + 1) − m(m ± 1), so that the sequence terminates at j = m and j = −m for J+ and J− , respectively. For both sequences to terminate properly, we must have j integral or half integral, because this is the only way to make the diﬀerence between the maximum and minimum value of m integral. Our claim is proven, and in fact we have also proved our theorem. To sum up: j may only take the values 1 3 j = 0, , 1, , . . . , 2 2 while m ranges like m = −j, −j + 1, . . . , j − 1, j. We see that the eigenvalue j(j + 1) for J 2 is has a degeneracy of 2j + 1.9 For a ﬁxed value of j, the eigenspace has dimension 2j + 1. This eigenspace has a basis of eigenvectors of each Ji , but since they do not commute their eigenvectors are not the same. It is clear that if j is ﬁxed we may describe the operators Ji as square Hermitian matrices of dimension 2j + 1, once we have chosen a basis for the eigenspace. It is rather surprising that we may get so much information just from the commutation relations (1.16), and Theorem 7 has a wide range of applications; from ﬁnding the energy eigenstates of the hydrogen atom and other spherical symmetric systems to the theory on eigenspin of elementary particles such as electrons, the latter which we will turn to right now. 9 At least 2j + 1 is more correct to say; it may happen that the eigenvalues of J again are degen3 erated. 20 1.5 – Important Consequences of the Postulates 1.5.6 Eigenspin of the Electron A surprising feature of the electron theory is the necessity of introducing an additional degree of freedom called eigenspin or just spin. As mentioned in section 1.3, the Hilbert spaces for electrons, protons, neutrons and many other particles are not only composed of L2 (R3 ), but also Cn , the latter ascribed to the spin degrees of freedom. One of the experimental facts that lead to the discovery of electron spin was the famous Stern-Garlach experiment, in which silver ions were accelerated through a magnetic ﬁeld. If the electrons where spinning in a classical sense, they would be deﬂected according to the distribution of the spin. It turned out however, that the deﬂected ions were concentrated into two spots, indicating a discrete nature of the eigenspin. See for example Ref. [11] for a complete account of the eigenspin discovery. The eigenspin is described through angular momentum operators acting on vectors in Cn . For a particle of spin s, we have the spin operator S , and S 2 has by deﬁnition only one eigenvalue, namely 2 s(s + 1). The eigenspace of each Si is then n = 2s + 1 dimensional, according to Theorem 7, and we may choose Cn as the Hilbert space connected to eigenspin. The spin components becomes spin matrices, and we choose S3 to be diagonal. In other words, the eigenvectors of S3 are chosen as a basis for the spin part of Hilbert space, and the spin matrices are described in this representation. Let us turn to the spin-1/2 particles, such as the electron. Since s = 1/2 we get C2 as our Hilbert space. Let S3 be diagonal and write 1 χ+ := , 0 0 . and χ− := 1 Deﬁning S3 as the 2 × 2 matrix S3 = 2 1 0 , 0 −1 makes χ+ have an S3 eigenvalue of +/2 and χ− an eigenvalue of −/2, viz., S 3 χ+ and S 3 χ− χ+ 2 = − χ− . 2 = Taking S3 as diagonal is just a conventional choice: Later, when discussing the hydrogen atom, we use spherical coordinates in which the z-axis has a special meaning. In addition, interactions of both orbital momentum and eigenspin with a magnetic ﬁeld becomes simpler to describe mathematically if we direct the ﬁeld along the diagonal axis of spin, see chapter 3. Using the raising and lowering operators of Theorem 7, we may ﬁnd S1 and S2 as well: 0 1 S1 = 2 1 0 0 −i and S2 = . 2 i 0 The matrices are easily seen to obey Eqn. (1.16). Note that Si does not act on any spatial degree of freedom. Thus, Si commutes with both Xi and Pi . If the Hamiltonian does not contain some interaction term such as a magnetic ﬁeld, then the spin operators also commute with H. Then the spin state 21 A Brief Introduction to Quantum Mechanics z C S r r y x Figure 1.5: The “spinning top electron” of the particle does not aﬀect the time development of the spatial wave function, and we may ignore it altogether as it has no physical signiﬁcance. On the other hand, if H contains a magnetic ﬁeld, then the magnetic moment associated with S will interact with the magnetic ﬁeld and disturb this ignorance of the spatial wave function of the spin, such as demonstrated in the Stern-Garlach experiment. Let us explain this a little further. Classically, when a charged particle of orbital momentum L is placed in a magnetic ﬁeld B, it gains an additional potential energy V =− qg µ · B, L · B = −µ 2mc where q is the particle charge. The dimensionless number g is called the gyromagnetic factor and classically this is related to the charge distribution of the particle. We write Γ = qg/2mc as a shorthand later when discussing the particular form of the time dependent Schrödinger equation to be solved, in chapter 3. The quantity µ deﬁned through µ := ΓL is called the magnetic moment. When this is nonzero we have coupling between the magnetic ﬁeld that may be present and the orbital momentum of the electron. We assume that the eigenspin also has an associated magnetic moment µ and a gyromagnetic factor to be determined experimentally. After all, we imagine the spin as some spinning property (albeit non-classical) of the electron which is a charged particle. Perhaps we may view the electron as a pure impossible-to-perceive spinning top. Thus, when electromagnetic forces are present in our system we must account for the spin state of the particle as well as the spatial state. Can we get some sort of picture of the “spinning top”? If we imagine that the electron is in the state χ+ , then the z component of the eigenspin has a sharp value of /2. The x and y components however, are not possible to determine. But the norm of the spin is 32 /4. Consider Fig. 1.5, where we have depicted everything we know about the spin state of the electron. Even though the spin vector S is not possible to measure directly, we may imagine that this “actual spin” lies on the circle C with √ radius r = / 2 at a distance /2 above the xy plane. 1.5.7 Picture Transformations In this section we shall consider so-called picture transformations, in which the vectors are transformed by a (possibly time dependent) unitary operator T (t). It corresponds 22 1.5 – Important Consequences of the Postulates to changing the frame of reference in Newtonian mechanics. We view our quantum states from another point of view, so to speak. Assume that T (t) is a unitary operator on Hilbert space with a possible explicit time dependence. Consider the transformation Ψ(t) −→ ΨT (t) := T (t)Ψ(t). (1.18) This transformation is invertible due to T (t) being unitary, i.e., T −1 = T † . The time dependence of Ψ is governed by the time dependent Schrödinger equation i ∂ Ψ(t) = H(t)Ψ(t), ∂t (1.19) and we ask: What is the governing equation for ΨT (t)? The left hand side of Eqn. (1.19) may be rewritten as † ∂T ∂ † † ∂ T (t)ΨT (t) = i i ΨT (t) + T ΨT (t) . ∂t ∂t ∂t We rewrite the right hand side in a similar way, viz., H(t)Ψ(t) = H(t)T † (t)ΨT (t). Multiplying Eqn. (1.19) with T (t) from the left and reorganizing the equation we get i ∂ ΨT (t) = HT (t)ΨT (t), ∂t with HT (t) := T (t)H(t)T † (t) + i We have also used that (1.20) ∂T † T (t). ∂t ∂T † ∂T T = −T † , ∂t ∂t which follows from diﬀerentiating T T † = 1 with respect to t. We may also ﬁnd the picture transformation of an arbitrary operator A. Let Φ = AΨ, and using Eqn. (1.18) we get ΦT = T Φ = T AΨ = T AT † ΨT , and thus A(t) −→ T (t)A(t)T † (t), (1.21) is the picture transformation of an operator A. As a consequence of the deﬁnition of the picture transformed operator, we get conservation of expectation values in the diﬀerent pictures: A = (Ψ, AΨ) = (T † ΨT , AT † ΨT ) = (ΨT , T AT † ΨT ) = AT . We summarize these ﬁndings in a theorem. Theorem 8 Consider a picture transformation given by Ψ(t) −→ ΨT (t) := T (t)Ψ(t) 23 A Brief Introduction to Quantum Mechanics where T (t)T † (t) = 1. Then ΨT obeys the picture transformed Schrödinger equation i ∂ ΨT (t) = HT (t)ΨT (t), ∂t and the picture transformed operators obey A(t) −→ T (t)A(t)T † (t). The two pictures are physically equivalent, in the sense that expectation values are conserved, viz., A = AT . Note that there is no reason to think of one picture as more fundamental than another. The original “Schrödinger picture” may be obtained from the picture given by T (t) with the picture transformation ΨT (t) −→ Ψ(t) = T † (t)ΨT (t). If T is unitary, so is of course T † ! In other words: For every unitary operator there exists a picture. And when a unitary operator acts on an orthonormal basis, we get another orthonormal basis. Hence, changing the picture may be looked upon as simply changing the frame of reference, if we consider the frame of reference to be the basis we describe our states in. To conclude we remark that through the following chain of identities, ΨT (t ) = T (t )Ψ(t ) = T (t )U(t , t)Ψ(t) = T (t )U(t , t)T † (t)ΨT (t), we obtain a relation for the transformed propagator UT , viz., UT (t , t) = T (t )U(t , t)T † (t). 1.5.8 Many Particle Theory Quantum mechanics is not only able to describe a single particle. It is also applicable to systems with many particles (i.e., material systems) in the same way as Hamiltonian mechanics is. We will here give a brief introduction to the non-relativistic quantum description of many particles. A thorough exposition is out of scope for this thesis; see Ref. [7]. Many-body theory is one of the most fundamental ingredients of various disciplines in physics such as nuclear physics, solid state physics and particle physics. The quantum mechanical formalism for many particles is obtained from assuming that the wave function depends on all the particle coordinates xi ∈ R3 , and the Hilbert space for N particles then becomes H = L2 (R3N ) ⊗ C2s1 +1 ⊗ C2s2 +1 ⊗ . . . ⊗ C2sN +1 , The fundamental commutator is [Xi,r , Pj,s ] = iδij δrs , i, j = 1 . . . N and r, s = 1 . . . 3, and we must bear in mind that there are 3N degrees of freedom. The commutators are thus deﬁned for coordinates and momenta belonging to diﬀerent particles as well. Most applications of many-body theory involves so-called identical particles. Identical particles have the property that there is nothing distinguishing the states of systems obtained from one another by permuting the positions of the particles. For a quantum mechanical system we may describe it in this way. If we by SN denote the set of permutations of N symbols, and if we by σ(x1 , . . . xN ) denote this permutation applied to the N coordinate vectors, then the physical properties of Ψ(x1 , . . . , xN ) 24 1.5 – Important Consequences of the Postulates should be identical to those of Ψ(σ(x1 , . . . , xN )). Here, we include the spin degrees of freedom in xi , since it is a coordinate on equal footing as the cartesian position even though it is just an integer. Note that this is an additional postulate to the four already given in section 1.3. It is diﬃcult to make this notion precise. Most textbooks use the above deﬁnition of identical particles based on the permutations of positions in conﬁguration space. Unfortunately this is physically inconsistent and should be avoided. Indeed, if the particles are distinguishable, the permutations of the particles is a non-observable concept. Indistinguishability is fundamental. Introducing the postulate leads to theoretical predictions in perfect agreement with experiments. The physical identiﬁcation of states obtained by permuting the particles is equivalent to the statement Ψ(pij (x1 , . . . , xN )) = ±Ψ(x1 , . . . , xN ), where pij is a transposition of particle i and j. In other words, Ψ is either symmetric or anti-symmetric with respect to particle exchanges. Particles of nature can thus be divided into two categories; those with symmetric wave functions and those with anti-symmetric wave functions. The former particles are referred to as bosons and the latter as fermions. This is an experimental fact not to be overseen and indeed it is fundamental in most branches of microphysics. A further experimental fact is that bosons always have integral spin and fermions always half-integral spin. Thus electrons, protons and neutrons are fermions, while photons are bosons. For fermions we may immediately identify the so-called Pauli principle, stating that two (identical) fermions cannot occupy the same quantum state. If we assume that two particles i and j occupy the same place in space, then Ψ(pij (x1 , . . . , xN )) = Ψ(x1 , . . . , xN ) = −Ψ(pij (x1 , . . . , xN )), so that Ψ(x1 , . . . , xN ) = 0 whenever two of the coordinates coincide. It is not obvious that this implies that two fermions are not allowed to be in the same state in general (and what is meant by “the same state” for two particles in a state for an N -particle system), but we have not yet developed the necessary quantum mechanical results to establish this. In Ref. [19], J.M. Leinaas and J. Myrheim shows that the identical particle postulate is actually superﬂuous. By carefully treating the transition from classical mechanics to quantum mechanics they managed to obtain the quantum mechanical notion of identical particles from the corresponding classical notion which is easier to describe accurately. The basic idea is that because of the indistinguishability of the classical particles their conﬁguration space is not R3N but rather R3N /SN ; the quotient space obtained by identifying conﬁgurations which diﬀer only by a permutation of the positions of the particles. Such quotient spaces are more complicated than the Euclidean space we normally use, as they may have non-zero curvature and singularities. For classical systems particles are not usually allowed to inhabit the same locations in space, simply because they are mutually repelling, such as electrons. In that case the eﬀect of using the modiﬁed conﬁguration space is not seen. In quantum mechanics the eﬀect is profound, and for three dimensional systems the notion coincides with the one sketched above. For lower-dimensional systems however, Leinaas and Myrheim showed that a diﬀerent behavior was possible as well; the sign change when transposing two particles instead becomes a phase change eia , with a real. 25 A Brief Introduction to Quantum Mechanics 1.5.9 Entanglement A famous quote by Einstein says that “God does not play dice.” There are many possible ways to interpret this, but the original context in which it was uttered was theoretical physics. In this short section we will discuss some of the rather esoteric features of quantum physics. Ref. [20] contains a concise yet precise treatment of the subjects of this section. Considering the fourth postulate, quantum physics is deterministic in the sense that the state evolves in time according to a diﬀerential equation. On the other hand, it is non-deterministic in the way that the outcome of an experiment is completely random. In 1935, Einstein, Podolsky and Rosen proposed a famous gedanken experiment in which the non-deterministic features yielded somewhat absurd consequences. Because of this Einstein felt that quantum physics must be incomplete; that there had to be a bigger “super-theory” that included the present form of quantum theory. To understand this paradox we ﬁrst need to furnish the concept of entanglement for manybody systems. Consider a particle with Hamiltonian H, which we for simplicity assume is independent of time. We diagonalize H, i.e., ﬁnd orthonormal φn such that Hφn = n φn , and φn form a basis for Hilbert space H. For arbitrary one-particle functions we have c n φn . ψ= n If we now consider an N -particle system with a Hamiltonian given by H = N H(i), i=1 where H(i) denotes that only operators concerning degrees of freedom belonging to particle number i is present in the term, the state Φ = φn1 (1) ⊗ φn2 (2) ⊗ · · · ⊗ φnN (N ), is easily seen to be an eigenstate of H , viz., H Φ = (n1 + · · · + nN )Φ. The Hilbert space for the N -particle system may be written as H = H · · · ⊗ H, ⊗H⊗ N terms where H is the Hilbert space associated with one particle. The states Φ above is easily seen to constitute a basis for H . Thus we may construct quite general states of the N -particle system by considering the direct product of arbitrary one-particle functions, viz., Ψ = ψ1 (1) ⊗ ψ2 (2) ⊗ · · · ⊗ ψN (N ). These states are however not the most general ones. Indeed, a simple superposition of two such states cannot be factorized in this way. If we considered product states exclusively, then we would actually simply study the one-particle system. But any square-integrable function of the diﬀerent coordinates may be used as a state and such non-product states are called entangled states. 26 1.6 – Electromagnetism and Quantum Physics As an example let us consider a one-particle Hamiltonian with only two eigenstates, viz., Hψi = i ψi , i = 1, 2. A basis of eigenstates for the two-particle Hamiltonian H = H(1) + H(2) is then given by the four states ψ1 (1)ψ1 (2), ψ1 (1)ψ2 (2), ψ2 (1)ψ1 (2) and ψ2 (1)ψ2 (2). (We have omitted the direct product symbols to economize.) Their energies are 21 , 1 +2 (for two states) and 22 , respectively. If we measure the total energy H(1)+H(2) of the system, then these are the values we may obtain. The energy of one single particle, say H(1), is also an observable, and upon measurement it will yield either 1 or 2 . Consider then the state 1 Ψ = √ (ψ1 (1)ψ2 (1) − ψ2 (1)ψ1 (2)) , 2 which is not an eigenfunction for the total energy, but rather an (anti-symmetric) linear combination of such. It is an entangled state. If we measure the energy of particle 1 and obtain 1 the state collapses into Ψ −→ Ψ1 (1)Ψ2 (2). But then we know the energy of particle 2 because the new state is an eigenstate for H(2). In other words, there is a perfect correlation between the energies of the particles. Measuring one particle’s energy will also determine the energy of the other. This is the essential content of the Einstein-Podolsky-Rosen paradox, but what is so paradoxical about this result, besides the fact that operating on one particle’s degrees of freedom seemingly aﬀects the other, even though H(1) and H(2) commute? One may prepare a quantum system in many ways. For example, one may prepare two particles to be localized in space; one here on earth and one on the moon. We may also prepare these particles’ spins to be in an entangled state at the same time. Substituting “energy” with “spin” in the above argument then leads to the fact that there may be perfect correlation between the spins of two particles very far away from each other. A measurement on the particle at home will instantaneously aﬀect the one at the moon. Einstein somewhat humorously called this phenomenon “spooky action at a distance.” Einstein and others felt that such behavior was absurd; that quantum mechanics should be local. An operation at a point should not immediately aﬀect other points in space due to the limited speed of light. On these grounds, one may hope for some kind of local “super-theory” that contains quantum mechanics as a special case. Mathematically, such a super-theory is called a hidden variable theory. The British mathematician John Bell derived a series of inequalities that any hidden variable theory must obey. These may actually be tested experimentally, and indeed experimental results tend to invalidate Bell’s inequalities for quantum mechanics. Then quantum mechanics is a complete theory and the “spooky action” must be a real phenomenon. 1.6 Electromagnetism and Quantum Physics An interesting class of time dependent Hamiltonians are the ones describing the interaction between a charged particle such as an electron and a classical electromagnetic ﬁeld. In this section we will give a very brief summary of this semiclassical theory. It is called semiclassical because only the charged particle is described in quantum mechanical terms; the electromagnetic ﬁeld is still a classical ﬁeld which really should 27 A Brief Introduction to Quantum Mechanics be quantized to make a consistent theory. However, when the electromagnetic ﬁeld has macroscopic magnitudes, the quantum behavior becomes neglible compared to the classical features, as dictated by the correspondence principle, see section 2.4. 1.6.1 Classical Electrodynamics Let us ﬁrst describe the electrodynamic theory in classical terms. On one hand we have a charged particle of charge q. This particle interacts with an electric ﬁeld E and a magnetic ﬁeld B. In this context, a ﬁeld means a vector quantity with three components varying in space. We refer to E and B jointly as the electromagnetic ﬁeld. The response of a charged particle of charge q due to the electromagnetic ﬁeld is given by the Lorentz force:10 1 (1.22) F =q E + v ×B . c The number c is the speed of light in vacuum. On the other hand the famous Maxwell’s equations give the response of the electromagnetic ﬁeld due to a charge, or a charge distribution ρ in general: ∇ · E = 4πρ 1 ∂B ∇×E + =0 c ∂t ∇·B =0 4π 1 ∂E = j ∇×B − c ∂t c (1.23) (1.24) (1.25) (1.26) The charge density ρ(x ) represents charges distributed in space. For a point particle of charge q situated at r = r0 we have11 ρ = qδ(r − r0 ), since the total charge then becomes ρ d3 r = q. all space The current density j deﬁnes the movement of the charge density through the continuity equation ∂ρ + ∇ · j = 0. (1.27) ∂t This equation must hold if Maxwell’s equations is to be fulﬁlled at the same time. For a charged particle we obtain j = qv δ(r − r0 ), where v = ṙ0 is the velocity of the particle. Note that the Lorentz force and Maxwell’s equations are connected through the appearance of the charge density and the current density. If we assume that no charges are present, i.e., ρ = j = 0, we may easily combine Maxwell’s equations and obtain the well-known wave equations and ∂2E ∂x2 ∂2B ∂x2 = = 1 ∂2E , c2 ∂t2 1 ∂2B , c2 ∂t2 10 In this thesis we employ Gaussian units for the electromagnetic ﬁeld, which are deﬁned through the proportionality constants in front of each term in Eqn. (1.22). See Ref. [21]. 11 Here we use the three dimensional Dirac delta function δ(x ) = δ(x )δ(x )δ(x ). 1 2 3 28 1.6 – Electromagnetism and Quantum Physics which shows that the electromagnetic ﬁelds propagate as waves through charge-free space with velocity c. Working with the ﬁelds E and B directly may be cumbersome, and more insight is gained if we introduce the potentials φ and A as follows: Eqn. (1.24) implies that B may be written as a curl, viz., B = ∇ × A. We insert this into Eqn. (1.25), and get 1 ∂A ∇× E + = 0, c ∂t which implies that the expression in parenthesis may be written as a gradient, viz., −∇φ = E + 1 ∂A , c ∂t or 1 ∂A − ∇φ. c ∂t Note that we have reduced the six degrees of freedom of E and B to four degrees of freedom, a considerable simpliﬁcation. The potentials A and φ are not unique. We may add to A the gradient of an arbitrary function Λ and still get the same ﬁelds E and B, and thus no physical distinction between the transformed potentials and the old ones. Let us prove this claim. Let A −→ A = A − ∇Λ, E =− and let 1 ∂Λ . c ∂t This transformation is called a gauge transformation and the function Λ is called a gauge parameter. The physical ﬁeld B of the transformed potential A becomes φ −→ φ = φ + B = ∇ × (A − ∇Λ) = ∇ × A − ∇ × ∇Λ = ∇ × A = B, since the curl of a gradient vanishes identically. For the transformed electric ﬁeld E we obtain 1 ∂Λ 1 ∂ (A − ∇Λ) − ∇ φ + E = − c ∂t c ∂t 1 ∂Λ 1 ∂Λ 1 ∂A − ∇φ + ∇ − ∇ = E. =− c ∂t c ∂t c ∂t Thus we have shown the gauge invariance of the electric ﬁelds, and hence the physical laws, under a gauge transformation. It is then clear that the gauge parameter Λ cannot have any physical signiﬁcance. Gauge invariance is a necessity of any physical theory utilizing the potentials A and φ directly. (If only the electromagnetic ﬁelds E and B themselves occur, then gauge invariance is automatically obtained.) The choice of Λ is rather arbitrary. The proof rested on the diﬀerentiability of Λ and the commutability of diﬀerentiation with respect to time and with respect to space. Hence, at least continuously diﬀerentiable functions Λ may be used. The gauge parameter may be chosen in diﬀerent applications at our convenience to simplify calculations and derivations. We will return to this later on, when discussing the interaction of a charged quantum particle such as an electron and the classical 29 A Brief Introduction to Quantum Mechanics electromagnetic ﬁeld. We mention here however that two gauges are quite common in classical electromagnetic theory and in the quantum description of this (also called QED; quantum electro-dynamics). These are the Coulomb-gauge and the Lorentzgauge. In the former we require that ∇ · A = 0 and in the latter that c∇ · A = ∂φ/∂t. These conditions may always be fulﬁlled if we have full freedom over Λ. See Ref. [7] for a discussion. We state gauge invariance as a theorem. Theorem 9 The electromagnetic ﬁelds E and B consistent with Maxwell’s equations (1.23) can be represented by the vector potential A and scalar potential φ such that B = ∇×A and E = −∇φ − 1 ∂A . c ∂t Furthermore, the electromagnetic ﬁelds are gauge invariant, i.e., they are invariant under the transformation −→ A − ∇Λ 1 ∂Λ φ −→ φ + , c ∂t A and where Λ(x , t) is any continuously diﬀerentiable real function. We may formulate the Lorentz force within the Hamiltonian framework in addition to the Newtonian framework, in which the force originally was given. In order to achieve this we must redeﬁne our (classical) Hamiltonian and the canonical momenta. (This is due to the fact that the electromagnetic forces are not conservative.) Hem := 1 q 2 pem − A + V (x ) + qφ(x ), 2m c (1.28) where q (1.29) pem := p + A c is the canonical momentum for the electromagnetic Hamiltonian. Thus, the canonical momentum is no longer the regular linear momentum, but has an additional term proportional to the vector potential. It is easy to prove that this reproduces Newton’s second law and the Lorentz force for the charged particle. Note also that the ﬁrst term of Hem is exactly the kinetic energy, so that Hem = total energy as we are used to. 1.6.2 Semiclassical Electrodynamics The quantization process, i.e., the process of applying the postulates to the semiclassical system, follows easily. The postulates were formulated for canonical variables, i.e., generalized coordinates and their corresponding momenta, and we now have acquired momenta that look just a little bit diﬀerent than usual. The canonical coordinates for the electromagnetic Hamiltonian, Hem , are pi,em and xi , where pi,em is diﬀerent from the linear momentum mvi (in Cartesian coordinates). This means that the quantum mechanical operator Pi,em does not correspond to the ith linear momentum component, but rather q Pi,em = Pi + Ai . c 30 1.6 – Electromagnetism and Quantum Physics Thus for the position representation, when we view our quantum states as L2 functions, Pi −→ −i ∂ q − Ai ∂xi c (1.30) is the ith component of the linear momentum in semiclassical electrodynamics. What about gauge invariance in this semiclassical theory? It is not obvious that a gauge transformation will leave the physics invariant. Indeed, the vector potential A appears explicitly in Eqn. (1.30), the expression for the linear momentum operator in the position representation. Of course, when we transform A and φ, we transform the Hamiltonian, so the Schrödinger equation also changes. We hope that this leaves room for gauge invariance, and indeed it is the case. To be more precise, if A −→ AΛ = A − ∇Λ and 1 ∂Λ c ∂t are gauge transformations of the potentials, this induces a gauge transformation of the Hamiltonian, viz., H = H(A, φ) −→ HΛ = H(AΛ , φΛ ). φ −→ φΛ = φ + If the original Schrödinger equation read HΨ = i ∂ Ψ, ∂t the new gauge-transformed Schrödinger equation will read HΛ ΨΛ = i ∂ ΨΛ . ∂t The theorem below states that the gauge transformation is just a picture transformation, and we identify the unitary operator T (t) in this case. Then the physics of the diﬀerent gauges cannot be diﬀerent, since picture transformations conserve physical measurements according to Theorem 8. Theorem 10 Let Ψ be the solution to the time dependent Schrödinger equation with the semiclassical Hamiltonian H. Let ΨΛ be the solution to the gauge transformed Schrödinger equation corresponding to the gauge transformed H −→ HΛ . Then ΨΛ is related to Ψ by a unitary transformation, viz., iq Ψ −→ ΨΛ = exp − Λ(X , t) Ψ. c Furthermore, HΛ (t) = T (t)H(t)T † (t) + i with ∂T † T (t), ∂t (1.31) iq T (t) = exp − Λ(X , t) . c In short: The gauge transformation corresponds to a picture transformation. Proof: We will prove the theorem by working in the position representation, i.e., expressing the state vector in the coordinate basis, because it is the natural choice as Λ is expressed as a function of X . 31 A Brief Introduction to Quantum Mechanics It is suﬃcient to prove that HΛ is given by Eqn. (1.31), because then T (t)Ψ is governed by the corresponding Schrödinger equation, and then ΨΛ must be given by this unitary transformation. Consider the picture transformation HT = T HT † + i ∂T † T . ∂t The ﬁrst term yields T HT † = q 2 1 T P − A T † + qT φT † , 2m c where the ﬁrst part is the kinetic energy. The second term is easy to calculate, since φ commutes with T as they are pure functions of X . The kinetic energy, however, is more complicated. We consider each term in the kinetic energy when the square is expanded: q q2 q 2 (1.32) T P − A T † = T P 2 − (AP + PA) + 2 A2 T † . c c c The last A2 -term commutes with T † because it is also a function of X . We will then have to compute the two remaining terms. These two terms will, as we shall see, give rise to the terms arising from the gauge transformation. We calculate the operator PT † by operating on a L2 -function Ψ: iq † q T (∇Λ)Ψ + T † ∇Ψ = T † (∇Λ) + T † P Ψ. PT † Ψ = −i∇(T † Ψ) = −i c c Hence, q PT † = T † (∇Λ) + T † P. c The middle term in Eqn. (1.32) is then easily calculated: q q2 q PT † A + APT † = T † 2 2 A(∇Λ) + T † (AP + PA). c c c For the P 2 term we note ﬁrst that q P 2 T † = P(T † (∇Λ) + T † P). c The second term yields q PT † P = T † (∇Λ)P + T † P 2 , c while the ﬁrst term is found by operating on Ψ, viz., −iq † 2 q q2 † q † † 2 P(∇Λ)T Ψ = T (∇ Λ) + 2 T (∇Λ) + T (∇Λ)P Ψ. c c c c Collecting all the terms we obtain iq q T T † (∇2 Λ) T (P − A)2 T † = T T † P 2 − c c q2 q + 2 T T † (∇Λ)2 + 2T T † (∇Λ)P c c 2 q2 q + 2 T T † A2 − 2T T † 2 (∇Λ)A c c q † − T T (AP + PA) c 32 1.6 – Electromagnetism and Quantum Physics Using the unitarity of T and reorganizing we get q q2 q T (P − A)2 T † = P 2 − (AP + PA) + 2 A2 c c c iq 2 q q2 q2 (∇ Λ) − 2 2 A(∇Λ) + 2 (∇Λ)2 . + 2 (∇Λ)P − c c c c Note that (∇Λ)P = P(∇Λ) + i(∇2 Λ), which we use to transform one of the (∇Λ)P terms into P(∇Λ). This cancels the ∇2 Λ term. Then the expression reads q q q2 T (P − A)2 T † = P 2 − (AP + PA) + 2 A2 c c c q q2 q q2 + (∇Λ)P + P(∇Λ) − 2 2 A(∇Λ) + 2 (∇Λ)2 c c c c 2 q q = (P − A) + (∇Λ) . c c The gauge transformed Hamiltonian reads HΛ = 1 ∂Λ 1 2 q P − (A − ∇Λ) + q φ + , 2m m c ∂t and we notice that we have successfully reproduced the gauge transformed kinetic energy. The scalar potential has an additional term proportional to ∂Λ/∂t, and this comes from the term (∂T /∂t)T † in HT , viz., i −iq ∂Λ q ∂Λ ∂T † T = i TT† = . ∂t c ∂t c ∂t Thus, we have proved that HΛ = H(AΛ , φΛ ) = T HT † + i ∂T † T = HT , ∂t and the proof is complete. We have established that performing a gauge transformation is in fact equivalent to performing a picture transformation. Hence, the gauge parameter Λ has no physical meaning in itself as in classical electrodynamics. However, the potentials A and φ have a physical meaning with somewhat surprising consequences. In classical mechanics a particle moves along a deﬁnite trajectory, sampling the electromagnetic ﬁelds E and B along the path, i.e., sampling ∇ × A and ∇φ. The quantum particle however moves according to a partial diﬀerential equation, namely the Schrödinger equation, and thus A and φ inﬂuence the movement of the particle (i.e., wave) globally. Thus in an area of space where the ﬁelds vanish and where a classical particle would feel no electromagnetic force, the quantum particle may be aﬀected from non-vanishing potentials, both in the same area of space and other areas of space! The simplest example of this striking fact is perhaps the so-called Ahoronov-Bohm eﬀect occurring in a modiﬁed double slit experiment. See Ref. [7] for a treatment. Gauge invariance is an important concept when we deal with numerical calculations. If our calculations show diﬀerent results in diﬀerent gauges, this indicates numerical errors arising in time. Indeed, we have an artiﬁcial physical eﬀect of the gauge parameter. If on the other hand our results turn out to be gauge independent, then we have an indication of a highly accurate calculation. 33 Chapter 2 Simple Quantum Mechanical Systems In this chapter we will have a look at some analytically solvable quantum mechanical systems. We do this to gain some further insight into the physics described in the previous chapter, and to show that even the simplest quantum mechanical systems are rather diﬃcult to solve. This points in the direction of numerical methods. We will have a look at the free particle; a particle propagating with no external forces acting on it. We will study the quantum mechanical harmonic oscillator, perhaps the most notorious and important system ever, regardless of physical discipline. The hydrogen atom is the most complicated system we will solve, and the techniques used are common. More complicated systems can be solved, but it should be clear after solving the hydrogen atom that numerical methods are well justiﬁed. In addition, when solving time dependent systems things gets even worse. There are very few systems with time dependent Hamiltonians that can be attacked with analytical means. In addition to the systems solved here, systems that are important when testing a numerical approach to the time independent Schrödinger equation are considered in chapter 6. These include a particle-in-box in two dimensions, a two-dimensional free particle in a constant magnetic ﬁeld and a two-dimensional hydrogen atom. 2.1 The Free Particle We begin with the free particle, which is the simplest spatial problem to solve. The free particle provides the ﬁrst system comparable to classical Newtonian mechanics and also oﬀers some insight into the behavior of the quantum particle. 2.1.1 The Classical Particle A free particle is a particle free to move without inﬂuence from external forces. The classical Hamiltonian reads p2 , H =T = 2m i.e., is it consists of kinetic energy only. Classically, this system has a very simple solution. Writing out Hamilton’s equations of motion (1.2), we get ẋ = p , m and ṗ = 0. 35 Simple Quantum Mechanical Systems Thus, the particle moves with constant momentum p and traces out a straight line (or a single point) in space, viz., p x (t) = x (0) + t. m 2.1.2 The Quantum Particle Let us turn to the quantum mechanical system and its solution. Let us ﬁrst note that the particle may have spin degrees of freedom, but since H commutes with any spin-dependent operator, we need not consider spin at all. Thus, H = L2 (R3 ). We will use the technique of section 1.4 to solve the Schrödinger equation. We then begin with the time independent Schrödinger equation, which is just the eigenvalue problem for the Hamiltonian, viz., P2 ψ = Eψ. 2m We have separated the time dependence e−iEt/ and the space dependence ψ(x ). Note that the Hamiltonian is proportional to the square of P. Thus, any eigenfunction of P is also an eigenfunction for H. Let us try to ﬁnd eigenfunctions ψ(x ) of Pi : ∂ i ψ = pi ψ. ∂xi The solution to this diﬀerential equation is easy do deduce: i 1 ψ(xi ) = √ e pi xi , 2π which are the eigenfunctions of the ith component of P with eigenvalue pi ∈ R. The factor (2π)−1/2 is chosen to achieve orthonormal eigenvectors, in the sense described in appendix A. The function may be scaled by an arbitrary number (that is, independent of xi ). Such a constant may be one of the eigenfunctions corresponding to other directions in space than the ith, viz., i i i i ψ(x )p = e p1 x1 e p1 x1 e p1 x1 = e p·x . We have added a subscript to indicate the eigenvalue belonging to the eigenfunction. Note that this function is an eigenfunction of all the components of P simultaneously. For the Hamiltonian we get Hψp = 1 1 2 (P 2 + P22 + P32 )ψp = p ψp , 2m 1 2m so that the eigenvalues of H become E= p2 , 2m p ∈ R3 . Thus, the energies corresponding to p of constant length are the same. The time dependence of ψp is given by i ψp (t) = ψp e− Et = 36 i i 1 1 e p·x e− Et = ei(k ·x −ωt) , 3/2 (2π) (2π)3/2 (2.1) 2.1 – The Free Particle where p and E p2 k 2 ω := = = . 2m 2m k := We recognize Eqn. (2.1) as a plane wave solution; a wave that travels with constant speed through space with wave number k . The speed of the wave is p/m, as in the classical case. The eigenvectors of the momentum operator are plane waves that are free to move with constant velocity in a ﬁxed direction. Compare this to the classical particle moving in a straight line with constant speed. In both cases we know the momentum p sharply – it has no uncertainty. Heisenberg’s uncertainty principle (1.12) then states that the position x of the particle in for the quantum mechanical wave should have inﬁnite uncertainty. Indeed, we have: |ψp (x )| = a constant, so that every point in space has the same probability density for ﬁnding the particle there upon measurement! Note that the eigenfunctions are not proper vectors of L2 , because they cannot be normalized to get unit norm. As described in appendix A this is because they are instead so-called improper vectors of Hilbert space. Improper vectors are not normalizable to a number, but they are distributions. Let us discuss arbitrary states of the free particle. We have found the eigenfunctions of H for the free particle, and we know that these constitute a basis for the Hilbert space in question. An arbitrary solution may then be found by superpositioning these, viz., i 1 Φ(p)e p·x d3 p Ψ(x ) = (2π)3/2 As described in appendix A, we recognize this as the inverse Fourier transform of Φ(p). Then we may write i 1 Ψ(x )e− p·x d3 x, Φ(p) = 3/2 (2π) which gives the momentum distribution in terms of the spatial distribution. It is important to be aware of the fact that the momentum distribution Φ(k ) and the spatial distribution Ψ(p) are equivalent. They both fully determine the wave function, the diﬀerence is that they are described in diﬀerent bases, i.e., Ψ(x ) = (ψx , Ψ) and Φ(p) = (ψp , Ψ), These expressions are the components of the vector Ψ along the eigenvector basis for the position and momentum operator, respectively. 2.1.3 The Gaussian Wave Packet The plane waves are not particularly suitable for describing a particle as we are used to experience it. Hardly ever do we deal with particles with equal probability of being found at any place in the universe. We need some kind of localization of the particle. Then it is natural to discuss the Gaussian wave packet. We will conﬁne this discussion to one dimension for simplicity and ease of visualization. 37 Simple Quantum Mechanical Systems p q q0 p0 Figure 2.1: Gaussian probability density for position and momentum The Gaussian wave packet is a spatial wave function whose absolute square is a Gaussian distribution. The wave function is given by 1 (q − q0 )2 ip0 q/ Ψ(q) = e exp − . (2.2) 4σq2 (2πσq2 )1/4 The factor exp(ip0 x/) gives rise to an average momentum p0 , as we shall see in a minute. The probability distribution in space is given by 1 (q − q0 )2 2 P (q) = |Ψ(q)| = exp − . 2σq2 2πσq2 Performing an (inverse) Fourier transform of Ψ to obtain the momentum distribution yields (p − p0 )2 1 −i(p−p0 )q/ e exp − , (2.3) Φ(p) = 4σp2 (2πσp2 )1/4 where σp = /2σq is the width of the momentum distribution which also happens to be a Gaussian distribution. Fig. 2.1 depicts these distributions, i.e., their absolute squares. Thus, the Gaussian wave packet represents a particle localized around the mean q = q0 with variance σq2 . The momentum has mean p0 and variance σp2 = 2 /4σ 2 . Hence ∆q∆p = σq σp = , 2 and we have achieved optimal uncertainty in q and p in the case of a Gaussian distribution. With Ehrenfest’s theorem we get at once d 1 q = p dt m and d p = 0. dt Hence, the wave packet’s mean position moves like a classical particle of constant momentum p0 . It can be shown that the time evolution of the free wave packet always assumes the shape of a gaussian. One may calculate the time evolution of σq . The width σp is constant in time by Eqn. (1.13), since P 2 commutes with the Hamiltonian. The time evolution of σq is given by 4σp4 2 2 2 2 σq (t) = t . 1 + 2 2t = σq (0) 1 + (2.4) 4σp2 m 4m2 σq (0)4 Thus, the wave packet spreads out in space as time increases. This phenomenon is called dispersion: The diﬀerent plane wave components of a wave packet have diﬀerent 38 2.2 – The Harmonic Oscillator velocities (i.e., diﬀerent phase velocities), so that the group velocity (the velocity of the envelope of the packet) diﬀers from the phase velocity, viz., vphase := ω p p dω = = = =: vgroup . k 2m m dk In classical terms we may view the Gaussian wave packet as an ensemble of travelling plane wave particles. Since they have diﬀering velocities their distribution in space must change with time: The faster waves outrun the slower ones. On the other hand, the particles are all free, so that their velocities does not change. Hence, the momentum distribution does not change with time. The Schrödinger equation is reversible, that is the time dependence of Eqn. (2.4) also holds for negative t. This means that it is possible to start with in some respects a more exotic wave packet U(−t0 , 0)Ψ whose space distribution gets sharper before it spreads out again. 2.2 The Harmonic Oscillator We will now study an ever-returning problem in both classical and quantum mechanics: The harmonic oscillator. The harmonic oscillator is in some respects the most fundamental physical mechanical system, because in very many applications it is a good ﬁrst approximation. There are dozens of examples of systems performing oscillatory behavior, and in very many of these the harmonic oscillator is a good choice of approximation. Consider Fig. 2.2 which shows some general one-dimensional potential V (q) with a local minimum at q0 . A Taylor expansion around q0 yields V (q) ≈ V (q0 ) + (q − q0 ) dV 1 d2 V + dq q0 2 dq 2 (q − q0 )2 q0 1 = V (q0 ) + mω 2 (q − q0 )2 , 2 with ω deﬁned through mω 2 = d2 V dq 2 q0 and where m is the mass of the particle moving in V . This approximation is the harmonic oscillator. (The constant term V (q0 ) is usually omitted.) We will conﬁne our discussion to the one-dimensional oscillator for clarity and simplicity. The full-blown general oscillator is readily obtained in more generalized arguments than those we present in this section. 2.2.1 The Classical Particle As stated above the Hamiltonian is given by H= 1 p2 + mω 2 q 2 . 2m 2 Here ω ∈ R is the frequency of oscillation in the classical sense. Sometimes the constant k = mω 2 is used instead, and one identiﬁes ω after the solution has been found. The coordinate name q is preferred over x in this section, because harmonic oscillators often occur when q is not a cartesian coordinate. 39 Simple Quantum Mechanical Systems V(q) q q0 Figure 2.2: Harmonic oscillator approximation Let us solve the harmonic oscillator with Hamilton’s equations (1.2). Writing these out yields p q̇ = m and ṗ = −ω 2 q, and thus q̈ = −ω 2 q. This is a standard diﬀerential equation, and we know that its solution is q(t) = C cos(ωt + φ), where C is the amplitude of the oscillations and φ is an arbitrary phase shift. The system thus undergoes periodic oscillations with period T = 2π/ω. From section 1.1 we know that ∂H dH = = 0, dt ∂t and hence the total energy is conserved for the harmonic oscillator. The situation may be illustrated through Fig. 2.3, which depicts a particle of mass m trapped in the V(q) H T m V -C C q Figure 2.3: The classical harmonic oscillator as a roller-coaster 40 2.2 – The Harmonic Oscillator harmonic oscillator potential; we display it here as sliding on a vertical wire shaped as the potential V (q) which makes a roller-coaster analogy.1 Since kinetic energy cannot be negative, the particle cannot be found at q coordinates outside the intersection of the graph of V (q) and the horizontal line of constant total energy T + V = H. The region R − [−C, C] is called the classically forbidden region. In quantum mechanics we shall see that the particle may still have a ﬁnite probability of being found outside this region. We have established the fact that a classical particle moving in an harmonic oscillator performs oscillations. Indeed, from the roller-coaster analogy of the system it should be obvious for anyone who has played with small marbles and fruit bowls. When we turn to the quantum mechanical system things get a little bit diﬀerent. Instead of a clear oscillatory behavior we ﬁnd discrete eigenstates of the Hamiltonian. These do not “move” in the classical sense, since they are stationary states, but for systems with very high energy (compared to ω) they correspond to the average behavior of the classical system, as we shall see. 2.2.2 The Quantum Particle We now have a good understanding of the classical harmonic oscillator. Turning to the quantum mechanical system, it turns out that it is a little bit more complicated to solve. Solving the time independent Schrödinger equation in position representation, that is as a diﬀerential equation eigenvalue problem, requires tedious work and knowledge of the so-called Hermite diﬀerential equation and the Hermite polynomials. We will instead choose an approach based on operator algebra which makes the solution readily obtainable. The trick is to deﬁne a lowering operator a as P mω Q + i√ . (2.5) a := 2 2mω The Hermitian adjoint of a, called the raising operator is easily seen to be P mω a† = . Q − i√ 2 2mω (2.6) The names of the operators will be justiﬁed in Theorem 11. A number x2 +y 2 may be factorized in the complex plane as (x−iy)(x+iy), and this is partly the idea behind the deﬁnition of a and a† : Could we write the Hamiltonian as a product in the same way? We deﬁne the Hermitian number operator N as N := a† a. The name will be justiﬁed in Theorem 12 below. When we compare H with N , we see that since a and a† do not commute, we get an extra term proportional to [Q, P ], viz., mω 2 2 1 1 mω P2 i † N =a a= + Q +√ [Q, P ] = H − 1. 2mω 2ω 2 ω 2 2mω Fortunately this term is just an energy shift which we may safely ignore.2 Analyzing the spectrum of N is much easier than analyzing H directly, as we will see. We calculate the product in reverse order, viz., aa† = 1 1 H + 1, ω 2 1 This analogy is perfectly acceptable. The acceleration of a particle sliding on a surface shaped like V (q) is proportional to V (q) as may easily be deduced. 2 Eigenvectors are unchanged if we make the transition H → H − σ1, where σ is any scalar. 41 Simple Quantum Mechanical Systems and this wields the commutator relation [a, a† ] = 1. The following theorem justiﬁes the names of the operators a and a† . Theorem 11 Let Φn be orthonormal eigenvectors of N := a† a with eigenvalue n. Assume [a, a† ] = 1. Then, √ a † Φn = n + 1Φn+1 (2.7) √ and aΦn = nΦn−1 . (2.8) In other words: The operator a transforms Φn into Φn−1 , and a† promotes Φn to Φn+1 . The lowering and raising operators produce a whole lot of eigenvectors of N once we are given one of them. A collective term for a and a† is ladder operators. Proof: Note that aΦn 2 = (Φn a† aΦn ) = n Φn 2 , and that † 2 a Φn = (Φn aa† Φn ) = (Φ, ([a, a† ] + a† a)Φn ) = (n + 1) Φn 2 . We investigate the action of N on aΦn and a† Φn : N aΨ = ([a† , a] + aa† )aΨ = (−1 + aa† )aΨ = (n − 1)aΨ N a† Ψ = a† ([a, a† ] + a† a)Ψ = a† (1 + N )Ψ = (n + 1)a† Ψ Hence aΦn is an eigenvector of N with eigenvalue n − 1, and a† Φn is an eigenvector of N with eigenvalue n + 1. With the assumption that Φn are orthonormal, the result follows at once. We will now prove that the only eigenvalues of N are exactly the non-negative integers. Note that Theorem 11 does not say this. It could happen that some eigenvalue existed between n and n + 1. Furthermore, the theorem does not say whether Φn are degenerated or not, that is if there exists more than one Φn with eigenvalue n. Theorem 12 Assume [a, a† ] = 1. The eigenvalues n of the number operator N := a† a are integral and non-negative. Proof: For any Ψ ∈ H, we deﬁne Φ = aΨ. The norm of Φ is Φ2 = (Ψ, a† aΨ) ≥ 0, so a† a is positive deﬁnite. (That N is also Hermitian is obvious.) Assume Φn to be an eigenvector of N with eigenvalue n (where n now is not necessarily integral.) From Theorem 11 we get that repeated use of a on Φn yields am Φn ∝ (n − m)Φn−m . But since N is positive deﬁnite the eigenvalue cannot be negative. The process must terminate so that n = m, for some integer m. Thus n is integral and non-negative. For the termination to occur, the lowest state must exist and have eigenvalue 0. 42 2.2 – The Harmonic Oscillator We still do not know whether all eigenvectors may be generated from only Φ0 by repeated use of a† . It could happen that we missed some eigenvector in case of a degenerate eigenvalue n. However, in one dimension we cannot have degeneracy, such as when applying the theorem to N = a† a in the present case.3 We have now deduced that the Hamiltonian has eigenvalues given by 1 En = ω(n + ), 2 n = 0, 1, 2, . . . Hence, the spectrum of the one-dimensional quantum mechanical harmonic oscillator is evenly spaced with spacing ω. Let us ﬁnd the ground state Φ0 , i.e., the state with the lowest energy E0 = ω/2). As noted in the proof of Theorem 12, operating with a will annihilate it, viz., mω P aΦ0 = Q + i√ Φ0 = 0. 2 2mω This yields a diﬀerential equation mω ∂Φ0 =− qΦ0 (q), ∂q whose solution is easily found by inspection: mω 1/4 mω 2 Φ0 = e− 2 q . π To produce excited states (that is, states of higher energy than the ground state) of the harmonic oscillator we may operate repeatedly on Φ0 with a† . n ip mω 1 1 † n q− √ Φ0 Φn = √ (a ) Φ0 = √ 2 2mω n! n! We get n/2 n mω ∂ q− Φ0 2mω ∂q n/2 n 2 mω ∂ 1 mω 1/4 q− =√ e−mωq /2 2mω ∂q n! π 1 Φn (q) = √ n! To simplify this expression, we introduce a change of variable from q to x, viz., mω x := q. This yields n 2 1 mω 1/4 ∂ √ Φn (x) = e−x /2 . x− n π ∂x 2 π Let us summarize the results in a theorem. In addition we want to relate our eigenfunctions to the so-called Hermite polynomials Hn (x), see Ref. [22]. A basic relation concerning these is n 2 2 d x− e−x /2 = Hn (x)e−x /2 . dx Using this, we state our summarizing theorem. 3 A proof of this can be found in Ref. [7], and rests upon uniqueness of solutions of ordinary diﬀerential equations. 43 Simple Quantum Mechanical Systems V(q) ⌽3 ⌽2 ⌽1 ⌽0 Figure 2.4: Lowest eigenstates of the harmonic oscillator Theorem 13 The energies of the harmonic oscillator are given by 1 En = ω(n + ), 2 and the corresponding eigenfunctions are given by n 2 1 mω 1/4 ∂ e−x /2 Φn (x) = √ x− n π ∂x 2 π 2 1 mω 1/4 =√ Hn (x)e−x /2 , n π 2 π where Hn (x) are the Hermite polynomials and mω q. x= Fig. 2.4 shows the ﬁrst four eigenfunctions for the harmonic oscillator. The other functions look very similar: An even or odd number of squiggles that fade out at each side. It is instructive to see what happens with the probability density |Φn (q)|2 as n grows. According to the correspondence principle (see section 2.4) classical behavior should be obtained in the limit of large quantum numbers, that is, for large n. When we say “behavior’ in this context, we mean that the probability density should approach that of the classical harmonic oscillator. Let us derive this distribution. Imagine that the classical harmonic oscillator performs very rapid oscillations. When we measure position, our eyes do not have any chance of catching exactly where it is, so we cannot anticipate the position. Then it becomes clear that the probability density P (q)dq for ﬁnding the particle between q and q + dq must be given by P (q)dq = |dt| . T /2 This is because during one half period of oscillation which takes the time T /2, the particle will have visited all q ∈ [−C, C]. The time spent between q and q + dq is |dt|. Diﬀerentiating q(t), we get dq = −Cω sin(ωt) = −ω C 2 − q(t)2 . dt 44 2.3 – The Hydrogen Atom 0.3 0.2 0.1 -8 -6 -4 0 -2 2 4 6 8 Figure 2.5: Probability density for n = 20 together with classical probability. Dimensionless units are applied. Horizontal unit is (/mω)1/2 and vertical unit is (mω/)1/2 . Note the dashed lines that mark the borders of the classically allowed regime. Thus, |dt| = ω dq . − q(t)2 C2 Remembering that T = 2π/ω we get P (q)dq = π dq C 2 − q2 (2.9) for the probability density. In Fig. 2.5 we see the classical probability density along with the probability density for n = 20. The probability density clearly approaches that of the classical oscillator as we hoped. As seen in Fig. 2.3 in the discussion of the classical harmonic oscillator, q is classically conﬁned to the region [−C, C]. On the other hand we see that the probability density |Ψ20 (q)|2 is nonzero outside this region for the quantum mechanical system. This always happens in quantum systems, when V is ﬁnite: There is always some probability that the particle will be found in the classically forbidden region dictated by the total energy H of the bound system. This is an example of the characteristic phenomenon in quantum mechanics called tunnelling. Quantum particles tunnel through classically forbidden regions. 2.3 The Hydrogen Atom The hydrogen atom is a two-body problem in which an electron orbits a proton. Classically the forces are given by the Coulomb force which is attractive for a system of two opposite charges. To study the hydrogen atom is interesting in several ways. It is one of the few analytically solvable three-dimensional problems in quantum mechanics, and ﬁnding the structure of the eigenstates of a non-trivial system could be illuminating. In 1913 Niels Bohr proposed a modiﬁed version of Rutherford’s hydrogen atom in which quantization principles were incorporated (though in a rather ad hoc manner) to get rid of certain classical inconsistencies. Rutherford’s model assumed a massive and dense nucleus of positive charge and classically orbiting electrons. (See Refs. [10, 11].) Orbiting classical electrons are accelerating. Accelerating charges emit radiation according to Maxwell’s equations, and hence loss of energy is inevitable. The orbit is not stable, making the electron eventually collide with the heavy nucleus. Indeed, the large charge-to-mass ratio of the electron makes the lifetime of the atom rather short, 45 Simple Quantum Mechanical Systems viz., τ ≈ 1.5 · 10−11 s. See Ref. [23] for a discussion of the classical instability of the orbiting electron. Bohr proposed a model in which the electron was allowed to orbit only in circular paths whose angular momentum was an integral multiple of , viz., L = n. (For this reason is also called the quantum of action.) The paths were still classical, as x and p were well-deﬁned quantities. In addition to the quantization hypothesis Bohr assumed that radiation did not occur during orbital motion; only in transitions between diﬀerent orbits. In that case, photons of energy equal to the energy diﬀerence between orbits were emitted or absorbed. However unsatisfactory for various reasons, Bohr’s model featured several important ideas, among these quantization of energy for a material system, interaction with a quantized electromagnetic ﬁeld and excellent correspondence with experimental data.4 To deﬁne the problem at hand, assume that the nucleus, i.e., a proton with mass mp , is at rest at the origin in R3 . Let the electron with mass me be located at the coordinates r relative to the proton. As the proton mass is much higher than the electron mass, viz., mp ≈ 1836, me we may neglect the motion of the proton on classical grounds. The large mass ratio leads to small errors. However, a two-body problem such as the hydrogen atom is best studied in the center of mass reference frame, i.e., the moving frame in which the center of mass is at rest. Describing the electron’s position relative to the center is equivalent to using the so-called reduced mass µ instead of the electron mass in the equations of motion for the electron. Instead of neglecting the proton motion in our equations we may instead use the center of mass description, and thus improve or description both theoretically and numerically. The center of mass Hamiltonian is identical to the original proton-neglecting Hamiltonian, but with the electron mass replaced with the reduced mass µ. In other words we will make the substitution me −→ µ := me · mp . me + mp See for example Refs. [10, 22] for a thorough discussion of the center of mass system and the Hydrogen atom. The Coulomb force is a central force, i.e., its line of action is the line joining the proton and the electron. Its value depends only on the distance between the particles. Such a conservative force may be expressed as a gradient, viz., F = −∇V (r), in which the potential V only depends on r, the distance between the particles, i.e. the distance from the origin. For the Coulomb force we have 1 V (r) = −ke2 , r (2.10) 4 Notice that quantization of light was proposed before non-relativistic quantum mechanics even though photons belong to a super-theory incorporating relativity. 46 2.3 – The Hydrogen Atom z v vr v θ r φ y x Figure 2.6: Spherical coordinates where e is the fundamental charge and k is a constant whose value is such that ke2 ≈ 1.44 eV · nm. Spherical symmetry suggests that we use spherical coordinates in our description, viz., x = r cos φ sin θ y = r sin φ sin θ z = r cos θ See Fig. 2.6 for a visualization. We will study the general case with arbitrary potentials to emphasize that the methods are applicable to other systems as well, such as the three dimensional harmonic oscillator. 2.3.1 The Classical System We will consider the classical system only brieﬂy to provide means for comparing the quantum mechanical system to the classical system. The Hamiltonian of the problem at hand is given by H= p2 + V (r). 2µ Let us ﬁnd a constant of motion for such a system, namely the angular momentum L. The torque on the particle is zero, viz., τ = r × F = 0, because the F = −∇V is parallel to r . Thus, angular momentum is conserved, viz., dL = v × (mv ) + r × F = 0. dt We may decompose the velocity into a radial part vr parallel to r and a part v⊥ in the plane orthogonal to r , see Fig. 2.6. Then it is easily seen that L = µrv⊥ , and thus the Hamiltonian may be rewritten as H= p2r L2 + + V (r) . 2µ 2µr 2 (2.11) Veff (r) 47 Simple Quantum Mechanical Systems This is the Hamiltonian of a one-dimensional system in which r is the (generalized) coordinate and where Veﬀ (r) is the potential. Note that since ∂H/∂t = 0 the total energy H is conserved in this system. It can be shown that the motion of the electron in the Coulomb potential lies on a conic section, depending on the total energy H: For H < 0 the motion traces an ellipse, for H = 0 the motion traces a parabola and for H > 0 the trace is a hyperbola. Thus all systems in which H < 0 yields bound motion with bounded conic sections, and H ≥ 0 yields unbound motion. 2.3.2 The Quantum System The derivation in this section will not be complete. We include only the most important aspects of the method and leave it to the reader to ﬁll in the details referred to. A good account is given in Ref. [22], and is recommended for further reading. Turning to the Hamiltonian again, we have in the position representation H =− 2 2 ∇ + V (r), 2µ and we need the Laplacian in spherical coordinates. This is given by 1 ∂2 1 ∂2 ∂2 2 ∂ ∂ + 2 + + cot θ ∇2 = 2 + ∂r r ∂r r ∂θ 2 ∂θ sin2 θ ∂φ2 as can be looked up in for example Ref. [24]. The expression in brackets turns out to be the quantum mechanical orbital momentum operator L2 as described in section 1.5.5.5 Thus, our Hamiltonian reads 2 ∂ 2 ∂ 2 L2 + + V (r). (2.12) H =− + 2 2µ ∂r r ∂r 2µr 2 The ﬁrst term is just the radial kinetic energy, viz., 2 ∂ 2 ∂ 2 2 pr = − + . ∂r 2 r ∂r We see from Eqn. (2.12) that L2 commutes with H, that is we may ﬁnd a set of simultaneous orthonormal eigenvectors to both operators. (Thus L2 is also a quantum mechanical constant of motion according to Theorem 4, section 1.5.4.) Since L2 commutes with the angular momentum components Li , these are also constants of motion. We know that the eigenstates of L2 and Lz are parameterized with two integral (or half-integral) quantum numbers l and m. For the orbital angular momentum it turns out that l may take on all integral values and no half-integral values. To be more speciﬁc, the eigenfunctions Ylm (θ, φ) of L2 and Lz are the so-called spherical harmonics; a basis for the square-integrable functions deﬁned on a sphere. For a discussion of these functions see for example Refs. [10, 22]. We may try separation of variables to obtain the eigenstates Ψnlm of H. Let us write Ψnlm = R(r)Ylm (θ, φ). This works perfectly well, and upon insertion into the time independent Schrödinger equation (1.7) we end up with the equation 2µ l(l + 1) ∂2 (rR) + [E − V (r)] − (rR) = 0. ∂r 2 2 r2 5 Given the diﬀerential operators in Cartesian coordinates, it is no diﬃcult task (albeit tedious) to calculate the spherical coordinate version of Li . 48 2.3 – The Hydrogen Atom The auxiliary function u = rR(r) therefore satisﬁes a modiﬁed time independent Schrödinger equation called the radial equation, viz., 2 l(l + 1) 2 ∂ 2 u + V (r) + u(r) = Eu(r). (2.13) − 2µ ∂r 2 2µr 2 Veff (r) This is clearly the quantum mechanical counterpart of Eqn. (2.11).6 The extra term in the eﬀective potential Veﬀ is called the centrifugal term. We see that for diﬀerent l we have diﬀerent radial equations and hence diﬀerent solutions R(r). We expect therefore that R must be labelled both with n and l, the nature of the former may be discrete or continuous. It turns out that n is discrete for E < 0 and continuous otherwise. We may at once devise some constraints on u(r) for the actual solution Ψ to be physical (or rather mathematical). First of all we must have u(0) = 0, because we want R(r) = u(r)/r not to be divergent. Second, if we let r approach inﬁnity, then the radial equation becomes ∂2u 2µE ∼ − 2 u(r). ∂r 2 For E < 0 the only acceptable solution is u(r) ∼ e−αr , where α = −2µE−2 . If we on the other hand let r approach zero, and assume that V (r) = O(r −k ) with k < 2, the centrifugal term will dominate the radial equation, giving ∂2u l(l + 1) ∼ u(r). ∂r 2 r2 The acceptable solution giving bounded R(r) in this case is u(r) ∼ r l+1 , or R(r) ∼ r l , r small. Let us summarize what we have found so far in a theorem. Theorem 14 Assume that we have a central symmetric Hamiltonian on the form H= p2 + V (r) 2µ where V (r) = O(r −k ), k < 2. Then the solutions to the time independent Schrödinger equation are on the form Ψ(r, θ, ψ) = R(r)Ylm (θ, φ), 6 The correspondence between the classical radial equation and the quantum mechanical equation is a mathematical coincidence. As shown in Ref. [25] it is for example not true in two dimensions. 49 Simple Quantum Mechanical Systems where Ylm are the spherical harmonics and where R(r) satisﬁes the radial equation 2 ∂ 2 (rR) 2 l(l + 1) − + V (r) + (rR) = E(rR). 2µ ∂r 2 2µr 2 Furthermore, the asymptotic behavior of R(r) for bound states (i.e., E < 0) is R(r) ∼ r l , r small, and R(r) ∼ exp − −2µE−2 r , r large. We will now consider the Coulomb potential (2.10) in particular. We will consider bound states, states in which E < 0. These are exactly the states that classically traces out a bounded orbit. We expect these to remain bounded, that is the solutions to the time independent Schrödinger equation are expected to be square integrable. First we rewrite Eqn. (2.13) on dimensionless form, introducing the scalings (2.14) ρ = r −8µE/2 µ . (2.15) and λ = ke2 −2E This yields ∂2u λ l(l + 1) 1 − + − u(ρ) = 0. ∂ρ2 ρ ρ2 4 (2.16) The strategy uses Theorem 14: We try to incorporate the asymptotic behavior. For large r we have u(ρ) ∼ e−ρ/2 , and therefore we write u(ρ) = e−ρ/2 v(ρ). Inserting this into the new radial equation yields a new diﬀerential equation for v(ρ), viz., ∂ 2 v ∂v λ l(l + 1) + v− − v = 0. ∂ρ2 ∂ρ ρ ρ2 We will not solve this equation explicitly as the arguments are a bit lengthy. See for example Ref. [22]. However, the acceptable solutions turn out to be polynomials of degree λ − 1, thereby imposing a quantization on the energy E through Eqn. (2.15). We deﬁne the principal quantum number n = λ, and we have n ≥ l + 1, so that for each n, the orbital momentum quantum number is limited. Note that the radial function R depends on both n and l, but that the energy E only depends on n. For an arbitrary spherical symmetric potential we would expect a dependence on l as well. We summarize the properties of the radial solution R(r) in a theorem, which we leave without further proof. Theorem 15 The hydrogen atom’s radial wave function is on the form Rnl (r) = Cρl e−ρ/2 L2l+1 n+l (ρ), where ρ is given by ρ=r· 50 2 , a0 n a0 := 2 , ke2 µ 2.3 – The Hydrogen Atom where a0 is called the Bohr radius. C is a normalization constant. The energy is given by (ke2 )2 µ 1 . En = − 22 n2 The function L2l+1 n+l (ρ) is the 2l + 1th associated polynomial of the n + l’th Laguerre polynomial, and it has degree n − l + 1. We have now found all the solutions to the time independent Schrödinger equation, viz., Ψnlm (r, θ, φ) = Rnl (r)Ylm (θ, φ). Visualizing these three dimensional functions is not very easy, but in Ref. [26] there is a whole chapter devoted to this. This concludes our exploration of the hydrogen atom. There are however some features of the solution to notice which aids us with a general understanding of bounded states in three dimensions: – The stationary states, i.e. the wave functions Ψnlm , and the energy eigenvalues are indexed by three quantum numbers. We choose a numbering such that higher quantum numbers yield higher energy. See also section 2.4 on the correspondence principle. – There are an inﬁnite number of bound states with E < 0 for the hydrogen atom. This is not a general fact. There are potentials without even one bound state even though the potential has a global minimum. – We also have eigenstates for the hydrogen atom with energy E > 0. Classically, these are not bound, and this is also the case for the quantum mechanical system: The eigenstates of H turn out to be non-normalizable, just as the eigenstates of the free particle. For the unbound states the energy spectrum is continuous, contrary to the bound states. – As for the one-dimensional harmonic oscillator there is a ﬁnite probability of ﬁnding the particle at arbitrary large r, thus violating the strict boundaries of the classically allowed regime. – The energies obtained are actually identical to those found by Bohr in his atomic model. – For large n, the energy diﬀerence En − En−1 approach zero, making in a sense a classical limit in which the energy is continuous. This is an example of the correspondence principle, which Bohr proposed for his quantum theory. We will say more on this in section 2.4. – As we have explained, the square magnitude |Ψ(x )|2 of the wave function is interpreted as the probability density of locating the electron at x upon measurement. In our case we have |Ψnlm (r, θ, φ)|2 = Rnl (r)2 |Ylm (θ, φ)|2 . Since |Y00 (θ, φ)| = 1, we get for the l = m = 0 states |Ψnlm (r, θ, φ)|2 = Rn0 (r)2 = C 2 e−ρ [L1n+1 (ρ)]2 . It is |Ψ(x )|2 dxdydz that measures the probability as opposed to the probability density. With spherical coordinates we have |Ψ(x )|2 dxdydz = |Ψ(r, θ, φ)|2 r 2 drdθdφ, 51 Simple Quantum Mechanical Systems and thus |Ψnlm (r, θ, φ)|2 r 2 dr = Rn0 (r)2 r 2 dr = C r 2 e−ρ [L1n+1 (ρ)]2 dr where C is a normalization constant is the probability density of ﬁnding the electron at a distance r from the nucleus. Recall that ρ = 2r/a0 n, and thus the exponential term falls oﬀ slower with higher energy (i.e., higher n). Without analyzing the Laguerre-polynomial term further, it is reasonable to accept that the electron’s average distance from the nucleus increases with energy, a feature expected from Bohr’s model. Indeed, it turns out that r is identical to the model’s postulated quantized radii. 2.4 The Correspondence Principle After analyzing a few simple quantum mechanical systems we are in position to state the correspondence principle, ﬁrst put forward by Bohr. The principle is quite simple: In the limit of high quantum numbers classical behavior should be reproduced. Alternatively, we may take the limit = 0 as the energies always will contain Planck’s constant as a factor. If we for instance consider the harmonic oscillator, the energies are given by 1 En = ω(n + ), 2 and in letting → 0 we must compensate with letting n → ∞ to keep the energy of the system ﬁxed. As seen in Fig. 2.5 on page 45, the quantum mechanical probability density is approaching the classical density in the limit n → ∞,7 equivalently if → 0 and the energy E is kept constant. The principle simply expresses that quantum mechanics is supposed to be a supertheory for Newtonian mechanics in the sense that everything obeying Newton’s laws is supposed to also obey non-relativistic quantum mechanics, but with the non-classical features of the dynamics obscured by the fact that is very small compared to macroscopic quantities. On the other hand, if the correspondence principle is violated for some system, then we may ﬁnd a macroscopic realization of a system in which quantum eﬀects is visible. 7 We must choose a numbering of the energy eigenvalues such that energy is increasing with higher quantum numbers. 52 Chapter 3 The Time Dependent Schrödinger Equation This chapter is a detailed description of the time dependent Schrödinger equation for a single particle in a classical electromagnetic ﬁeld. This is the domain of the systems we wish to study with our numerical methods. 3.1 The General One-Particle Problem We study the quantum description of a single particle. We shall in general let µ be the mass of the particle. In addition to existing in (at most) three dimensions the particle also has spin s. Hence, the wave function for our particle is a square integrable function of (at most) three real variables (i.e., Ω ⊂ Rd , d ≤ 3) into the complex vector space Cn , with n = 2s + 1. For each point in Ω, the wave function has n complex components. We denote each component function by Ψ(σ) , where σ = 1, . . . , n. We let S be the spin operators Si for the particle. In our chosen representation (σ,τ ) these are n × n matrices with components Si . We take the standard basis vectors to be eigenvectors of S3 (i.e., the spin along the z-axis) with increasing eigenvalues. Hence, |Ψ(σ) (x , t)|2 is the probability density of ﬁnding the particle at x at time t with a spin eigenvalue m = (σ − 1 − s) along the z-axis.1 The particle will in general have charge q and move in an electromagnetic ﬁeld, i.e., in an electric ﬁeld E and a magnetic ﬁeld B. Equivalently, the ﬁelds are described by the potentials A and φ, as described in section 1.6. A classically spinning particle gains an additional potential energy −ΓB ·S from its rotational motion, where Γ = qg/2µc is the scaled gyromagnetic factor. For an electron we have g ≈ 2.2 For a quantum particle we assume the same formal expression for the potential energy of a spin, and thus we do in a way envisage the electron as spinning, as described in section 1.5.6. In fact, the coupling with B is the only observable eﬀect of the spin of the particle. We call this the Zeemann eﬀect.3 The full-blown Hamiltonian of this system then reads H= 1 As 2 1 q −i∇ − A(x , t) + qφ(x , t) + Vext (x , t) − ΓB(x , t) · S . 2µ c (3.1) m = −s . . . s, σ = m + s + 1 = 1 . . . 2s + 1. may be determined experimentally or by means of relativistic corrections to the quantum 2 This theory. 3 See Ref. [22]. 53 The Time Dependent Schrödinger Equation Note that all but the last term commute with Si , and it becomes natural to write H = Hspatial + Hspin , with Hspin = −ΓB · S . However, Hspin does not commute with neither the momentum nor the position operator due to the magnetic ﬁeld which may vary in space. For each component Ψ(σ) of the quantum state we have the equation i ∂Ψ(σ) = Hspatial Ψ(σ) − Γ B · S (σ,τ ) Ψ(τ ) . ∂t τ (3.2) Clearly, Hspin couples the diﬀerential equations for each component wave function. If B = 0 this coupling vanishes and the equations become identical. Hence, there is no need for implementing a simulator for coupled PDEs in this case. In the case of a constant (in both space and time) magnetic ﬁeld B we may align it along the z-axis, and this makes Hspin diagonal and again the equations become decoupled. Eqn. (3.2) is the most general problem we would wish to solve numerically. There are however many special cases yielding considerable simpliﬁcations to the problem such as the above-mentioned decoupling of the PDEs for Ψ(σ) or gauge-transformations of the potentials. As discussed in section 1.6 we may perform a unitary transformation on the form iq T = exp − Λ(x , t) , c which is equivalent to the gauge transformation A −→ AΛ = A − ∇Λ, φ −→ φΛ = φ + 1 ∂Λ . c ∂t The transformed quantum state ΨΛ = T Ψ obeys the Schrödinger equation obtained by substituting the potentials with the transformed potentials. (Theorem 10, section 1.6, which also holds for the case in which spin is incorporated because T commutes with the spin operators S .) We can of course not eliminate A or φ in general and in this way obtain a simpler description of our particle. But in several circumstances, such as in the dipole approximation discussed below, some gauges have advantages over others. In the below discussions of diﬀerent magnetic ﬁelds it is important to bear in mind that for the potentials A and φ to have any meaning they must represent ﬁelds consistent with Maxwell’s equations (1.23). For example, a magnetic ﬁeld depending on time alone is not consistent.4 We discuss diﬀerent kinds of magnetic ﬁelds, such as a uni-directional ﬁeld, but it may happen that they are inconsistent. In that case we must be somewhat careful in our manipulations of the ﬁelds and potentials. We must have some kind of justiﬁcation for using such a ﬁeld. One such justiﬁcation is that our system is small compared to the spatial variations of the ﬁeld so that we may neglect the spatial dependence of the ﬁelds. 3.1.1 Uni-Directional Magnetic Field A uni-directional magnetic ﬁeld is a ﬁeld given by B = B(x , t)n(t), 4 It is easily seen that B = B(t) implies that B = const is a contradiction through Maxwell’s equations. B = B(t) implies Et = c∇ × B = 0, which again implies E = const. This in turn implies Bt = −c∇ × E = 0, and ﬁnally B = const. 54 3.1 – The General One-Particle Problem where n(t) is a unit vector dependent on time only. The ﬁeld strength B may vary in space as well. Hence, the direction of B only varies with time. Assume that the unit normal is independent of time. If the magnetic ﬁeld B is directed along the z-axis (if not, we just rotate the frame of reference) we note a considerable simpliﬁcation in the Schrödinger equation (3.2). Let B = B(x , t)k̂, where k̂ is the unit vector in the z-direction. Then Hspin = −ΓB(X , t)S3 , and Eqn. (3.2) becomes i ∂Ψ(σ) (σ,σ) = Hspatial Ψ(σ) − ΓB(X , t)S3 Ψ(σ) . ∂t (3.3) Hence, the diﬀerent equations are no longer coupled due to S3 being diagonal. We may not do this if n varies with time because the spin-diagonal Schrödinger equation (3.3) is not valid at all times. One could imagine an attempt at diagonalizing Hspin with an operator T deﬁning a picture transformation. The columns of T would be the eigenvectors of n · S , and HT = Hspatial − ΓBS3 + i ∂T † T , ∂t but the last term will not in general be diagonal. There is no easy way to decouple the PDEs for Ψ(σ) whenever n varies with time. If B is allowed to vary even more arbitrarly the diagonalization of Hspin of course also breaks down. Since the diagonal elements of S3 are just real numbers, each PDE diﬀers only by a scaling of the magnetic ﬁeld. Thus, each component Ψ(σ) evolves like the other components except for a magnetic ﬁeld of diﬀering strength. For example, if s = 1/2 we have (1,1) (2,2) and S3 =− =+ , S3 2 2 and thereby ∂Ψ(σ) = Hspatial Ψ(σ) ± ΓB(x , t)Ψ(σ) . i ∂t 2 If B in addition is independent of x (i.e., a dipole-approximation; see below), then the PDEs become even simpler. In fact, we may integrate each PDE analytically, if we know the eigenvectors of Hspatial . It is easy to see that if H = H0 + f (t), where a basis Ψn of eigenvectors of H0 are known and where f (t) is a scalar function of time alone, then Ψn are also eigenvectors of H. The time development of each eigenvector is given by t f (t ) dt ) Ψn (0), Ψn (t) = exp −i(En t + 0 where En are the eigenvalues of H0 , and hence we can calculate the time development of any linear combination of the Ψn s. In fact, if Ψ = n cn Ψn , then by linearity of the evolution operator we get t Ψ(t) = e−i 0 f (t ) dt e−iEn t Ψn . n In other words; the wave function diﬀers from the one found by solving the time dependent Schrödinger equations with H = H0 only by a phase factor. This means that it is only necessary to solve one PDE in the uni-directional case with a time dependent homogenous magnetic ﬁeld. Note that we must require the direction of the ﬁeld to be constant in time in order to de-couple the PDEs. 55 The Time Dependent Schrödinger Equation 3.1.2 A Particle Conﬁned to a Small Volume Imagine a particle conﬁned in a very small volume, i.e., approximately to a point in space. The particle is then approximately in an eigenstate of both the position operator and the momentum operator. The position is x0 and the momentum is zero.5 Then we may take H = Hspin = −ΓB(x0 , t) · S as our Hamiltonian and ignore the spatial dependence altogether, leaving us with an element of Cn whose components are Ψ(σ) (x0 ) as the full quantum description of the particle. In this case it is not necessary to solve a PDE; indeed the Schrödinger equation becomes an ODE. Note that we may choose the spin s to be arbitrary, making arbitrary large systems of ODEs. This system may be a good test case for ODE integrators that should preserve qualitative features of the full Schrödinger equation. There are also physically interesting systems of this type which are analytically solvable, in which a time dependent magnetic ﬁeld represents an explicit time dependence in the Schrödinger equation. For example, if B(t) = B0 (ak̂ + b(cos(ωt)î + sin(ωt)ĵ)) = B0 n(t), the spin state performs so-called Rabi oscillations. This magnetic ﬁeld rotates with constant angular frequency ω around the z-axis. The Hamiltonian reads H = −ΓB(t) · S . This system may be solved analytically for spin-1/2 particles, i.e., for a two-dimensional ODE; see Ref. [27]. This is one of very few solvable time-dependent quantum mechanical problems. For diﬀerent angular frequencies ω the Rabi oscillations display resonances, and such systems are utilized in both the theory of quantum computing and in laser cooling techniques. 3.1.3 The Dipole-Approximation Imagine that our particle is under inﬂuence of a source of electromagnetic radiation from a long distance r. The power emitted from the source is then inversely proportional to the distance, and we say that the radiation is of a dipole type. If our system is small of extent compared to the distance from the source it is a good approximation to set the electromagnetic ﬁelds to be independent of position, i.e., E = E (t), B = B(t). Clearly, this is a very simple form for the electromagnetic ﬁelds as they are independent of position. We stress that these ﬁelds are not consistent with Maxwell’s equations (1.23) as proven on page 54. Our model allows us to neglect the necessary spatial dependence of the ﬁelds. Care must be taken when considering potentials A and φ, since they must be derived according to the full model and not the simpliﬁed ﬁelds. 3.2 Physical Problems We have described the features of various simpliﬁcations of the full problem represented by the Hamiltonian (3.1). In this section we will brieﬂy describe some actual and interesting physical systems that is worth investigating numerically. Many applications are possible to ﬁnd in atomic physics and laser physics as well as solid state physics. 5 Of course it is not possible to perfectly conﬁne a particle in this way due to Heisenberg’s uncertainty principle, but it may be a very good approximation. 56 3.2 – Physical Problems atom at xij Figure 3.1: Finite element grid corresponding to a two-dimensional model of a solid with node perturbation and reﬁnement 3.2.1 Two-Dimensional Models of Solids We have not yet described the ﬁnite element method, but it is nevertheless interesting to hint at a stationary model that directly may exploit the features of the ﬁnite element discretization in a numerical solution. As a model of a solid (e.g., a metal, crystal or similar) we may study a twodimensional system in which the atoms of the solid are arranged in a simple grid. We imagine a quadratic slab of length L with N 2 atoms, ions or similar distributed evenly, i.e., at positions i−1 j−1 , xij = , i, j = 1, . . . , N. N −1 N −1 An electron inserted into the system experiences a potential Vij (x ) from each lattice site, e.g., an eﬀective repulsive Coulomb force. The one-particle Hamiltonian then reads N 2 Vij (x ). H = − ∇2 + 2µ i,j=1 Such models are popular and some of them are in fact possible to solve analytically. One feature of the model is that, roughly speaking, the inserted electrons surprisingly enough acts as if the potentials were absent; they move unhinderedly at constant velocity through the grid, see for example Ref. [22] in which a one dimensional model with inﬁnitely many atoms is considered. The eigenfunctions in an inﬁnite periodic model are plane waves exp(ikx) modulated with periodic functions. (Not all wave numbers k are allowed, however.) In practice, such behavior is not found in real systems, and this is due to irregularities in the grid-like structure. To solve the system numerically we may use a rectangular ﬁnite element grid, placing the nodes at xij and then reﬁne the grid to provide suﬃcient accuracy to the interpolating functions between the nodes. This is illustrated in Fig. 3.1. The ﬁnite element method ﬁnds a piecewise polynomial approximation over this grid; ﬁner grid means a better piecewise approximation. We will return to the ﬁnite element discretization in section 4.4; at this point we will just consider the method as one with very ﬂexible possibilities with respect to the geometry of the problem. The discrete Hamiltonian implies a discrete time independent Schrödinger equation. Diagonalizing the discrete Hamiltonian arising from this process will then intuitively give an approximation to the exact wave function and energy levels of such a system. This is what we will do in chapter 6 for other systems. If we perturb the positions xij randomly the discrete Hamiltonian will similarly be perturbed, yielding new eigenvalues 57 The Time Dependent Schrödinger Equation and eigenfunctions. The eﬀect is however to perturb the locations of the atoms or ions, and we have in addition obtained a numerical approximation. Doing simulations on an ensemble of such perturbed systems may yield statistical information on the distribution of for example energies. Systems such as those described here are very hot topics in solid state physics. For an introduction, see for example Ref. [28]. 3.2.2 Two-Dimensional Hydrogenic Systems An interesting problem is a two-dimensional hydrogenic system with an applied magnetic ﬁeld. Such a system has a Hamiltonian given by H= 1 e 2 ke2 −i∇ + A − − ΓB · S , 2µ c r where −e is the electron charge. The external potential −ke2 /r is an attractive Coulomb force.6 The full three-dimensional system has proven to be interesting in several major research ﬁelds, such as plasma physics, astrophysics and solid state physics. The solid state interest centers on the eﬀect of a magnetic ﬁeld on shallow impurity levels in a bulk semiconductor, see Ref. [29] and references therein. Ref. [30] is also a comprehensive treatment of the system. We will concentrate on the two-dimensional model of a hydrogenic system. This system arises as a limit in the case of a bulk semiconductor with impurities, but it also constitutes an interesting system in itself as a fairly complicated system that may display many features of quantum mechanics. Furthermore, such electronic systems in two dimensions constitute a hot topic when viewed as so-called quantum dots (one or more electrons conﬁned in small two-dimensional areas). In Ref. [29] one concentrates on a vertical time-dependent magnetic ﬁeld arising from for example the dipole approximation, viz., B = γ(t)k̂, and a candidate for the vector potential is then the so-called symmetric gauge, viz., A= γ(t) (−y, x, 0). 2 Note that ∇ · A = 0, hence we have the Coulomb gauge and the vector potential commutes with the momentum, viz., [A, P] = A · P − P · A = 0. The spin-dependent term in the Hamiltonian becomes Hspin = −Γγ(t)S3 , and the PDEs for diﬀerent spin components gets decoupled. In fact, Hspin commutes with Hspatial (i.e., with the rest of the Hamiltonian), and hence we may ignore the spin degrees of freedom altogether. If En are the energies of Hspatial and σ are the energies of Hspin , the energies of Hspatial + Hspin are Enσ = En + σ . In Ref. [29] the eigenvalues of the (spinless) Hamiltonian are discussed as a function of the applied magnetic ﬁeld strength. In the limits of a vanishing ﬁeld and of a strong ﬁeld the eigenvalues are found analytically. Perturbative methods combined with Padé interpolation for analytic continuation of the perturbative results are used to ﬁnd expressions for the approximate eigenvalues in the intermediate regime. 6 Note 58 that in Gaussian units k = 1. We keep such constants however for completeness. 3.2 – Physical Problems In Ref. [31] one mentions the possibility of studying the moderate magnetic ﬁeld region by use of ﬁnite element methods to achieve eigenvalues and eigenvectors with a high degree of accuracy. The focus here is to perform a ﬁnite element approximation to this system and compare the results with those of the Padé approximants of Ref. [29]. To study the two-dimensional hydrogen atom numerically we need to write the Schrödinger equation on dimensionless form. The numerical values of for example in cgs or SI units is extremely small, and using such quantities in numerical computations easily lead to round-oﬀ errors. Furthermore they are less intuitive to work with. It is easier to study quantities of order unity instead of order 10−34 . The Hamiltonian is given by H = Hspatial + Hspin where Hspin = −ΓB · S and Hspatial = e 2 1 1 −i∇ + A − ke2 . 2µ c r Let us ﬁrst focus on Hspatial , dropping the subscript for ease of notation. We introduce a new length scale to our problem, writing x = αx , where α is a numerical constant with the dimension of length, hence x is a dimensionless vector. Then, ∇= 1 ∇, α where ∇ denotes diﬀerentiation with respect to the components of x . If we insert this into our Hamiltonian, we get H= αe 2 ke2 1 2 A − + . −i∇ 2µα2 c α r Here, r = αr . Note that αe/c · A must be dimensionless, so we introduce A = αe/c · A. Next we introduce a scaled dimensionless Hamiltonian H given by H = 2µα2 µαke2 2 2 H = (−i∇ + A ) − . 2 2 r If we require the factor in front of 2/r to be unity, we obtain the length scale α= (197.3 eV · nm)2 2 = 0.0529 nm, ≈ 2 µke 0.5107 · 106 eV · 1.440 eV · nm and accordingly H = 22 H = β −1 H, m(ke2 )2 where β is the energy scale. The numerical value of β is β ≈ 0.5107 · 106 eV (1.440 eV · nm)2 = 13.603 eV. 2(197.3 eV · nm)2 When studying the time dependent Schrödinger equation we also need a time scale, i.e., t = τ t . Upon insertion into the Schrödinger equation we have i where τ= ∂Ψ = H Ψ, βτ ∂t 23 6.582 · 10−16 eV · s = = 4.839 · 10−17 s = β µ(ke2 )2 13.603 eV 59 The Time Dependent Schrödinger Equation quantity length deﬁnition 2 α = µke 2 energy time mag. ﬁeld β= τ= µ(ke2 )2 22 23 µ(ke2 )2 β gµB numerical value 0.0529 nm 13.603 eV 4.839 · 10−17 s 4.282 · 10−8 gauss Table 3.1: Units for the two dimensional hydrogen atom is the natural time scale.7 Thus, i ∂Ψ = H Ψ(x , t ), ∂t with 2 , (3.4) r is the dimensionless form of the time dependent Schrödinger equation. Still we need to ﬁnd a suitable scale for the magnetic ﬁeld and the spin operators in Hspin to complete our discussion. The natural unit for spin is , as the spin matrices are given in terms of this constant, i.e., S = S . Hence, H = (−i∇ + A ) − 2 β −1 Hspin = gβ −1 µB δB · S , where we have introduced δ as the natural scale of the magnetic ﬁeld. The constant µB is called the Bohr magneton and has the value ge = 5.789 · 10−9 eV/gauss. 2µc Requiring gµB δ/β = 1 yields δ= β = 4.282 · 10−8 gauss gµB where we have used g = 2.00232 for the electron, quoted from Ref. [12]. From now on, we will drop the primes as the conversion in units is unambiguously given by the tabulated quantities in Table 3.1. We use the symmetric gauge A = γ(t)/2 · (−y, x, 0) for the vector potential. Note that by the chain rule, ∂x ∂ ∂y ∂ ∂ ∂ ∂ = + = −y +x , ∂φ ∂φ ∂x ∂φ ∂y ∂x ∂y and that A2 = γ(t)2 (x2 + y 2 )/4 = γ(t)2 r 2 /4. We obtain for the kinetic energy term in Eqn. (3.4) (−i∇ + A)2 = −∇2 − 2iA · ∇ + A2 = −∇2 − iγ(t) γ(t)2 2 ∂ + r . ∂φ 4 The spinless Hamiltonian now reads H = −∇2 − iγ(t) γ(t)2 2 2 ∂ + r − . ∂φ 4 r (3.5) In section 6.1.3 we deal with the time independent version of this problem numerically. 7 Compare the time scale with the life time of the classical hydrogen atom losing its energy due to radiation, τ ≈ 1.5 · 10−11 s. 60 Chapter 4 Numerical Methods for Partial Diﬀerential Equations This chapter is devoted to numerical solutions of partial diﬀerential equations (PDEs). Most such are very hard to solve analytically, implying the need for approximate solution methods. The ﬁrst section describes the main features of partial diﬀerential equations in general and ordinary diﬀerential equations (ODEs) in particular. Then we turn to diﬀerent numerical methods for approximating these. The main diﬀerence between PDEs and ODEs is that in the ODE the unknown function we search for depends only on one parameter, i.e., the equation contains only derivatives with respect to one variable. Let us outline the general strategy used when attacking a PDE numerically. Consider as an example a simpliﬁed version of the Schrödinger equation, viz.,1 iut = −uxx + V (x)u. This diﬀerential equation contains partial derivatives of ﬁrst order with respect to time and of second order with respect to spatial coordinates. We use u(t = 0) = f as initial condition and there are boundary conditions that must be fulﬁlled in the spatial directions. How do we solve such a problem on a computer? Clearly, the problem must be ﬁnite in extent in order to be calculable. We must in some way transform diﬀerential operators and functions into discrete and ﬁnite representations. The unknown function u(x) at time t has inﬁnitely many degrees of freedom; one for each x. Imagine that we instead assume that x only can take a ﬁnite number of values xj , j = 1, 2, . . . , N , which is the main idea of ﬁnite diﬀerence approximations among others. Clearly, u(t) becomes an element of CN . As a consequence the diﬀerential operator ∂ 2 /∂x2 must be replaced by an algebraic relation between the diﬀerent function values u(xj , t). The multiplicative operator V (x) must be represented by some appropriate variant; clearly also an algebraic relation. The PDE is by these means converted into an ordinary diﬀerential equation of dimension N , viz., iut = D(u) + V (u), u(t) ∈ CN , where D and V are mappings from CN into CN . Since the Hamiltonian −∂ 2 /∂x2 +V (x) is an Hermitian operator, we also wish that our numerical methods yield Hermitian discrete versions, i.e., that D and V can be represented by linear Hermitian matrices. This ensures that unitarity of the Schrödinger equation is transferred to the ODE as well. 1 Note the subscript notation for partial derivatives. 61 Numerical Methods for Partial Diﬀerential Equations Our attention may now be focused on solving the ODE. The following sections outline some widely used methods for performing the conversion of a PDE to an ODE. After this we outline methods for solving ODEs, focusing on methods applicable to the Schrödinger equation. This is typically done with a ﬁnite diﬀerence method of some kind of consideration of the exact ﬂow of the ODE such as operator splitting methods. In other words, the temporal degree of freedom is also converted into a discrete approximation. Ultimately, we have a complete discrete formulation of our diﬀerential equation. There is nothing wrong by going the other way round, i.e., ﬁrst discretize the time dependence and then apply the spatial approximation. Indeed, in some cases it is clearer to take this alternate approach.2 Ultimately, the PDE is reduced to an algebraic problem, or if we wish to view it as such, a sequence of such; one problem at each time level. 4.1 Diﬀerential Equations In this section we review some basic concepts regarding diﬀerential equations; both ordinary and partial diﬀerential equations. We will not employ mathematical rigor, because it is not necessary for our applications. There are numerous standard texts to consult, see for example Refs. [17, 32, 33]. A diﬀerential equation is an equation in which the unknown is a function. The equation relates the unknown’s partial derivatives. We distinguish between ordinary diﬀerential equations (ODEs) and partial diﬀerential equations (PDEs). Partial diﬀerential equations are an extremely rich class of problems. They are divided into diﬀerent types, each displaying diﬀerent behavior. The diﬀerent kinds of equations may behave diﬀerently with diﬀerent numerical methods. Examples of PDE types are parabolic equations, elliptic equations (such as the Schrödinger equation) and hyperbolic equations. 4.1.1 Ordinary Diﬀerential Equations The unknown in an ODE is a function of only one variable t. In general we search for a function y : I ⊂ R −→ V, where V is some vector space called the phase space and I = [t0 , t1 ] is some interval, and the ODE reads ẏ = f (y, t). This is a ﬁrst order equation because only derivatives of ﬁrst order enter the equation. This is not a limitation, since all higher-order ODEs may be rewritten as ﬁrst-order ODEs by extending V . The function y(·, t) is a function from V into V , i.e., a vector ﬁeld. We say that an ODE is autonomous if f has no explicit dependence on time. Any non-autonomous ODE may be turned into an autonomous ODE by extending the phase space with one dimension, viz., d y f (y, τ ) = . 1 dt τ If f is suﬃciently nice it is easy to see that the solution (if it exists) is unique, once given the initial condition y(t0 ). We will not prove existence or uniqueness here, see Ref. [15]. 2 If discretizing the time domain ﬁrst, we must be aware of some existence problems in the formulation. If we ignore this, our derivations become purely formal, and the formality of the calculations vanish when we introduce a spatial discretization. We will see this in section 4.5. 62 4.1 – Diﬀerential Equations In the case of a unique solution we have a mapping U (t, s) that takes an initial condition y(s) into the solution of the diﬀerential equation at time t, i.e., U (t, s)y(s) = y(t). By ﬁrst propagating to t and then to t we get the composition property U (t , s) = U (t , t)U (t, s). For t suﬃciently close to s we may invert the time propagation, yielding U (t, s) = U (s, t)−1 . Of course, U (t, t) = 1. This mapping is called the ﬂow, the propagator or the evolution operator. It corresponds to the propagator of the Schrödinger equation. This means that ordinary diﬀerential equations describe the motion of a point in phase space. The image y(I) is called the trajectory. One might consider more general ODEs in which y is a point on a diﬀerentiable manifold instead of a vector space, and in which f is a tangent vector ﬁeld. The diﬀerential equation for the propagator is such a generalized ODE. By the following identities, ẏ(t) = d U (t, s)y(s) = U̇ (t, s)y(s) = f (U (t, s)y(s)) = f ◦ U (t, s)y(s), dt which must hold for all y(s), we get U̇ (t, s) = f ◦ U (t, s), U (s, s) = 1. Hamilton’s equations of motion (1.2) is a very important example of an ordinary diﬀerential equation. 4.1.2 Partial Diﬀerential Equations Where ODEs searched for a (multidimensional) function of a single variable, partial diﬀerential equations (PDEs) search for a (possibly multidimensional) function of several variables. Hence, the partial derivatives with respect to all the arguments of the unknown appear in the equation. Assume that the function u we are seeking is deﬁned on some subset of Rn , viz., u : Ω ⊂ Rn −→ V, where V again is some vector space. It is useful to consider u as an element in some linear space, such as a Banach space or a Hilbert space. In general we write our PDE as L(u) = f, where f is a function in the mentioned space, on equal footing with u. The operator L may be complicated, but in case of a linear operator we say that the PDE is linear. (Note that linear equations may be harder to solve than non-linear ones! Linear PDEs form a huge class of problems.) Furthermore, the equation is said to be homogenous if f = 0. Here are some simple yet important examples of PDEs. – ut + ux = 0: A transport equation. 63 Numerical Methods for Partial Diﬀerential Equations – ut − uxx = 0: The one-dimensional diﬀusion equation. – iut − uxx = 0: The one-dimensional time dependent Schrödinger equation for a free particle. – utt − uxx = 0: The one-dimensional wave equation. – uxx + uyy = 0: Laplace’ equation. Common to all these examples is that they are homogenous. The time dependent Schrödinger equation is in general also homogenous. There is some missing information in the examples for them to represent well-deﬁned physical problems with a unique solution. What is the initial distribution of temperature u in the heat equation? Assuming that we model a vibrating string with the wave equation, what are the reﬂection coeﬃcients at the ends of the string? Typically we separate between spatial dependence and time dependence. This is connected to our physical way of thinking. Thus, often one silently assumes that an initial condition problem is at hand when the PDE contains t as a parameter, and some sort of boundary condition problem whenever x occurs. This is particularly clear in the last two examples above. The wave equation and Laplace’ equation are almost identical, but the presence of y instead of t in the last implies a problem of completely diﬀerent character. Laplace’ equation is a stationary problem while the wave equation is a time-dependent problem. For equations of order n in time (i.e., when n is the highest order of the time derivative that occurs) we need n initial conditions to “set oﬀ” the solutions, i.e., we must supply ∂ k u/∂tk for k = 1, 2, . . . , n at t = 0. Boundary conditions specify the asymptotic behavior of u. When solving PDEs numerically we consider only compact domains (i.e., closed and bounded domains), and in that case the boundary conditions specify the behavior of the solution at the boundary ∂Ω. Most important for our purposes are Dirichlet boundary conditions in which we specify the value of u at the boundary, i.e. we specify u(∂Ω). Other kinds of conditions include Neumann conditions, in which the normal component of the gradient of u is speciﬁed, i.e., n · ∇u is supplied, where n is a unit normal ﬁeld of ∂Ω. Usually we formulate the Schrödinger equation with Ω = Rn , but for computational purposes this is not suitable. Physically, limiting Ω to some compact region in eﬀect sets the potential to inﬁnity there since the particle is not allowed to penetrate into the region outside Ω. Then the wave function vanishes at the boundary for the equation to be fulﬁlled. In other words, when solving the time dependent Schrödinger equation we use homogenous Dirichlet boundary conditions, viz., u(∂Ω) ≡ 0. PDEs and ODEs are closely related, especially for computational purposes. On one hand, an ODE is a PDE in which the unknown depends only on t. On the other hand, if we can specify u as being an element in some (inﬁnite-dimensional) Hilbert space and if we can ﬁnd a countable basis, then a time-dependent PDE is equivalent to an inﬁnite-dimensional ODE. An example of this is seen in the discussion on the time dependent Schrödinger equation in section 1.4. The typical computational approach is then to in some way limit the dimensionality of the Hilbert space by ignoring all but a ﬁnite subspace, as indicated in the introduction. 4.2 Finite Diﬀerence Methods Perhaps the most obvious and intuitive way of discretely approximating diﬀerential operators is by means of ﬁnite diﬀerences. This method replaces the continuous function 64 4.2 – Finite Diﬀerence Methods h1 x(2,4) (N1,N2) L2 h2 (0,0) L1 Figure 4.1: A uniform grid by a discrete function deﬁned at evenly spaced grid points in space. Partial derivatives are approximated with diﬀerence expressions derived from considerations of Taylor expansions of the original function. 4.2.1 The Grid and the Discrete Functions Assume that we are given a PDE with unknown u, viz., u : Ω ⊂ Rn −→ V. Let us introduce some notation. A multi-index α is an ordered sequence of n integers, viz., α = (α1 , α2 , · · · , αn ), where n is the dimension of Ω in our applications. We may add and subtract multiindices in a component-wise fashion, i.e., (α + β)i = αi + βi , and without danger of confusion, if we specify a multi-index α + βi , then βj is supposed to have zero components βi for i = j. Suppose that Ω is a rectangular domain, i.e., it is the cartesian product of intervals, viz., Ω = I1 × I2 × · · · × In . If we subdivide each interval Ii into Ni subintervals of uniform length hi , we get Ni + 1 well-deﬁned “joints” xij , j = 0, · · · , Ni . See Fig. 4.1 for an illustration. The grid G now consists of the points xi1 i2 ...in given by xi1 i2 ...in = (x1i1 , x2i2 , . . . , xnin ), ij = 0, . . . , Nj , j = 1, . . . , n. Thus, the grid G is given as G = {xα : αj = 0, 1, . . . , Nj }. Such a G is referred to as a uniform grid. The mesh width is the largest of the spacings hi and intuitively measures the “ﬁneness” of the approximation. Assuming that Ω is rectangular is not really hampering, because one may specify boundary conditions in the discrete formulation mimicking the real shape of Ω. This is illustrated in Fig. 4.2. Our computational algorithm gets much more complicated when employing such complicated geometries, one of the major drawbacks with ﬁnite diﬀerence methods. Furthermore, if the discretized boundary has for example “staircase shape” artiﬁcial physical eﬀects may be introduced to the numerical solution. 65 Numerical Methods for Partial Diﬀerential Equations Ω Figure 4.2: Discretizing a non-rectangular domain As for the function u and other functions in the problem deﬁnition such as initial conditions, we deﬁne a discrete approximation uh deﬁned in the grid points of G, viz., u(x ) −→ uh (xα ). Thus, uh may be viewed as an N1 N2 . . . Nn -dimensional vector whose components are elements in V , or as a tensor of rank n with Ni components of type V in each direction. Indeed, we write uα := uh (xα ) for the components of this tensor or discrete function. When doing practical computations in for example two dimensions, we simplify the notation and write xij for the grid points and uij for the components of uh , where i and j are indices in the x and y directions, respectively. 4.2.2 Finite Diﬀerences The diﬀerential operators occurring in the PDE problem are approximated by diﬀerence expressions called ﬁnite diﬀerences. At a given grid point xα the partial derivatives are approximated by a (linear) function of uα and the neighboring grid points. There is no unique way to deﬁne ﬁnite diﬀerence expressions for the diﬀerent partial derivatives. The choice of the approximation is highly problem dependent. In addition, more complicated compositions of operators such as (f (x)ux )x may be approximated by choosing a perhaps non-obvious but “smart” ﬁnite diﬀerence. There are however some ﬁnite diﬀerences that are more widely used than others, and we shall introduce a special notation for these. This notation makes it easy to develop diﬀerence schemes for various PDEs and the notation follows closely that of the actual partial derivatives. We introduce the notation in the setting of a one-dimensional problem. The generalization to several dimensions is obvious. First we deﬁne a centered diﬀerence for the partial derivative ∂/∂x: [δnx u]j := 1 (uj+n/2 − uj−n/2 ). nh (4.1) Here, h is the grid spacing and n > 0 scales the step size in the ﬁnite diﬀerence. Note that this is clearly inspired by the deﬁnition of the partial derivative. The same goes for the one-sided forward and backward diﬀerences, viz., + [δnx u] := 1 (uj+n − uj ), nh (4.2) − [δnx u] := 1 (uj − uj−n ). nh (4.3) and 66 4.2 – Finite Diﬀerence Methods In all three cases, the distance along the axis of the sampling points are nh. These three ﬁnite diﬀerence expressions all approximate the ﬁrst derivative. A widely used expression for the second derivative is given by combining δx twice, viz., [δx δx u]j = 1 (uj+1 − 2uj + uj−1 ). h2 (4.4) It is fundamental to be able to analyze the error in the numerical discretizations. One of the concepts to consider is the truncation error, deﬁned as the error in the numerical discretization when we insert an exact solution. Thus, if L is a diﬀerential operator and if Lh is a numerical approximation, then the truncation error is deﬁned as τ := Lh (u) − L(u). For example, the approximation (4.4) of the second derivative has truncation error τ= 1 (u(x + h) − 2u(x) + u(x − h)) − u (x). h2 When we expand u(x ± h) in Taylor series we easily obtain τ= 1 h2 (2 u (x) + O(h4 ) − u (x) = O(h2 ), h2 2 and so the truncation error is of second order in the step size. When τ vanishes with vanishing mesh width we say that we have a consistent approximation. The name of τ is related to the fact that it is the error arising from truncating the Taylor series after a few terms. The truncation errors for the other ﬁnite diﬀerences presented above is easily derived, viz., 1 (nh)2 u (x) + O(h4 ), 24 1 + [δnh u]j = u (xj ) + nhu (x) + O(h2 ), 2 1 − and [δnh u]j = u (xj ) − nhu (x) + O(h2 ). 2 [δnh u]j = u (xj ) + (4.5) (4.6) (4.7) The centered diﬀerence provides a better approximation to u (xj ) than the one-sided diﬀerences. Intuitively this is so because it utilizes information from both sides of xj . 4.2.3 Simple Examples Let us apply the ﬁnite diﬀerences to some simple model problems. We will see that the notation introduced makes the similarity between the PDE formulation and ﬁnite diﬀerence formulation obvious. The examples will also illustrate some important concepts, such as incorporating boundary conditions and initial conditions. The Heat Equation. Consider the two-dimensional heat equation, viz., ut = uxx + uyy . We assume that the spatial grid is given and has points which we denote as xi,j = (xi , yj ). Time is discretized as t = ∆t and although we according to the above description of the grid should include it in x it is somewhat clearer to separate the time-dependent part of the grid. We also assume that hx = hy = h. First, let us consider a ﬁnite diﬀerence scheme applied to only the spatial degrees of freedom, viz., [u̇(t) = δx δx u + δy δy u]j . 67 Numerical Methods for Partial Diﬀerential Equations This clearly turns the spatial derivatives into algebraic operators on the discrete function. Since we still have a time derivative in the equation we now have an ordinary diﬀerential equation. We will employ two diﬀerent diﬀerence schemes for resolving the derivative; namely a forward diﬀerence and an backward diﬀerence approximation. A forward diﬀerence yields [δt+ u]i,j = [(δx δx + δy δy )u]i,j . Note that we have placed the time level index as a superscript. When performing an implementation we seldom keep the solution at all time levels in memory. Writing out the diﬀerence scheme and reorganizing yields u+1 i,j = ui,j + ∆t ui+1,j + ui−1,j + ui,j+1 + ui,j−1 − 4ui,j . 2 h This scheme is called explicit, because updating the solution at the next time level is explicitly given as a function of the solution of the current time level. Time stepping with δt+ is called forward Euler or explicit Euler. Replacing δt+ with δt− yields a diﬀerent scheme. This way of time stepping is called backward Euler or implicit Euler. The scheme reads ui,j − ∆t ui+1,j + ui−1,j + ui,j+1 + ui,j−1 − 4ui,j = u−1 i,j . 2 h This scheme is called implicit because the solution at time level is given implicitly; we have to solve a non-trivial set of algebraic equations to ﬁnd u . It is easy to see that the system of equations is a linear algebraic system, i.e., we may rewrite it as a matrix equation. Boundary conditions were neglected in the above discussion, but we include them in the next example. The Wave Equation. Let the domain be given as Ω = [0, 1]. Let us describe a ﬁnite diﬀerence method for the one-dimensional wave equation, i.e., utt = uxx , with boundary conditions u(0, t) = u(1, t) = 0. The initial condition is u(x, 0) = f (x), ut (x, 0) = g(x). We use N + 1 grid points, i.e., G = {xj = jh : j = 0, . . . N }. The time levels are given as t = ∆t as in the previous example. We devise the scheme [δt δt u]j = [δx δx u]j , j = 1, . . . , N − 1. For j = 0 and j = N we impose the boundary conditions, i.e., that u0 = uN = 0. For = 0 we use the initial condition, viz., u0j = f (xj ). For = 1 we use the second part of the initial condition and a forward diﬀerence approximation, viz., u1j = u0j + ∆tg(xj ). 68 4.2 – Finite Diﬀerence Methods For the subsequent time levels we use the devised scheme, which when written out reads ∆t2 −1 uj+1 − 2uj + uj−1 . u+1 = 2u − u + j j j 2 h This is an explicit scheme. Note the way in which the initial conditions were incorporated by a special rule for the two ﬁrst time levels. Similarly, the boundary conditions were incorporated by an (obvious) special rule. Let us mention some stability properties of this scheme. It turns out that the scheme is stable if and only if the Courant number C := ∆t2 /h2 ≤ 1. In fact we obtain the exact solution at the grid points if C = 1. (This is not true for the two-dimensional generalization of the scheme.) Thus, taking too large time steps makes the process unstable, a feature that we also will see when studying the Schrödinger equation. See Ref. [34] for a thorough discussion. A Non-Linear PDE. Now for a more delicate treat. Consider the PDE ut = (f (u)ux )x , where f (u) may be any positive and smooth function. This equation is a non-linear Heat equation. We will ignore boundary conditions for now to focus on the nonlinearity, and we use the same grid in space and time as for the wave equation. We devise the following scheme: [δt− u]j = [δx (f (u)δx u]j . On the right hand side we have a small problem. When written out it reads 1 f (uj+1/2 )[δx u]j+1/2 − f (uj−1/2 )[δx u]j−1/2 , h and the trouble is we do not know uj±1/2 . Therefore, we choose an approximation given by uj+1 + uj ). f¯j := f ( 2 It is easy to see that f¯j = f (uj+1/2 ) + O(h2 ), so this approximation is of second order. When we write out the equation and gather the unknowns uj on one side we get u − ∆t ¯ fj (uj+1 − uj ) − f¯j−1 (uj − uj−1 ) = u−1 , j 2 h which is non-linear in uj . Therefore, to generate the solution on the next time step we must be able to solve a non-linear set of algebraic equations. There are several ways to do this, but perhaps the most popular one is Newton-Rhapson iteration. See Ref. [34] for a discussion. By choosing for example forward Euler instead of backward Euler one may get an explicit scheme instead, with no non-linear equations to solve. However, forward Euler tends to be unstable. In addition to being more complicated to solve, the non-linear schemes are more diﬃcult to analyze with respect to stability and convergence. Ref. [34] contains some material on the subject. 69 Numerical Methods for Partial Diﬀerential Equations The One-Dimensional Schrödinger Equation. As an appetizer to solving the time dependent Schrödinger equation in chapter 7 we will solve the simpliﬁed one-dimensional version for a spinless particle, i.e., iut = −uxx + V (x)u(x), where the potential V (x) is some scalar function that is independent of time. The Hamiltonian is then H = ∂ 2 /∂x2 +V . We choose for simplicity Ω = [0, 1] as for the wave equation and the corresponding grid. The standard ﬁnite diﬀerence approximation reads iu̇j = [−δx δx u]j + Vj uj , where Vj = V (xj ). Notice that δx δx can be written as a tridiagonal matrix when acting on uh . The big question is what discretization to use for time integration. In section 4.5 we will study the Crank-Nicholson scheme, a scheme ﬁrst introduced by Goldberg et al. in Ref. [35]. Let us write out this scheme for our example. It is deﬁned by i[δt+ u] = This yields 1 1 Hu+1 + Hu . 2 2 1 1 +1 = 1 − i∆tH u , 1 + i∆tH u 2 2 and we see that we have an implicit scheme. This scheme preserves the norm of uh ∈ CN exactly. In other words, the scheme is stable. The implicit nature of the scheme means that we have to solve a set of linear equations of dimension N at each time step. This system is easily seen to be tridiagonal, and hence it can be solved in O(N ) operations, see chapter 5. When going to higher dimensions the system no longer becomes tridiagonal ant it takes much longer to solve. 4.2.4 Incorporating Boundary and Initial Conditions As seen in the last example, incorporating boundary conditions was no big trouble. Let us however review the process in a more general setting. When devising a grid G corresponding to the domain Ω, we should choose a subset Γ ⊂ G corresponding to the boundary ∂Ω ⊂ Ω. On this set we impose our discrete boundary conditions. Choosing Γ is not always a simple task, but on a rectangular Ω it is of course in a natural way given by Γ = {xα ∈ G : αj = 0 or αj = Nj for at least one j}. Imposing Dirichlet boundary condition is the simplest task, because one can simply set uα = ψ(xα , t ) = ψα , xα ∈ Γ, = 0, 1, . . . , where ψ : ∂Ω → V is the boundary condition. Neumann conditions may impose diﬃculties. How do we interpret n, i.e., the unit normal, when considering the discrete boundary Γ? The answer is problem dependent. For now, let us only consider rectangular domains. In this case, the boundary of Ω is clearly an n − 1 dimensional box in Rn consisting of 2n sides (that is hyperplanes). If such a side is described by the condition xk = c with c a constant, then the unit normal is ±ek ; the kth standard basis vector in Rn , the sign depending on what side we are considering. Thus, ∂u n · ∇u = ± k ∂x along this side. 70 4.3 – The Spectral Method Let us consider a concrete example for illustration. Given a two dimensional PDE (with perhaps an additional time dependence) and assume that we want to apply the Neumann condition n · ∇u = a along the boundary of our rectangular domain Ω given by Ω = [x0 , x1 ] × [y0 , y1 ]. Along the boundary x = x0 the unit normal is −ex and thus n · ∇u = − ∂u = a. ∂x Similarly, ∂u =a ∂x along x = x1 . For the two remaining parts y = y0 and y = y1 we obtain n · ∇u = − ∂u =a ∂y ∂u = a, ∂y and respectively. Solving a transient problem of course means solving the PDE for t ∈ [t0 , t1 ], and so the time grid is always rectangular. The initial conditions are imposed by devising special rules for = 0, . . . , p − 1, where p is the order of the highest time derivative occurring in the PDE. In each case we must consider exactly how, but u0 is of course the initial function and u1 is usually calculated with either a forward Euler step and similarly for u1 up to up−1 . 4.3 The Spectral Method The ﬁnite diﬀerence approximations for spatial derivatives that are employed are typically of rather low order in the step size h, i.e., typically of order two. Based on the discrete Fourier transform (DFT) we may create a method that is of order N , where N is the number of grid points in one spatial direction. While this method is applicable up to three dimensions, and while it provides great accuracy, there are some catches. First of all the computational eﬀort is much higher. A ﬁnite diﬀerence approximation requires typically not much more than O(n) operations (where n is the total number of grid points), while the spectral method requires O(n log n) operations when we utilize the so-called fast Fourier transform algorithm (FFT) for performing the DFT.3 In addition, the FFT algorithm requires 2k grid points in each spatial direction with an equal spacing each, and the method requires periodic boundary conditions in a natural way. Thus the freedom of the geometry in question gets somewhat restricted. The idea behind the spectral method is that diﬀerentiation is a diagonal operator in the frequency domain. Recall that the Fourier transform of a function f (x) is given by ∞ 1 f (x)e−ikx dx, F [f ](k) := g(k) = 2π −∞ and the inverse Fourier transform is given by −1 F [g](x) := f (x) = ∞ g(k)eikx dx. −∞ 3 Se Ref. [...] for a discussion of the fast Fourier transform. 71 Numerical Methods for Partial Diﬀerential Equations Diﬀerentiating we get ∞ f (x) = ikg(k)eikx dx. −∞ Thus, F [f (x)](k) = ikF [f ](k) = ikg(k), and diﬀerentiating becomes multiplication with ik in the frequency domain, i.e., it is a diagonal operation. It is easy to see that if A is an operator combining diﬀerent ∂ ),4 then diﬀerentiations, i.e. A = A( ∂x F [A( For example, if A = ∂2 ∂x2 ∂ )f (x)](k) = A(ik)F [f ](k). ∂x ∂ + β ∂x , then F [Af (x)](k) = (−k2 + βik)F [f ](k). It is not diﬃcult to imagine that if we can make a good approximation to the Fourier transform of our discrete function, then we get a good approximation to any diﬀerentiation process with a simple diagonal (i.e., multiplicative) operator on the Fourier transformed function. Operating with A on f (x) is then equivalent to calculating A( ∂ )f = F −1 [A(ik)F [f ]] . ∂x 4.3.1 The Discrete Fourier Transform Assume that we have a discretely sampled function f (x) on [0, a], i.e., that we have a one-dimensional grid G of N + 1 points (with N even), viz., G = {xn = hn : n = 0, 1, . . . , N } with fn = f (xn ) and h = a/N . We shall assume f0 = fN , viz., that f is periodic. If we imagine the discrete f as being a superposition of plane waves, i.e. of diﬀerent eikx , then clearly we cannot have wave numbers greater than π/h, that is, waves with wavelength smaller than 2h. If the wavelength were smaller, then the discrete eikx would be equivalent to a wave of bigger wavelength. This phenomenon is called aliasing or folding. The critical wave number kc = π/h = N π/a is called the Nyquist wave number.5 As motivation for the discrete Fourier transform we need the so-called Nyquist theorem which we state without proof. Theorem 16 Assume that g(k) = F [f ](k) is identically zero for |k| ≥ kc = π/h, i.e., that f (x) is band-limited. Then f (x) is uniquely determined by the discrete version fn , n = 0, 1, . . . , N − 1. This remarkable theorem states that the information content in f (x) is very much less than the content in a function that is not band-limited. In fact, the complete information is given by a ﬁnite number of values, namely fn . On the other hand, this means that there is much redundant information in the Fourier transformed function g(k); it would be suﬃcient to devise N numbers gm = g(km ), m = 0, 1, . . . N − 1 that can be put in a one-to-one correspondence with fn . In other words, we may 4 Of course, some care should be taken when claiming this, but at least for simple polynomial expressions it holds trivially. 5 Some texts work with time t and angular frequency ω as conjugate variables. In that case one speaks of the Nyquist frequency. 72 4.3 – The Spectral Method devise a discrete Fourier transform (that of course in some sense should be a good approximation to the continuous transform) containing all the information in fn . The DFT is deﬁned by deﬁning a second grid G with N + 1 points, viz., G = {km = m N N 2πm = 2kc : m = − , . . . , }, Nh N 2 2 and by using the trapezoidal rule for approximating the Fourier transform at the grid points km , viz., gm = 1 2π a e−ikm x f (x) dx ≈ 0 N −1 h −ikm xn e fn 2π n=0 N −1 h −2πinm/N = e fn , 2π n=0 (4.8) where a = N h is the length of the interval, and where we have used the periodicity of f . If we deﬁne the matrix Z by Zmn = e−2πinm/N , then gm = n, m = 0, 1, . . . , N − 1, N −1 Zmn fn . n=0 Clearly, f has only N independent components due to f0 = fN and we also have ∗ . If we can prove that Z is invertible, then we have a well-deﬁned oneg−N/2 = gN/2 to-one mapping between fn and gm . It turns out that Z is in fact unitary (up to a multiplicative constant), so in the same way as in the original Fourier transform we may interpret the transformation as an orthogonal change of basis.6 To be more speciﬁc, ZZ † = N I ⇒ Z −1 = 1 † Z , N where I is the identity matrix. Let us prove this. Theorem 17 The N × N matrix Z given by Zmn = e−2πinm/N , n, m = 0, 1, . . . , N − 1, obeys ZZ † = N I, Z −1 = 1 † Z . N In other words, the column vectors form an orthogonal basis for CN , their length being √ N. Proof: To prove this we must compute [ZZ † ]mn = N −1 † Zmj Zjn = j=0 e−2πimj/N e2πinj/N j=0 = N −1 j=0 6 See N −1 2πij(n−m)/N e ≡ N −1 φj . j=0 Appendix A. 73 Numerical Methods for Partial Diﬀerential Equations Note that for n = m we have φj = 1, so that the diagonal elements are equal to N . For the oﬀ-diagonal elements, note that |n − m| < N and that if we deﬁne k = N − 1 − j, then we may write S= 0 e2πi(N −1−k)(n−m)/N k=N −1 = e2πi(n−m) e−2πi(n−m)/N N −1 e−2πik(n−m)/N = e−2πi(n−m)/N S. k=0 Since |n − m| < N then (n − m)/N is not an integer and so e2πi(n−m)/N = 1, and consequently we must have S = 0. Hence, ZZ † = N I, and if ψn is the n’th column vector, then (ψm , ψn ) is the inner product with the m’th row vector of Z † , and hence √ (φm , φn ) = δnm N ⇒ φn = N . By virtue of the unitarity property of Z the inverse transformation of Eqn. (4.8) is then given by N −1 N −1 2π † 2π 2πinm/N fn = Znm gm = e gm . (4.9) N h m=0 N h m=0 Note that considered as functions deﬁned on the whole real line, both fn and gm are periodic functions: fn ≡ fN +n and gm ≡ gN +m . Hence, the spectral range |k| ≤ kc is not unique. For an interpretation of the inverse DFT we turn to the inverse continuous Fourier transform and again use the trapezoidal rule for approximation, viz., fn = kc ikxn e −kc 2π g(k) dk ≈ Nh m =N/2−1 eikm xn gm . (4.10) m =−N/2 We have used the natural frequency range given by the Nyquist theorem. It is convenient however for both implementation purposes and for the intuitive appeal to the ﬁnal formulas of the DTFs to have the same summation range in both cases. By using Eqn. (4.10) we get the wave number range |k| ≤ kc , and by using Eqn. (4.9) we get the range 0 ≤ k ≤ 2kc . If we shift the negative indices in Eqn. (4.10) upward by N , we get Eqn. (4.9). But then we must bear in mind that the actual frequencies km are changed, according to Fig. 4.3. Taking −N/2 < m < 0 corresponds to k > 0, and taking 0 ≤ m < N/2 − 1 corresponds to k < 0. We may of course just use the range 0 ≤ k ≤ 2kc ; the numerical results are equivalent due to the periodicity of eikx . Note that taking −kc or kc at m = N/2 − 1 is equivalent. In a similar fashion, one sees that the restriction to the interval 0 ≤ x ≤ a is not necessary. Shifting away only leads to multiplying gm by a constant of magnitude 1. The number h is in computational settings a quite small number. If we deﬁne hGm := gm , we may rid ourselves of this factor in the transformations. We now have the pair N −1 2π † fn = Z Gm . (DFT) N m=0 nm and Gm = 74 N −1 1 Zmn fn 2π n=0 (inverse DFT). 4.3 – The Spectral Method wave number kc G´ 0 N/2–1 N –kc Figure 4.3: Adjusting the wave numbers Note that h does not appear anywhere in the formulas, implying that the coeﬃcients Gn is independent of h. Because of the trapezoidal approximation nature of the deﬁnition, it is natural to ∂ take ∂x to be a diagonal operator when applied to the transformed function (which is now a vector in CN ), viz., [f (x)]n := 1 † [Z D(Z f˜)]n , N where f˜ is the vector whose components are fn and D is a diagonal matrix whose ﬁrst N/2 diagonal elements Dmm , m = 0, . . . , N/2 − 1 are given by 2π/N h. The last N/2 − 1 diagonal elements are then given as Dm+N,m+N = 2π/N h. It can be shown that the DFT provides an approximation to f (x) of order N , i.e., we have exponential convergence of the diﬀerential operator. Intuitively this is so because the spectral method uses all values fn to compute the derivative, not just the ones in the immediate vicinity of xn . 4.3.2 A Simple Implementation in Matlab Here we present a very simple implementation of the spectral method applied to a time-independent model problem. More speciﬁcally, we wish to study the scattering of a wave packet onto a square barrier potential. We will also make a brief comparison of the DFT results with ﬁnite diﬀerence results. Further error analysis will not be performed in this example. The Hilbert space for this problem is given by H = L2 ([−c, c]), where c is chosen large enough so that we may neglect interference of the wave packet with itself due to the boundary conditions. The Hamiltonian is H = T + V, with T = −∂ 2 /∂x2 and V (x) = a x ∈ [−b, b], 0 otherwise. Here, a is a positive constant deﬁning the strength of the barrier, and 2b is the width. 75 Numerical Methods for Partial Diﬀerential Equations As initial condition we use a Gaussian wave packet centered at x0 with initial momentum k0 and width σ, i.e., ψ0 (x) = 2 2 1 e−(x−x0 ) /2σ e−ik0 x . 2 1/4 (2πσ ) We will choose x0 and k0 so that the wave is neglible outside the barrier and so that it is travelling towards it from the left. Since we are going to use the DFT for implementing the diﬀerential operator, we employ periodic boundary conditions. For integration in time of the Schrödinger equation we use a split-operator method (see Ref. [36]) which approximates the propagator to third order in the time step size τ , viz., Uτ = e−iτ T /2 e−iτ V e−iτ T /2 = U (t0 + τ, t0 ) + O(τ 3 ). Note that V is diagonal in position representation and that T is diagonal when applied to the Fourier transformed wave function. Therefore, the algorithm for obtaining the (approximate) wave function at time t + τ reads: 1. Use DFT on the numerical solution. 2. Apply e−iτ T /2 (a diagonal operator.) 3. Use inverse DFT. 4. Apply e−iτ V /2 (again a diagonal operator.) 5. Use DFT on the numerical solution. 6. Apply e−iτ T /2 (a diagonal operator.) 7. Use inverse DFT. If we perform this scheme several times in succession, we are doing quite a few redundant Fourier transforms. Usually one combines the last and the ﬁrst application of e−iτ T /2 if no output of the wave function is desired in between time steps. For this simple problem however, it is no problem to waste a few computer cycles. The integration scheme is also applicable to ﬁnite diﬀerence methods. If we use the −δx δx diﬀerence approximation to T , then we have by the deﬁnition of the exponential operator τ τ2 eiτ δx δx /2 = 1 + i δx δx − (δx δx )2 + O(τ 3 ), 2 4 and the error of this approximation is of the same order as the scheme itself. The δx δx operator with periodic boundary conditions is easily implemented with a sparse matrix in Matlab, making the implementation quick and easy to read. See appendix B for the code. In our program we implement both the spectral method and the ﬁnite diﬀerence method for comparison. Fig. 4.4 shows a series of numerical wave functions for a simulation with a = 28, b = 2.5, x0 = −10, k0 = 5 and σ 2 = 4. Parameters for the numerical methods were N = 1024 and τ = 0.00025. Note that the kinetic energy of the wave packet is k02 = 25 < a, so that classically the particle is not allowed to pass the barrier. Nevertheless we see that some of the packet passes the obstacle, i.e., there is some probability that the particle upon measurement of its position will be found in the classically forbidden regime. Performing numerical experiments of this kind can give very rich insight into the behavior of the system. This particular case can display resonance phenomena such as 76 4.4 – Finite Element Methods 2 2 t =0.0 1.6 1.2 1.2 0.8 0.8 0.4 0.4 0 -30 -20 -10 0 10 20 30 2 0 -30 -20 -10 0 10 20 30 2 t =0.5 1.6 1.2 0.8 0.8 0.4 0.4 -30 -20 -10 0 10 20 t =2.0 1.6 1.2 0 t =1.0 1.6 30 0 -30 -20 -10 0 10 20 30 Figure 4.4: Results from running a simple simulator using the spectral method and split-operator time stepping. some of the probability getting caught inside the barrier. (This is actually hinted at in the picture at t = 2.0.) Furthermore, we see an interference pattern at the moment of collision. What is the width of the fringes? What parameters in the simulation does it vary with? Another interesting experiment is to measure the particle’s mean position at each side of the wall as function of t and compare this with the classical results. Animations of systems of this kind can be reached from Ref. [5]. These are produced with ﬁnite diﬀerence methods and the leap-frog scheme for time integration. (See section 4.5 for details on the leap-frog scheme.) These simulations were used in lectures with the aim that the students should gain some insight into the behavior of quantum mechanical systems. In the last picture, the ﬁnite diﬀerence solution is also shown for comparison. Qualitatively we see that they match very well, indicating that both the ﬁnite diﬀerence method and the spectral method yield reasonable results. 4.4 Finite Element Methods In this section we introduce the ﬁnite element method; a powerful class of numerical methods for solving partial diﬀerential equations. The perhaps most intuitively appealing aspect of the method is the handling of complicated geometries. Furthermore, the method is formulated in a highly modularized way, giving object-oriented languages such as C++ a great advantage when one wishes to implement the methods. We start out by introducing the ﬁnite element method in a rather informal way, emphasizing the algorithm over the numerical properties of the method. We begin by introducing the weighted residual method of which the ﬁnite element method is a special case. Next, we compute an example in one dimension and brieﬂy describe 77 Numerical Methods for Partial Diﬀerential Equations instances of the method in higher dimensions. Finally, implementations of the ﬁnite element method with the aid of the programming library Diﬀpack for C++, which is the combination of tools used in this thesis, are discussed. Throughout the discussion we will use the typical Hamiltonian H = −∇2 + V (x ) as a prototype for diﬀerential operators that we need to discretize. (We omit constants such as and µ.) Finite elements are typically used in the spatial domain. For time dependent problems we use some diﬀerence scheme in time, leading to a sequence of spatial problems that we may solve with the ﬁnite element method. Hence, we will consider stationary problems in the introduction to keep it simple. Consider for simplicity the backward Euler method in time, i.e., 1 +1 u − u = −iHu , ∆t where u denotes the (complex) wave function at time t = ∆t. This equation may be written (1 + ∆tiH)u+1 = u , or Au = f, (4.11) where A = 1+∆tiH, u = u+1 is the unknown and f = u . Eqn. (4.11) is the prototype of the kind of equations we solve in both ﬁnite diﬀerence and ﬁnite element methods. We see that the operator A is a simple function of the Hamiltonian, and this is the case in all our applications. Hence, it is important to understand how operators such as ∇2 , V (x ) and so on are discretized with the ﬁnite element method. The PDE is deﬁned on some domain Ω ⊂ Rd , where d is the dimension of our system. Finite element methods require Ω to be a compact domain, i.e., closed, bounded and path-connected. For unbounded quantum mechanical systems we must in some way choose an appropriate bounded representative of whole space. We also have some kind of boundary conditions in our PDE. Typically, we may have Dirichlet boundary conditions or Neumann conditions. Divide the boundary Γ = ∂Ω into two parts: ΓD and ΓN , on which we impose Dirichlet and Neumann conditions, respectively: and u(x ) = g(x ), ∂u = n · ∇u(x ) = h(x ), ∂n x ∈ ΓD , x ∈ ΓN , where n is the unit normal on ΓN . 4.4.1 The Weighted Residual Method In the weighted residual method we deﬁne a subspace Vh ⊂ V as Vh dim Vh = sp {N1 (x ), N2 (x ), . . . , Nm (x )} , = m. Hence, Ni form a basis for Vh .7 In Vh we seek our discrete approximate solution uh . 7 The subscript h usually denotes some kind of discretization. Recall that h is usually used for the mesh width in discretization formulations. 78 4.4 – Finite Element Methods Any element uh ∈ Vh can be written uh = m uj Nj , i=1 and we deﬁne a vector U ∈ Cm by letting Uj = uj . Note the fundamental diﬀerence from ﬁnite diﬀerence methods in which we ignore the behavior of u in between grid points. In the weighted residual method the discrete function is exactly that; a function deﬁned in the whole domain. Next, for any u ∈ V deﬁne the residual R through R := Au − f. The residual vanishes if and only if u is a solution of our prototype linear PDE (4.11). The idea of the weighted residual method is to choose m functions Wi ∈ V with which we weight the residual R, i.e., we take the inner product. Requiring this inner product to vanish for each i then forces R to be small (in some sense) and leads to m equations for the m unknown coeﬃcients ui . In other words, we require that (Wi , R) = 0, i = 1, . . . , m. By linearity of A we obtain m uj (Wi , ANj ) = (Wi , f ). j=0 Clearly, this is a matrix equation of ﬁnite dimension. Deﬁne the matrix Ah as Wi∗ (x )ANj (x ) dΩ (Ah )ij := (Wi , ANj ) = Ω and the vector b as bi := (Wi , f ) = Wi∗ (x )f (x ) dΩ, Ω and the weighted residual method becomes equivalent to the matrix equation Ah U = b. If this equation has a unique solution, we have found the unique discrete function uh ∈ Vh such that the residual is orthogonal to all weighting functions Wi . If we deﬁne W = span {W1 , W2 , . . . Wm } ⊂ V, then R is orthogonal to all vectors in W , i.e., it belongs to the orthogonal complement W ⊥ of W . If W spans a large portion of V then W ⊥ must be only a small portion of V (in some intuitive sense; dim W ⊥ may easily be inﬁnite!) If our PDE is non-linear our system of algebraic equations also becomes non-linear. This is a more complicated problem to solve, and methods such as the Newton-Rhapson iteration or successive substitution are popular solution methods, see Ref. [34]. There are of course many diﬀerent ways of choosing the weighting functions Wi . We mention two popular choices here. – The collocation method, in which we require the residual to vanish at m speciﬁed points, i.e., the weighting functions are given by Wi (x ) = δ(x − x [i] ) for each of the m points x [i] . 79 Numerical Methods for Partial Diﬀerential Equations – Galerkin’s method, in which we require R to be orthogonal to Vh which is equivalent to R being orthogonal to the basis functions Ni . Hence, Wi = Ni are the weighting functions. We will exclusively use Galerkin’s method in this text. When considering the numerical properties of the weighted residual method and the ﬁnite element method, Galerkin’s method shows some remarkable and fortunate properties, such as best approximation properties in diﬀerent norms and equivalence with minimation of a functional over V and Vh , respectively, in the continuous and discrete problems. Galerkin’s method written out in terms of our prototype diﬀerential equation then reads m uj (Ni , ANj ) = (Ni , f ), i = 1, . . . , m. j=1 4.4.2 A One-Dimensional Example We have introduced the weighted residual method, of which the ﬁnite element method is a special case. The ﬁnite element method deﬁnes the basis functions Ni and in a quick introduction such as this it is best explained through an example. When solving the discrete equations numerically the particular form of Ah matters. First, if Ni are far from orthogonal we will in general obtain a dense matrix Ah which requires a lot of computer time and storage to process. Second, the matrix may become ill-conditioned, rendering the solution sensitive to round-oﬀ errors. Hence, we should choose orthogonal (or nearly orthogonal) basis functions. The ﬁnite element method in a natural way ensures this. The ﬁnite element basis functions Ni are deﬁned in conjunction with the grid. The grid in ﬁnite element methods is somewhat more complicated than in ﬁnite diﬀerence methods. In addition to deﬁning grid points called nodes, the domain is also divided into disjoint elements with which the nodes are associated. The elements play a fundamental role when deﬁning the basis functions. For the one-dimensional example, consider Ω = [0, 1] and choose m points x[k] such that 0 = x[1] < x[2] < · · · < x[m] = 1. The grid points, or nodes, naturally divide Ω into sub-intervals Ωe = [x[e] , x[e+1] ]. These sub-intervals are our elements, and we see that we have ne = m − 1 elements. Now we deﬁne our m basis functions Ni : – Each Ni should be a simple polynomial over each element Ωe . Thus, uh becomes a piecewise polynomial function. In our case we choose linear functions. – Fundamental in the ﬁnite element method is the requirement Ni (x[j] ) = δij . This imposes as many conditions on each polynomial as there are nodes in an element. In our case we have two nodes per element, which implies that Ni must be linear over each element. Note that uh (x[i] ) = uj Nj (x[i] ) = j i.e., ui is uh evaluated at a node point. 80 j uj δij = ui , 4.4 – Finite Element Methods uh(x) Ne(x) x[e-1] x[e] Ωe-1 x[e+1] x Ωe Figure 4.5: Linear elements in one dimension. A sample basis function Nk is showed together with a piecewise linear function uh . Fig. 4.5 illustrates our one-dimensional grid. If we let he = |Ωe |, then it is easy to see that for i = 2, . . . , m − 1, Ni (x) is given by 0 Ni (x) = ) hi−1 (x − x 1 [i] 1 − hi (x − x ) 1 [i−1] x∈ / Ωi−1 ∪ Ωi x ∈ Ωi−1 . x ∈ Ωi For i = 1 or i = m the deﬁnition is similar, i.e., 0 N1 (x) = 1− and Nm (x) = 1 h1 (x x∈ / Ω1 − x ) x ∈ Ω1 [1] 0 1 hm−1 (x −x [m−1] ) x∈ / Ωm−1 x ∈ Ωm−1 . Hence, the element functions vanish identically over most of the grid. The “tent functions” are also almost orthogonal, as (Ni , Nj ) = 0 whenever x[i] and x[j] does not belong to the same element. It is also easy to see that they can be combined to create any piecewise linear function with x[k] deﬁning the joints of the linear functions. It is easy to see that the nodal point condition implies that (Ni , Nj ) = Ni Nj dΩ Ω vanishes when i and j are nodal indices belonging to diﬀerent elements. Looking forward, we see that matrices whose elements are on the form (ANi , BNj ) with A and B arbitrary linear operators obtain the same sparse structure. Let us sketch a weighted residual statement with these basis functions. We wish to solve Au = f, A = 1 − ∇2 . We choose as boundary conditions u(0) = C1 , u (1) = C2 , a Dirichlet and a Neumann condition, respectively. The Dirichlet condition automatically implies u1 = C1 . As we will see the Neumann condition enters when we rephrase our equations into weak form; see also section 4.6 for more on the weak form. 81 Numerical Methods for Partial Diﬀerential Equations Note that Ni is not deﬁned, since Ni is piecewise linear and hence cannot be diﬀerentiated.8 The weighted residual statement reads m uj (Ni , Nj − Nj ) = (Ni , f ), i = 1, . . . , m. j=1 In other words, m Ni (x)Nj (x) − uj Ω j=1 Ni (x)Nj (x) Ω = Ni (x)f (x), Ω where we have used that Ni are real functions. The second order derivative is eliminated by integrating by parts, viz., ! "1 Ni (x)Nj (x) = − Ni (x)Nj (x) + Ni (x)Nj (x) 0 . Ω Ω This gives m j=1 uj Ni (x)Nj (x) + Ω Ni (x)Nj (x) + Ni (0)uh (0) − Ni (1)uh (1) Ω Ni (x)f (x). = Ω The boundary term at x = 0 seems troublesome as we do not know uh (0). However, equation i = 1 is eliminated due to the left boundary condition, a Dirichlet condition. As Ni (0) = 0 for the remaining equations, the term drops out. The boundary term at x = 1 involves the Neumann condition, and we have Ni (1)uh (1) = δim C2 . Not only did we get rid of the second derivatives by integration by parts, but we introduced the Neumann condition into the discrete system in a convenient way. Finally we obtain the equations m uj j=1 Ni (x)Nj (x) + Ω u1 = C 1 , Ni (x)Nj (x) = Ni (x)f (x) − δim C2 , Ω (4.12) i = 2, . . . , m. Ω (4.13) This is a set of m linear equations in the unknowns ui . There are two matrices that appear naturally in the equations; the stiﬀness matrix K and the mass matrix M . They are deﬁned by ∇Ni · ∇Nj dΩ (4.14) Kij := Ω and Mij := Ni Nj dΩ. (4.15) Ω They have a tendency to appear in ﬁnite element formulations and it is wise to get to know them for the most used element types, such as the linear elements in one dimension. They are very sparse in large systems, and this indicates that iterative methods could be used to solve them with great eﬃciency, see chapter 5. Let us sum up the general results from these calculations: 8 Strictly speaking, neither can Ni be diﬀerentiated as Ni is undeﬁned as the nodes. Whatever value we choose for Ni (x[j] ) does however not contribute to the integrals. This is connected with the concept of weak derivatives, see Ref. [17]. 82 4.4 – Finite Element Methods – Galerkin’s method leads to an m-dimensional system of linear equations, Ah U = b, where A = M + K and bj = (Nj , f ) except for modiﬁcations due to boundary conditions. – A Dirichlet boundary condition at node x [k] is enforced by replacing equation k in the system with the boundary condition, i.e., Akk = 1, Akj = 0 whenever j = k, and bk = g(x [k] ), where g is the prescription of u at the boundary. Dirichlet conditions are called essential boundary conditions in ﬁnite element contexts because their incorporation is done immediately in the linear system. – Neumann boundary conditions are imposed by integration by parts of second order derivatives. This leads to extra terms in the right hand side vector b. Neumann conditions and other boundary conditions involving derivatives are called natural boundary conditions in ﬁnite element contexts. They “appear naturally” in the process of integrating by parts. 4.4.3 More on Elements and the Element-By-Element Formulation In this section we will brieﬂy describe the generalization of the ideas presented in the one-dimensional example above. In the one-dimensional case the shape of the elements Ωe becomes rather limited. They are simply intervals of varying length. In higher dimensions we have greater freedom of choosing element shapes. For example, if linear elements (i.e., linear Ni ) are employed, the shape of Ωe may be any quadrangle. Furthermore, triangular shapes may be used, and indeed this is a very popular choice because very good algorithms exists for dividing a region Ω into triangles. In three dimensions we would use deformed parallelepipeds and tetrahedrons. A simple example is depicted in Fig. 4.6. A simple rectangular grid is subdivided into equal square slabs. The nodes are located at the corners of each slab, and an element function is depicted at a particular node. Note the obvious tensor-generalization of the one-dimensional case, see Ref. [34] for details on tensor-product generalizations of one-dimensional element types. Clearly, the process of calculating the element basis becomes rather complicated with increasing complexity of the geometries, location and shapes of elements and so on. All geometric variants of an element are reﬂected in the expressions for the corresponding basis functions over that element. Hence, it is natural to use a reference element and map the results from this element back to the real geometry. This is done in the element-by-element formulation, which we brieﬂy describe here. For more details, see Ref. [34]. The idea is to ﬁrst rewrite the integral over Ω as a sum of the integrals over each element Ωe , viz., A= Ae , and b = be , e e where each term in the sums are by deﬁnition identical to A and b but with integration only over Ωe . The mass matrix for example becomes e Mij = Mij = Ni Nj dΩe . (4.16) e e Ωe Next, one observes that due to the localized character of each basis function Ni , almost all of the components of Ae and be are zero. In fact, if i and j are indices that correspond to nodes outside of element e, then the basis functions Ni are zero and hence (Ae )ij and 83 Numerical Methods for Partial Diﬀerential Equations L Nk(x) L 0 0 0 x[k] 0 L L Figure 4.6: Linear elements in two dimensions. The nodes are located at line intersections. (be )i are also automatically zero. Each element in our one-dimensional example has two nodes, hence Ae can be represented by a 2 × 2-matrix Ãe and be can be represented by a two-dimensional vector b̃e . In our example, we choose a reference element ξ ∈ [−1, 1] = Ω̃. We use a linear change of coordinates to map Ωe into Ω̃. Then only two basis functions do not vanish over Ω̃. The two nodes ξ = ±1 correspond to x[e] and x[e+1] , respectively. Clearly, there is a mapping q(e, r), where r = 1, 2 is the local node number, that maps the element number e and node number r into the global node number k. This mapping exists in more general cases as well, but usually it is only known from a table due to complex geometries. The mapping from local coordinates to global coordinates is 1 x(e) (ξ) = x[e] + (ξ + 1)(x[e+1] − x[e] ). 2 Hence, Ni (x(e) (ξ)) = Nq(e,r) (x) = Ñr (ξ) is the global basis function in terms of the local basis function Ñr (ξ). In Fig. 4.7 this is illustrated. Integration over the reference element introduces the Jacobian of the coordinate change into the integrand. The linear one-dimensional elements are examples of so-called isoparametric elements. Such elements are characterized by the fact that the same mapping is used both for interpolating uh and for mapping from local to global coordinates. I.e., Ñr (ξξ ) deﬁnes both the basis functions and the mapping from local to global coordinates. The ˜1() N x ⍀1 ˜2() N +1 –1 ⍀̃ ⍀2 coordinate change Figure 4.7: Illustration of local coordinates for one-dimensional linear elements. 84 4.5 – Time Integration Methods (inverse) coordinate change in isoparametric elements are in general given by x (e) (ξξ ) = nno Ñr (ξξ )x [q(e,r)] , r=1 where nno is the number of nodes in each element. The introduction of local coordinates eases the implementation of a ﬁnite element solver and we also realize the modularized nature of the ﬁnite element method and hence also the appropriateness of object-oriented programming techniques in this case. Everything from the linear equations, via the grid and the elements to the assembly routines are actually deﬁned in an object-oriented manner, so to speak. 4.5 Time Integration Methods In its simplest form, we again state the time dependent Schrödinger equation: ∂ Ψ = −iH(t)Ψ. ∂t Again, Ψ is a complete quantum state, encapsulating space- and spin-degrees of freedom that the particle might have. The Hamiltonian H(t) contains some spatial dependencies through the kinetic energy term (i.e., spatial second-order diﬀerentiation) and the potential energy. For example, H(t) = −∇2 + V (x , t), in the case of a spinless particle under the inﬂuence of the time dependent potential V (x , t). The Schrödinger equation imposes stronger constraints on the state than the states allowed in quantum theory. It must be at least twice diﬀerentiable with respect to spatial coordinates, for example. The time dependence is our concern in this section. Naturally, we will use some kind of ﬁnite diﬀerence approximation to the time derivative, which yields a sequence of discrete problems, regardless of how we choose to handle the space dependencies and their approximations. Our weapons of choice are the leap-frog scheme, an explicit integration scheme whose name points to the staggered nature of the time stepping, and the theta-rule, an implicit scheme covering both the forwards and backwards Euler schemes in addition to the popular Crank-Nicholson scheme. What characterizes the quality of a numerical scheme? How do we decide whether it is a good scheme or not? What do we mean by “a good approximation?” Several factors contribute, among them the accuracy, the stability and the computational cost. Clearly, when our time step becomes small enough our scheme should become more and more like the continuous diﬀerential equation. This is measured through the socalled truncation error τ . When we write our PDE in homogenous form, i.e., L(Ψ) = 0, and formulate a discretization, viz., L∆ (Ψh ) = 0, we deﬁne the truncation error τ as the residual we are left with when we insert a solution of the continuous PDE into the discrete version, viz., τ = L∆ (Ψ), where L(Ψ) = 0. 85 Numerical Methods for Partial Diﬀerential Equations We use Ψh for the time-discretized wave function. It should not be confused with the spatially discretized wave function from the previous sections. The expression for τ and also for the integration scheme are purely formal. The iteration process implies that we must apply operators like the diﬀerential operator more then once. We have not assumed that Ψ is more than two times diﬀerentiable. However, the formality vanishes when we apply a spatial discretization. For instance, the operator δx δx may always be applied to a discrete function. Clearly, if τ does not vanish when the temporal mesh width becomes inﬁnitely small, L∆ cannot represent a good approximation to the equation. In the opposite case of a vanishing τ , we say that the numerical scheme is consistent. We also need stability. Stability may be deﬁned as the property that the numerical solution Ψh reﬂects the qualitative properties, i.e., the physical properties, of the continuous solution Ψ. If this is not fulﬁlled, then our approximation is useless! We want physical solutions. This demand is somewhat informal, but it is almost always easy to devise the proper requirement for a given PDE. If both stability and consistency is present, then a famous theorem by Lax states that our numerical solution is convergent, that is the solution to the numerical equations converges to the continuous ones in the limit of vanishing grid spacings. The converse is also true. See Ref. [34]. We study this further in section 4.6. The stability criterion we are searching for in the case of the time dependent Schrödinger equation is conservation of probability, i.e., the norm of the wave function. Basic quantum mechanics teaches us that the norm of the wave function Ψ is conserved at all times, and this property we also demand from the numerical solution in order to ensure that it converges to the true solution by Lax’ theorem. 4.5.1 The Theta-Rule The theta-rule is a means for discretizing a ﬁrst order derivative on the form ∂ψ = G. ∂t (4.17) Here, G is some arbitrary expression that we assume to be diﬀerentiable so we may take the Taylor expansion of G around some arbitrary time to obtain the expression at some other time. The θ-rule is an interpolation between the forwards and backwards Euler discretizations and is deﬁned through 1 +1 Ψh − Ψh = θG+1 + (1 − θ)Gh . h ∆t (4.18) We comment that the derivations in this section are formal, in the sense that we do not actually know whether the quantities are deﬁned in all circumstances. For example, we do not know whether G2 Ψh is a function. If G = −∇2 then Ψh must be four times diﬀerentiable and so on. On the other hand, when introducing spatial discretization the problem vanishes. For example, the diﬀerence operator is always deﬁned on any discrete function. Reorganizing Eqn. (4.18) yields the updating scheme for the numerical solution: " " ! ! = 1 − i(1 − θ)∆tH Ψh . (4.19) 1 + iθ∆tH +1 Ψ+1 h When discretizing a diﬀerential equation on the form of Eqn. (4.17) it is common to consider the forwards and backwards Euler schemes. For the Schrödinger equation though, neither one is particularly suitable. Forwards Euler is notoriously unstable and the backwards variant contains damping, making the scheme non-unitary. We 86 4.5 – Time Integration Methods recover forwards and backwards Euler with θ = 0 and θ = 1, respectively. (The claimed stability and damping will be clear after we have analyzed the θ-rule.) The case θ = 1/2 gives us the Crank-Nicholson scheme and is of special interest which will be clear later. Let us derive the truncation error of the θ-rule. We start by considering the left hand side of Eqn. (4.18) and insert an exact solution. We formally expand the Taylor series of Ψ+1 around t , viz., 1 ∆t∂t 1 +1 Ψ e − Ψ = − 1 Ψ . (4.20) ∆t ∆t Note the notation for the Taylor series. To second order we obtain ∂Ψ 1 +1 ∆t ∂ 2 Ψ ∆t2 ∂ 3 Ψ Ψ + − Ψ = + + O(∆t3 ). (4.21) ∆t ∂t 2 ∂t2 6 ∂t3 This is nothing but the familiar truncation error for a forward diﬀerence, see section 4.2. Consider the right hand side in a similar fashion, viz., θG+1 + (1 − θ)G = θe∆t∂t G + (1 − θ)G . (4.22) Expanding the series we obtain ∂Ψ ∆t ∂ 2 Ψ ∆t2 ∂ 2 G ∆t2 ∂ 3 Ψ ∂G + + θ + = G + θ∆t + O(∆t3 ). ∂t 2 ∂t2 6 ∂t3 ∂t 2 ∂t2 (4.23) The PDE (4.17) written in homogenous form is ∂Ψ − G(Ψ, t) = 0, ∂t and our discretization is correspondingly 1 +1 Ψh − Ψh − θG+1 − (1 − θ)G = 0. L∆ (Ψh ) = ∆t The truncation error τ is deﬁned as L(Ψ) = τ = L∆ (Ψ). (4.24) We insert the results obtained in Eqns. (4.21) and (4.23) and when noting that ∂ n G ∂ n+1 Ψ = , ∂tn ∂tn+1 we obtain 1 1 θ ∂ 3 Ψ ∂ 2 Ψ τ = L∆ (Ψ ) = ∆t( − θ) 2 + ∆t2 ( − ) 3 + O(∆t3 ). 2 ∂t 6 2 ∂t (4.25) Hence, the theta-rule has truncation error of order O(∆t) for θ = 1/2, but in the case of θ = 1/2 we obtain a truncation error of order O(∆t2 ). This particular case is called the Crank-Nicholson-scheme and is very well suited for solving the time dependent Schrödinger equation; not only because of its higher accuracy, but we shall also see that we obtain unitarity of the scheme. Thinking about it, since the forward Euler scheme makes the solution unstable in the sense that the norm of the solution is growing, and the backwards Euler scheme behaves in the opposite way, it is not unnatural to expect that for some θ these eﬀects are exactly balanced. That our scheme is of second order is ﬁne, but another important question is: What is the error in Ψh after time integration steps in time? How much does it diﬀer from Ψ , which is the exact solution found from solving the continuous diﬀerential equation (4.17), when we in both cases start with the same initial condition? In the case of the θ-rule the answer is somewhat delicate, but in the leap-frog case it is rather trivial. is implicitly deﬁned in Eqn. (4.19). The trouble is that Ψ+1 h 87 Numerical Methods for Partial Diﬀerential Equations 4.5.2 The Leap-Frog Scheme We now turn to the leap-frog scheme. In the context of the time dependent Schrödinger equation this scheme was ﬁrst proposed by Askar and Cakmak in 1978 (see Ref. [37]) as an alternative to the implicit Crank-Nicholson scheme. Given a ﬁrst order diﬀerential equation on the form of Eqn. (4.17) we simply use a centered diﬀerence on the left hand side and evaluate the right hand side in-between, viz., 1 +1 Ψh − Ψ−1 = G(Ψh , t ). h 2∆t (4.26) Reorganizing this equation yields for the new wave function Ψ+1 Ψ+1 = Ψ−1 − 2i∆tH Ψh . h h (4.27) to ﬁnd the new Ψ+1 Notice that we need two earlier wave functions Ψh and Ψ−1 h h , i.e., twice the information needed in the theta-rule. In reality we will not need this double information. If we separate the real and imaginary parts of the wave function, i.e., write Ψh = Rh + Ih , we obtain R+1 I +2 = R−1 + 2∆tH I = I − 2∆tH +1 R+1 . If we assume that H applied to a real vector again is a real vector, we have created an algorithm to update R and I alternately. In this case we say that we are using a staggered grid.9 On homogenous form we have L∆ (Ψh ) = 1 +1 Ψh − Ψ−1 − G(Ψh , t ). h 2∆t To ﬁnd the truncation error we insert a solution Ψ of the continuous problem in L∆ and expand Ψ±1 in Taylor series around t . This yields for the time diﬀerence #∞ $ 1 +1 1 ∆t2n+1 ∂ 2n+1 Ψ −1 Ψ = −Ψ 2∆t ∆t n=0 (2n + 1)! ∂t2n+1 (4.28) ∂Ψ ∆t2 ∂ 3 Ψ ∆t4 ∂ 5 Ψ = + + + O(∆t6 ). ∂t 6 ∂t3 120 ∂t5 Hence, the truncation error is of second order also for the leap-frog scheme, viz., τ= ∆t2 ∂ 3 Ψ + O(∆t4 ). 6 ∂t3 (4.29) Note that in both the Crank-Nicholson case and the current leap-frog scheme is not only τ of second order, but the dominating term in the truncation error is also proportional to the third time derivative of Ψ, i.e., τ∼ ∂2G ∂3Ψ = . 3 ∂t ∂t2 9 The assumption does not hold in general. If ψ is a real function and if H = −i∂/∂x, Hψ is purely imaginary. 88 4.5 – Time Integration Methods 4.5.3 Stability Analysis of the Theta-Rule Recall from section 1.3 that the norm of the wave function Ψ is conserved at all times due to the unitarity of the evolution operator. In light of the probabilistic interpretation of the wave function this means that the probability of ﬁnding the particle at some place with some spin orientation at any time is unity. The unitarity of the evolution operator should of course be reﬂected in the numerical solution of our discretized equations, i.e., in the numerical evolution operator. The updating rules for the theta-rule and the leapfrog scheme in reality are approximations to the time evolution operator U (t+1 , t ). In particular, for the theta-rule we have −1 1 − i(1 − θ)∆tH . = 1 + iθ∆tH +1 U∆ (4.30) A simple special case is the case where H is independent of time. If n is its discrete set of eigenvalues,10 then the eigenvalues of U∆ clearly are λn = 1 + iθ∆tn , 1 − i(1 − θ)∆tn and the eigenvectors coincide with those of H. Furthermore, the norm of the eigenvalues are 1/2 1 + θ 2 ∆t2 2n |λn | = . 1 + (1 − θ)2 ∆t2 2n It is easy to see that for θ = 1/2, |λn | ≡ 1. Furthermore, θ < 1/2 yields |λn | > 1 and θ > 1/2 yields |λn | < 1, in other words unconditionally unstable and stable schemes, respectively, in the sense of ampliﬁcation and damping of the solution. For time dependent Hamiltonians however, the matter is more complicated. Let us deﬁne two operators A and B so that U∆ = A−1 B, viz., = 1 + iθ∆tH +1 −1 1 − i(1 − θ)∆tH . U∆ A B Given some wave function Ψh , the norm of the new wave function after evolving it is with U∆ +1 2 † Ψ = (Ψh , (U∆ ) U∆ Ψh ) = (Ψh , B † (AA† )−1 BΨh ). h † If the scheme is to be perfectly unitary, then (U∆ ) U∆ must be the identity operator 1. This happens if the Hamiltonian is independent of time and θ = 1/2. For the nonautonomous cases we cannot expect this to happen. We can only hope for unitarity up to some order of ∆t. First we note that AA† = 1 + ∆t2 θ 2 (H +1 )2 . We need the inverse of this operator. In general, in a similar fashion as with ordinary scalar numbers, we have the power series expansion: (1 − X)−1 = ∞ Xk, k=0 provided that all the eigenvalues of X is less than 1 in magnitude. This puts a restriction on ∆t for the series to converge, i.e., 2 ∆t2 θ 2 (+1 n ) < 1, ∀n, 10 Even though H may have a continuous spectrum, any spatially discretized version will have a discrete spectrum with a ﬁnite number of eigenvalues. 89 Numerical Methods for Partial Diﬀerential Equations where n are the (real) eigenvalues of H . We cannot invert A with a series in a general inﬁnite dimensional Hilbert space, since the eigenvalues are not bounded in general.11 However, we are going to do some kind of discretization in space, leaving the Hilbert space with a ﬁnite dimension and thus a bounded spectrum. We may then choose ∆t small enough and perform the expansion of A−1 . Increasing the dimension of the discrete Hilbert space will typically lower the typical mesh width h of the spatial grid used. This leaves us with a better approximation of the complete, inﬁnite dimensional space and also more eigenvalues, which of course will grow in magnitude as we get more of them. Hence, the ∆t-criterion must contain h in such a way that lowering h will automatically require a lower ∆t. Assume that the series is convergent, i.e., † −1 (AA ) = ∞ (−θ∆tH +1 )2m = 1 − ∆t2 θ 2 (H +1 )2 + ∆t4 θ 4 (H +1 )4 + O(∆t6 ). m=0 Multiplying with B from the right yields (AA† )−1 B = 1 − i(1 − θ)∆tH − θ 2 ∆t2 (H +1 )2 + i(1 − θ)θ 2 ∆t3 (H +1 )2 H +θ 4 ∆t4 (H +1 )4 − i(1 − θ)θ 4 ∆t5 (H +1 )4 H + O(∆t6 ). Then, multiply with B † from the left and note that the ﬁrst order terms cancel, viz., B † (AA† )−1 B = 1 + (1 − θ)2 ∆t2 (H )2 − θ 2 ∆t2 (H +1 )2 −i(1 − θ)θ 2 ∆t3 H (H +1 )2 + i(1 − θ)θ 2 ∆t3 (H +1 )2 H −(1 − θ)2 θ 2 ∆t4 H (H +1 )2 H (4.31) +θ 4 ∆t4 + O(∆t5 ) This ﬁnal expression shows that the operator is of order O(∆t2 ). However, an examination of the terms of this power reveals that if the Hamiltonian varies slowly enough, then they are of order O(∆t3 ) in the θ = 1/2 case. This is because H +1 = e∆t∂t H , so we get (H +1 )2 = (H )2 + ∆t(H ∂H ∂H + H ) + O(∆t2 ). ∂t ∂t Inserting this into Eqn. (4.31) yields " ∆t2 ! +1 2 (H ) − (H )2 + O(∆t3 ) 4 ∂H ∂H ∆t3 + H + O(∆t4 ). =1− H 4 ∂t ∂t † U∆ U∆ = 1 − (4.32) If we insert Eqn. (4.32) into the norm of Ψ+1 h , we obtain +1 2 Ψ = 1 + O(∆t3 ) Ψh 2 h in the Crank-Nicholson case. In the other cases like forward Euler, we cannot erase the O(∆t2 ) term, and we know that this integration is unstable and explodes after just a few integration steps. Hence, the crucial point in the stability is this term, which we managed to get rid of. 11 For example the hydrogen atom has a spectrum divided into two parts: A discrete bounded spectrum part of energies below zero, and an unbounded continuous spectrum part above zero. 90 4.5 – Time Integration Methods 4.5.4 Stability Analysis of the Leap-Frog Scheme We will now analyze the stability of the leap-frog scheme, i.e., of Ψ+1 = Ψ−1 − 2i∆tH Ψh . h (4.33) Where we have reorganized Eqn. (4.26) to obtain the rule that updates Ψh . Using the truncation error τ = O(∆t2 ) yields Ψ+1 = Ψ+1 + O(∆t3 ). h Hence, the numerical development of the wave function in time has an error of third , viz., order in ∆t. We may use this in Eqn. (4.33) to ﬁnd a good approximation to U∆ Ψ+1 = Ψ−1 − 2i∆tH Ψ + O(∆t3 ) h h −1 = Ψ−1 − 2i∆tH e∆t∂t Ψ−1 + O(∆t4 ) h (4.34) −1 ∆t∂t−1 3 = Ψ−1 − 2i∆tH (Ψ + O(∆t )) + O(∆t4 ) e h h −1 Ψ−1 = 1 − 2i∆tH e∆t∂t + O(∆t4 ) h We expand the exponential operator to second order, making our operator correct to third order. To easily see the result, we operate on an arbitrary function Ψ, viz., ∂Ψ ∆t2 ∂Ψ2 ∆t∂t−1 e Ψ = Ψ + ∆t + + O(∆t3 ) ∂t t−1 2 ∂t2 t−1 ∆t2 ∂ = Ψ − i∆tH −1 Ψ + (−iHΨ) + O(∆t3 ) 2 ∂t t−1 (4.35) ∆t2 ∂H −1 −1 −1 2 3 = Ψ − i∆tH Ψ + Ψ − (H ) Ψ + O(∆t ) −i 2 ∂t ∆t2 ∂H −1 = 1 − i∆tH −1 − i + (H −1 )2 + O(∆t3 ) Ψ. 2 ∂t With the deﬁnition ∂H −1 + (H −1 )2 , ∂t our discretized time evolution operation becomes ! " = 1 − 2i∆tH − 2∆t2 H H −1 + i∆t3 H A Ψ−1 + O(∆t4 ). Ψ+1 h h A=i (4.36) U∆ in terms of Ψ−1 we must calculate the product In order to ﬁnd the norm of Ψ+1 h h † (U∆ ) U∆ , where the operator U∆ is deﬁned in Eqn. (4.36). It is clear that the order )†U∆ is also the order of the unitarity of the leap-frog scheme. of 1 − (U∆ The ﬁrst order terms cancel immediately, viz., † (U∆ ) U∆ = 1 − i∆t3 A† H + 4∆t2 (H )2 + 4i∆t3 H −1 (H )2 − 2∆t2 H H −1 − 4i∆t3 (H )2 H −1 + i∆t3 H A + O(∆t)4 Introducing a commutator relation yields † ) U∆ =1 − 2∆t2 (H −1 H + H H −1 ) + 4∆t2 (H )2 (U∆ + i∆t3 (H A − A† H ) + 4i∆t3 [H −1 , (H )2 ] + O(∆t4 ). 91 Numerical Methods for Partial Diﬀerential Equations The second order terms should “almost” cancel because H −1 is equal to H to ﬁrst order, viz., ∂H + O(∆t2 ). H −1 = H − ∆t ∂t Introducing this relation transforms the second order terms into third order terms, and this ∂H ∂H 2∆t2 (H −1 H + H H −1 ) = 4∆t2 (H )2 − 2∆t3 H + H . ∂t ∂t The other third order term may be simpliﬁed, viz., ∂H −1 ∂H −1 H A − A† H = i H + H + [H , (H −1 )2 ] ∂t ∂t The commutator [H , (H −1 )2 ] is of ﬁrst order in ∆t. To see this we Taylor expand and use H (H −1 )2 = (H )2 + O(∆t), −1 and thereby, [H , (H −1 )2 ] = [H , (H )2 ] + O(∆t) = O(∆t). Thus, we may ignore the commutator altogether as the third order term becomes a fourth order term. By the same reasoning we also ignore the other commutator [H −1 , (H )2 ]. Furthermore, ∂H −1 ∂H = + O(∆t). ∂t ∂t Combining these considerations yields ∂H † 3 ∂H (U∆ ) U∆ =1 + ∆t H + H + O(∆t4 ). ∂t ∂t Hence we have secured unitarity to third order. For time independent problems, we see that we have at least fourth order unitarity. Simpliﬁed Analysis in One Dimension. We will use a variant of the von Neumann stability analysis, see Ref. [34]. Let us assume that the Hamiltonian is independent of time. Assume an initial condition on the form of an eigenfunction for the discrete Hamiltonian, viz., u0j = sin(kπxj ). The eigenvalues for the Hamiltonian of a particle-in-box (in the unit interval) are = FDM k 4 sin2 (kπh/2), h2 FEM = k 12 sin2 (kπh/2) , h2 2 + cos(kπh) and for the ﬁnite diﬀerence method and ﬁnite element method with linear elements, respectively. See section 6.4. The number h is the mesh width. The leap-frog scheme reads u+1 = u−1 − 2i∆tHu . We postulate that the wave function at time t is given by u = ξ sin(kπx), 92 4.5 – Time Integration Methods where ξ is a complex number. It is clear that the scheme is then stable if and only if |ξ| ≤ 1 and unitary if and only if |ξ| ≡ 1. Inserting uj into the time stepping scheme yields ξu0j = ξ −1 u0j − 2i∆tk u0j . As the eigenvectors u0j for the Hamiltonian form a basis for the set of discrete functions this means that ξ 2 + 2i∆tk ξ − 1 = 0. This yields ξ = −i∆tk ± 1 − (∆tk )2 . Assume that the radicand is positive, i.e., that 1 − ∆t2 2k ≥ 0, ∀k. Then we have |ξ|2 = ∆t2 2k + 1 − ∆t2 2k = 1, and the scheme is unitary. This imposes the constraint 1 ∆t ≤ maxk {k } on the time step. On the other hand, assume that there exists a k such that the radicand is negative, i.e., that ξ is purely imaginary, viz., ξ = −i∆tk ± i 1 − ∆t2 2k . Unless ∆t2 2k = 0 there is no way to make |ξ| = 1. But this is impossible since both the square energies 2k and the time step must be positive quantities. Hence, the leap-frog scheme is stable if and only if the time step is restricted by the inverse of the largest eigenvalue, i.e., 1 ∆t ≤ max . For the ﬁnite diﬀerence method we obtain for the particle-in-box ∆t ≤ h2 h2 = . 4 4 sin (N πh/2) 2 For the ﬁnite element method we obtain another estimate, viz., ∆t ≤ h2 2 + cos(N πh) h2 . = 2 12 sin (N πh/2) 12 If we have an additional potential, i.e., that H = Hpib + V (x), and if we assume that V (x) varies slowly in space, the stability criterion should not be severely modiﬁed. If on the other hand V (x) varies rapidly, we must lower ∆t. We see that the ﬁnite element method actually requires a smaller step size than the ﬁnite diﬀerence method. If we use a two-dimensional uniform grid, the largest eigenvalue has twice the magnitude as in the one-dimensional grid. Hence, ∆t must be halved in the two-dimensional case. Actually, when looking at Fig. 6.2 we see that the fact that the ﬁnite element method produces qualitatively better eigenvalues and that the eigenvalues are overestimated leads to a more strict stability criterion. Notice that the number of time steps required is not linear in the number of grid points N = 1/h, but quadratic. For the ﬁnite element method we must solve a linear system at each time step, each requiring perhaps O(N 2 ) operations. This totals O(N 4 ) operations for the complete simulation! For the theta-rule the time step may be chosen more arbitrarily. On the other hand, it is clear that to improve the quality of the numerical solution we must also lower ∆t if we lower h in the theta-rule as well, so it is not clear at this point what method is best. 93 Numerical Methods for Partial Diﬀerential Equations 4.5.5 Properties of the ODE Arising From Space Discretizations We have seen that in general we have represented the wave function Ψ with a ﬁnitedimensional vector which we call y ∈ CN . Accordingly, the spatial part of the PDE is represented by N algebraic equations in N unknowns. Spatially linear PDEs become linear systems, i.e., the spatial operators become N × N matrices. For the time dependent Schrödinger equation, the spatial operator is just the Hamiltonian; an Hermitian operator. When solving the resulting ODE it would be fortunate if this Hermiticity is preserved, i.e., that the discretized operator is still Hermitian. This implies that some qualitative properties such as unitarity also is inherited by the ODE. Let us be more speciﬁc. The time dependent Schrödinger equation (1.5) has been converted into a linear ODE, viz., iẏ = H(t)y, y ∈ CN , (4.37) where N is the total number of degrees of freedom in our discretization. Let y be the Euclidean norm on CN . Diﬀerentiation yields d y2 = ẏ † y + y † ẏ = iy † H † y − iy † Hy = iy † (H † − H)y, dt i.e., the ODE conserves the Euclidean norm of the solution if and only if H † = H. Can we be sure of that all the numerical methods described in this chapter yield Hermitian H? Let us for simplicity consider the case where the PDEs for the diﬀerent spin orientations Ψ(σ) are decoupled. The coupled case yields the same results but with a little bit more notation. Let us also consider only the operators i∂/∂xk and xk , that is the momentum component k and the kth position operator. If the discretizations yield Hermitian matrices Pk and Xk for these, then any Hermitian combination A(i∂/∂xk , xk ) also yields Hermitian discretizations if this is deﬁned by A(Pk , Xk ). The spectral method is the simplest to analyze. For simplicity we conﬁne the discussion to one dimension. Of course x is Hermitian, since it is just multiplication by a diagonal matrix with real elements. The momentum operator i∂/∂x is given by P =i 1 1 † Z iKZ = − Z † KZ, N N where K is the diagonal matrix of real wave numbers. Then P is clearly Hermitian, and so is any power of P . For ﬁnite diﬀerence discretizations we must be a little more careful, since we are free to choose any consistent diﬀerence for i∇ and −∇2 as our approximation. The standard second order diﬀerences are however easily seen to be Hermitian. The matrix for δ2x is skew-symmetric; upon multiplication with i it becomes Hermitian. The matrix for δx δx is real and symmetric and hence Hermitian. The ﬁnite element method involves the stiﬀness matrix K representing −∇2 . This matrix is by deﬁnition Hermitian, see Eqn. (4.14). For the momentum operator i∇, we have by integration by parts the matrix elements ∗ Ni (∇Nj ) = −i (∇Ni )Nj + boundary terms = Pji , Pij := i Ω Ω because the boundary terms vanish due to homogenous Dirichlet boundary conditions. Hence, P is Hermitian. For an operator A that is a function of x we have Aij := Ni A(x )Nj Ω which clearly is Hermitian. 94 4.6 – Basic Stability Analysis 4.5.6 Equivalence With Hamilton’s Equations of Motion We may develop a useful analogy with Hamilton’s equations of motion of the discretized time dependent Schrödinger equation (4.37). This analogy will also hold for quantum systems of ﬁnite dimension such as pure spin systems. The analogy may be very useful when solving the Schrödinger equation, because many good integrators for classical Hamiltonian systems have been discovered. The vector y ∈ CN is complex. If we by q and p denote the real and imaginary parts, respectively, we have d (q + ip) = −iH(t)(q + ip) = −iH(t)q + H(t)p. dt If we assume that the action of the Hamiltonian on a real vector is again a real vector, we have dp dq = H(t)p, and = −H(t)q, dt dt where q and p are vectors in RN for all t. If we deﬁne the function H as H= 1 1 T q H(t)q + pT H(t)p, 2 2 then it is easy to see that q and p satisfy Hamilton’s equations of motion with H as the Hamiltonian, viz., q̇i = ∂H , ∂pi and ṗi = − ∂H , ∂qi i = 1, 2, . . . , N. Solving the Schrödinger equation with y(0) as initial condition is then equivalent to solving Hamilton’s equations of motion with p(0) = Re y(0) and q(0) = Im y(0) as initial conditions. The unitarity of Eqn. (4.37) is equivalent to y2 = q T q + pT p = constant. This is not a general feature of Hamiltonian mechanics. But Hamilton’s equations are known to be symplectic, that is area preserving. If we take a ball B ⊂ R2N of initial conditions, then the sum of the areas of the projections of B onto the qi pi planes is conserved, i.e., N A= Area Pi (B) = constant, i=1 where Pi is the projection onto the qi pi -plane. Furthermore, the volume of B is conserved. This is referred to as Liouville’s theorem, viz., V = . . . dq1 dp1 . . . dqN dpN = constant. B This property deﬁnes a conservative ODE. There are many useful theorems and interesting views on classical Hamiltonian mechanics. See Refs. [6, 23, 38] for a full treatment. 4.6 Basic Stability Analysis This short section is an introduction to the methods and concepts used when analyzing the numerical properties of time dependent and stationary ﬁnite element discretizations. The mathematical theory of ﬁnite element methods introduces advanced 95 Numerical Methods for Partial Diﬀerential Equations function spaces such as Sobolev spaces and also concepts such as weak derivatives. This is beyond the scope of this thesis, but we nevertheless aim at giving a sketch of how the analysis is performed. We focus on the basic ideas, leaving out many details and calculations. 4.6.1 Stationary Problems A PDE is an equation in which the unknown is a function. Mathematical analysis deﬁnes diﬀerent function spaces. These are linear spaces of functions, and the Hilbert spaces such as L2 of quantum mechanics are examples of such.12 The spaces Lp is deﬁned as the p-integrable functions (where p > 0 is an integer), i.e., the set of functions u : Ω ⊂ Rd → R for which the norm is ﬁnite, viz., 1/p u := |u| < ∞. p Ω The integral in question is the Lebesgue integral. These spaces are Banach spaces, i.e., complete normed spaces. The space L2 is a special case, and this space is also a Hilbert space with respect to the inner product (u, v)L2 := uv, Ω and hence the norm is u := (u, u). The Sobolev spaces Wpk are generalizations of Lp in which we assume that the partial derivatives of u up to order k are elements of Lp . The norm is deﬁned by uWpk := Dα uLp , |α|≤k where we have used the multi-index notation from section 4.2.1. See also See Refs. [16, 17] for details on Sobolev spaces and other Hilbert spaces, that are the function spaces most common in ﬁnite element analysis.13 The Sobolev spaces W2k =: H k also become Hilbert spaces with the inner product (Dα u, Dα v)L2 . (u, v)H k := |α|≤k The most common class of Hilbert spaces used in ﬁnite element analysis are the Sobolev spaces H k , of which L2 = H 0 is an example. Whatever space we are working with, let us call it V . A stationary PDE is then an equation on the form Au = f, u ∈ V, v ∈ W. Here, A is assumed to be a linear operator from V to W . The space W may be larger than V . Introducing a discretization, i.e., a subspace Vh ⊂ V , gives a discrete formulation of the problem, viz., Ah uh = f, uh ∈ Vh , where Ah is a linear operator on Vh into W . The subscript h indicates a discretization of some kind with parameter h. For ﬁnite diﬀerence methods it may be is the largest grid spacing and for ﬁnite element methods it may be the diameter of the largest 12 To comply with the notation of Sobolev spaces we will use L2 instead of L2 in this section. derivatives appearing in the deﬁnition of Sobolev spaces are actually so-called weak derivatives, i.e., derivatives of L2 -functions that do not necessarily have a classical derivative. The weak derivative coincides with the classical if the latter exists. 13 The 96 4.6 – Basic Stability Analysis element. If h → 0 then Vh → V (in some sense) and we expect that uh → u. (Unless otherwise stated, u → v means u − v → 0.) We assume for simplicity that Ah is deﬁned on all of V and that it has an inverse. Let us estimate the error of the discrete solution uh : −1 −1 u − uh V = A−1 h Ah (u − uh )V = Ah (Ah u − Au)V ≤ Ah L(W,V ) · Ah u − AuW . Hence, for the discrete solution uh to approach u it is enough to require a uniform −1 boundedness on A−1 h , i.e., that it exists a constant C > 0 such that Ah ≤ C for all h. This property is called stability. Furthermore, we must require that (Ah − A)u = Ah u − f → 0 as h → 0. This property is consistency. It measures the error in the equation, as we easily see. This formulation is the one usually used in the analysis of ﬁnite diﬀerence methods, and we easily recognize stability and consistency as they were introduced earlier. Both properties are easy to deduce in this case. For ﬁnite element methods, however, the usual approach is diﬀerent, as the truncation error Ah u − f is not easy to estimate. In ﬁnite element methods, our discrete problem is a discretization of a weak formulation of the PDE. For simplicity, we will conﬁne the discussion to real PDEs. The above formulation of the PDE was: Find u such that Au = f . This pointwise fulﬁllment (u is then called a classical solution) of the PDE is replaced with a fulﬁllment on average, i.e., ﬁnd u such that a(u, v) = (f, v), ∀v ∈ H. Here, a(·, ·) : H × H → R is a bilinear form on H, i.e., linear in both arguments. We have replaced V with another space H ⊃ V , as the weak formulation has lesser constraints on the solution than the classical, pointwise PDE. To see this, consider the PDE −∇2 u = f, i.e., Laplace’ equation. Note that we must require u to be at least twice continuously diﬀerentiable if f is continuous. Multiplying with a test function v and integrating yields ∂u 2 = v∇ u = ∇v · ∇u − v f v. − Ω Ω ∂Ω ∂n Ω Assuming that the boundary terms vanish as in the case of homogenous Neumann boundary conditions, we may identify the bilinear form ∇u · ∇v. a(u, v) = Ω Notice that we only need u and v to be diﬀerentiable. Hence, a solution of a(u, v) = (f, v) is weaker than the classical solution. Before integrating by parts, we had to require u ∈ H22 . Integrating by parts then reduced the constraints, i.e., u ∈ H21 . We must assume some fundamental properties of bilinear forms to carry out our discussion. First, we need14 a(·, ·) to be symmetric, i.e., a(u, v) = a(v, u). Second, we need boundedness, i.e., there exists a constant C1 ≥ 0 such that a(u, v) ≤ C1 uv, ∀u, v ∈ H. Boundedness of the bilinear form is easily seen to be equivalent with continuity. Third, we need coercivity, i.e., there exists a constant C2 > 0 such that a(u, u) ≥ C2 u2 , 14 Not ∀v ∈ H. really, but it makes life much simpler. 97 Numerical Methods for Partial Diﬀerential Equations The bilinear form in the example is easily seen to be symmetric, bounded and coercive. If a(·, ·) is symmetric and coercive the weak PDE is equivalent to the minimization of the functional 1 J(u) = a(u, u) − (f, u), 2 i.e., to ﬁnd u such that J(u) ≤ J(v) for all v ∈ H; see Ref. [34]. In our example, 1 J(u) = |∇u|2 − f u. 2 Ω Ω For this reason, the weak formulation is also called the variational formulation. When formulating a discrete problem one chooses Vh ⊂ H as for the classical PDE and seek uh such that a(uh , v) = (f, v), ∀v ∈ Vh . This is exactly Galerkin’s method in the weighted residual method. Let us analyze the error u − uh of the weak formulation. Note that by coercivity, we have C2 uh 2 ≤ a(uh , uh ) = (f, uh ) ≤ f ∗ uh , where f ∗ is the dual norm of f , i.e., the norm of f as a linear functional on H. Hence, 1 f ∗ , uh ≤ C2 and uh is bounded automatically by f . This is the stability criterion, and we see that in the weak formulation we get this for free, so to speak, as long as coercivity of a(·, ·) holds. Let us turn to consistency, i.e., if the discrete equation converges to the exact equation as h → 0. Let δh (u) be the distance from u to Vh , i.e., δh (u) := inf u − v. v∈Vh This really deﬁnes that Vh → V . If u is the solution to the exact problem, then δh (u) must vanish as h → 0 in order to make the discrete formulation consistent. We see that consistence depends on the choice of Vh . In ﬁnite element methods, it turns out that the error u − uh is proportional to δh (u). Hence, both stability and consistence is fulﬁlled for every ﬁnite element method. See for example Refs. [16, 17]. Finally we show the best approximation in norm property of Galerkin’s method. First note that for any v ∈ Vh , a(u − uh , v) = a(u, v) − a(uh , v) = (f, v) − (f, v) = 0. Hence, for any v ∈ Vh , C2 u − uh 2 ≤ a(u − uh , u − uh ) = a(u − uh , u − v) ≤ C1 u − uh · u − v. Hence, u − uh ≤ C1 u − v, C2 ∀v ∈ Vh . (4.38) We see that the discrete formulation actually ﬁnds uh ∈ Vh that is closest to u. If δh (u) → 0 as h → 0 then clearly uh → u. See also Ref. [34]. Let us introduce a theoretical tool which will become useful when we study the stability of time dependent problems. We deﬁne an operator Rh ∈ L(V, Vh ). This 98 4.6 – Basic Stability Analysis operator projects a function u onto the discrete space Vh with respect to the energy inner product a(·, ·), i.e., a(Rh u, v) = a(u, v)m, ∀v ∈ Vh . Eqn. (4.38) then becomes (1 − Rh )uH 1 ≤ C1 inf u − vH 1 . C2 v∈Vh 4.6.2 Time Dependent Problems Although we argued in the introduction to this chapter that discretization in time is independent from discretization in space this is not entirely true as far as the numerical convergence concerned. For example, when h → 0 in space, the resulting ODE becomes larger and larger. Hence, the convergence in time is aﬀected by the mesh width in space. We need some kind of uniformity to circumvent this. Let us perform an analysis similar to the one above for a time dependent problem. Will consider a PDE on the form ut + Au = f, u ∈ H 2, where we seek u : [0, T ] −→ H 1 , where T ≤ ∞. We are given an initial condition u(0) = g ∈ H 1 . A mathematical question we will not investigate here is whether u(t) for a given t exists. We will assume that it does. The weak formulation reads ∀v ∈ H 1 (ut , v) + a(u, v) = (f, v), (4.39) where we assume a(·, ·) to be a symmetric, bounded and coercive bilinear form on H 1 . In particular, the weak formulation must hold for v = u, i.e., (ut , u) + a(u, u) = (f, u). We have (ut , u) = ut u = Ω Ω 1 ∂ 2 1 d (u ) = 2 ∂t 2 dt u2 = Ω 1 d u2L2 , 2 dt where we assume that integration and diﬀerentiation with respect to time is interchangeable. This gives 1 d u2L2 ≤ (f, v), (4.40) 2 dt as a(u, u) ≥ 0 by coercivity. We note that by the triangle inequality, |(f, v)| ≤ f L2 uL2 ≤ 1 f 2L2 + u2L2 . 2 Integrating Eqn. (4.40) from t = 0 to t = T and using u(0) = g gives 1 1 1 u(T )2L2 ≤ gL2 + 2 2 2 T f L2 0 1 dt + 2 T u(t)L2 dt. 0 This is a stability criterion. To see this, consider Grönwall’s inequality, see Refs. [larsson2003,mcowen2003]. If x(t) ≥ 0 obeys x(T ) ≤ h(T ) + b T x(t) dt, 0 99 Numerical Methods for Partial Diﬀerential Equations with h(t) ≥ 0 and 0 ≤ T ≤ t, then T x(T ) ≤ eT x(0) + eT −t f (t) dt. 0 Turning to a discrete version of the weak formulation, we see that like in the stationary case we need an approximation property similar to Eqn. (4.38). Consistency is built into the formulation in the case of ﬁnite element methods. The discrete version of Eqn. (4.39) reads ((uh )t , v) + a(uh , v) = (f, v), ∀v ∈ Vh , with uh (0) = gh as initial condition. From this we immediately obtain uh (t)L2 ≤ gL2 , i.e., stability is built into the formulation. We wish to say something about the time development of the error u − uh . We introduce an operator Rh : H 1 → H 1 that takes an element u ∈ H 1 to its projection on Vh using a(·, ·) as inner product.15 I.e., Rh u is deﬁned by a(Rh u, v) = a(u, v), ∀v ∈ Vh . Hence, Rh u ∈ Vh . This operator is only a theoretical tool; we will never actually compute it (or it’s action on u). The time development of the error is given by ((u − uh )t , v) + a(u − uh , v) = 0, ∀v ∈ Vh . Writing u − uh = u − Rh u + Rh u − uh where Rh u − uh ∈ Vh yields ((Rh u − uh )t , v) + a(Rh u − uh , v) = ((Rh u − u)t , v), ∀v ∈ Vh , where the right hand side may be interpreted as a source term. We have used that by deﬁnition a(u − Rh u, v) = 0 for all v ∈ Vh . We see that the time development of u − uh is rewritten in terms of functions in Vh . Letting v = Rh u − uh yields 1 d Rh u − uh 2L2 ≤ (Rh ut − ut , Rh u − uh ), 2 dt in the same way as when we analyzed stability of u ∈ H 1 . Hence, if we deﬁne e = Rh u − uh ∈ Vh we have T T Rh ut − ut 2L2 dt + e(t)2L2 dt, e(T )2L2 ≤ 0 0 where we have used uh (0) = Rh u(0) by deﬁnition. By Grönwall’s inequality we have T e(T )2L2 ≤ eT (1 − Rh )ut 2L2 dt + e(0)2L2 0 ≤ Te T max (1 − Rh )ut 2L2 + e(0)2L2 , 0≤t≤T where e(0) is the initial error. The initial error is not zero because the discretized initial condition is not necessarily equal to the continuous initial condition. On the other hand, we see that if δh becomes smaller then so must e(0)L2 . In that case the stability criterion becomes stronger. The function (1 − Rh )w is the orthogonal complement to Rh w whose norm also must become smaller when δh decreases. We see that decreasing the mesh width improves the stability criterion. 15 An 100 inner product is a symmetric, positive deﬁnite bilinear form. Chapter 5 Numerical Methods for Linear Algebra This chapter is a rather brief overview of the various methods that are in use for solving (square) systems of linear equations and (standard and generalized) eigenvalue problems. Linear algebra is perhaps the most fundamental tool in numerical computation, and hence it is important to master the powerful tools that have been developed. The theoretical aspects of linear algebra may be found in Ref. [39]. The numerical methods are described well in Ref. [40] and Ref. [41]. In this section we are primarily concerned with square matrices. We shall consider two kinds of equations; square systems of linear equations and eigenvalue problems. Let A ∈ CN ×N be any square matrix of dimension n and let b ∈ CN be any n-dimensional vector. We then seek x ∈ CN such that Ax = b. (5.1) This equation has a unique solution if and only if A has an inverse A−1 and if and only if determinant does not vanish, i.e., det A = 0. The matrix is then called non-singular. If det A = 0, i.e., if A is singular, the set of solutions to the homogenous equation Ax = 0 is a subspace of CN . A solution to Eqn. (5.1) may happen not to exist in the non-singular case, but if it does then it is in general given by x + xp , where xp is any particular (i.e., an arbitrary) solution of Eqn. (5.1). We will assume that A is non-singular in the following discussion. The need for solving Eqn. (5.1) arise in a wide range of problems in computational physics. For our part, we need to solve large systems when using the ﬁnite element method and also when we are given implicit time integration schemes. In addition, solving eigenvalue problems also implies solving linear systems, as will be described later. In fact, most of the time spent in solving PDEs is invested in linear algebra problems, and so it is of major importance to choose fast and reliable methods. The generalized eigenvalue problem reads: Given a pair of square matrices A and B, ﬁnd the eigenpairs (λ, x), where λ is a scalar and x is a vector such that Ax = λBx.. (5.2) The scalar λ is called an eigenvalue and x is called an eigenvector. In the case B = I, i.e., the identity matrix, we obtain the standard eigenvalue problem, viz., Ax = λx. We shall only concern ourselves with the Hermitian eigenvalue problems, i.e., both A and B are Hermitian matrices. In this case one may always ﬁnd N eigenpairs with real eigenvalues and orthonormal eigenvectors, i.e., a basis for CN . 101 Numerical Methods for Linear Algebra 5.1 Introduction to Diﬀpack Before we discuss the numerical methods we give a brief introduction to Diﬀpack, the C++ library used in the ﬁnite element implementations in this thesis. It is natural to introduce Diﬀpack at this point, after having introduced the ﬁnite element discretizations. A basic knowledge of Diﬀpack is also useful to get an overview of how the numerical methods are used in the program. We cannot even scratch the surface of Diﬀpack in this thesis. We will however mention some of the key points in the structure, class hierarchy and implementations of ﬁnite element solvers in order to understand pros and cons, limitations and possibilities. Diﬀpack is a very complex library. In addition to deﬁning hundreds of classes and functions it also includes many of auxiliary utilities, performing a wide range of tasks such as generation of geometrically complicated grids, visualization of data and conversion of data to and from various common formats. The programming library and the utilities together constitute a complete environment for solving PDE related problems; including planning and preparation, implementation, simulation and data organizing, visualization and presentation of results. The Diﬀpack project was started in the mid 90’s by Sintef and the University of Oslo. As the project grew in size, complexity and reputation the ﬁrm Numerical Objects was founded in 1997. Diﬀpack is now owned by the German company inuTech and jointly developed with Simula Research Laboratory. See Refs. [42–44] for details. Diﬀpack is a library developed with extensive use of object-oriented techniques, as is a must when developing ﬂexible ﬁnite element solvers. Many classes are templated1 and a hierarchy of smart pointers (so-called handles) are implemented, easing the memory handling which otherwise have a tendency to require many debugging session in C++. There are class hierarchies for matrices, grids and grid generators, ﬁnite element abstractions, solvers for systems of linear equations and so on. Ref. [34] is an excellent introduction to both the ﬁnite element method and Diﬀpack programming. It is an easy-to-read and informative account and a good introduction and reference for practitioners at all levels. In addition, the online Diﬀpack documentation in Ref. [45] is recommended. 5.1.1 Finite Elements in Diﬀpack Diﬀpack implements many diﬀerent ﬁnite element types, including (but not limited to) linear and quadratic elements in one dimension, and tensor product generalizations to more dimensions. The subclasses of ElmTensorProd exemplify this. For example, the element class ElmB2n1D implements a linear element in one dimension, i.e., one-dimensional elements (intervals) with two nodes located at the endpoints, and the class ElmB4n2D is the two-dimensional generalization, by taking the tensor product of two one-dimensional elements. Hence, it is deﬁned on quadliterals and has four nodes located at the corners. The (piecewise bilinear) element functions are depicted in Fig. 4.6. The class ElmB3n2D is a one-dimensional quadratic element with three nodes in each element, two located at the endpoints of the interval and the third somewhere in-between. The two-dimensional generalization is ElmB9n2D with 9 nodes. In three dimensions we have the class ElmB27n3D with 27 nodes distributed in a parallelepiped. Besides methods that evaluate the basis functions and their partial derivatives, the element classes contain Gaussian integration rules that may be used when assembling the element matrices and vectors. This is done in the integrands() method of FEM, the class from which our ﬁnite element simulators are derived. This method evaluates the integrands in the one-dimensional ﬁnite element system example in Eqns. (4.12) and 1 Actually, templating is emulated with preprocessor macros and directives due to compilerdependent behavior the template feature in C++. 102 5.1 – Introduction to Diﬀpack (4.13). Assembling the element matrices and vectors is done when calling FEM::makeSystem(). Numerical integration takes part and FEM::integrands() evaluates the integrands at the points deﬁned by the current integration rules. Hence, numerical integration is a very important part of ﬁnite element implementations with Diﬀpack. A detailed description of Diﬀpack is out of the scope for this text. The code for the programs written in this thesis use many of Diﬀpack’s features than described here, but the code is well commented in order to compensate for this. 5.1.2 Grid Generation Diﬀpack is bundled with lots of utilities and scripts that typically call third-party grid generation software (called preprocessors) in order to produce compatible grids. Examples of such third-party software are Triangle and GeomPack. Diﬀpack comes with classes with interfaces to such utilities in addition to scripts and utilities. With these, both simple and complicated grids may easily be created. Ref. [46] is a comprehensive introduction to ﬁnite element grid preprocessors used in Diﬀpack. In this thesis, we employ square grids and disk-grids (i.e., approximations to circular regions). A typical way to generate a grid is by the external utility (written with Diﬀpack) makegrid. This is a command line based interface to the various grid preprocessors. Here is a sequence of commands that generate a unit box in two dimensions with 9 × 9 linear elements: set set ok set set ok preprocessor method = PreproStdGeom output gridfile name = box geometry = d=2 [0,1]x[0,1] partition = d=2 e=Elmb4n2D div=[10,10] grading=[1,1] For making a disk grid, one may create a ﬁle describing the outline of the circular region in terms of a polygon and then use Triangle to triangulate it. Here is a short snippet of Python code to create such a grid: import os from math import * # Generate a disk with radius r and mesh width h. # Uses the "Triangle" program. def makeDiskTri(r, h): div = floor(r/h + 1) # number of segments in outline polygon print "generating disk with div=%d, h=%f" % (div, h) nel = floor((div-1)*(div-1)/(2*pi)) # desired number of elms el_area = pi*r*r/nel # average element area # create the input file to Triangle. poly_file_text = "%d 2 0 1\n" % (div) for i in range(div): angle = i*2*pi/float(div) (x, y) = (cos(angle)*r, sin(angle)*r) poly_file_text = poly_file_text + "%d %g %g 20010\n" % (i+1, x, y) 103 Numerical Methods for Linear Algebra poly_file_text = poly_file_text + "%d 1\n" % (div) for i in range(div): if (i==div-1): i2 = 1 else: i2 = i+2 poly_file_text = poly_file_text + "%d %d %d 20010\n" % (i+1, i+1, i2) poly_file_text = poly_file_text + "0\n" f= open(".disk.poly", "w") f.write(poly_file_text) f.close() # Call Triangle. No angles should be less than 28 deg, # all triangle areas should be less than el_area # (yielding at least nel elements). os.system("triangle -q28Ipa%f .disk.poly" % (el_area)) os.system("triangle2dp .disk") fname = "disk_%d_%f" % (div, h) os.system("mv .disk.grid %s.grid" % (fname)) A grid generated with this script can be seen in Fig. 5.1. 5.1.3 Linear Algebra in Diﬀpack It is important to be aware of the diﬀerent kinds of matrices that Diﬀpack oﬀers with their advantages and disadvantages. Here we give a brief overview of the matrix types and their corresponding classes. Each matrix type may be complex or real. As the matrix classes in Diﬀpack are templated this is indicated as a template parameter, e.g., Matrix(Complex) or Matrix(real). In general we write NUMT for the type. All matrix classes are derived from Matrix(NUMT). See the online documentation in Ref. [45] for details. The important diﬀerences between the matrix types lies in that we assume diﬀerent structures of the matrices. This allows for optimizations when calculating for example matrix-vector products. There is no need to store a full 1000 by 1000 matrix if we know that only the diagonal is diﬀerent from zero, and performing a matrix-vector product with nearly a million multiplications with zero is a waste of resources. As a matter of fact, the matrix-vector product is very important to optimize for this thesis, as it is the most fundamental operation when both solving linear systems of equations iteratively and when solving eigenvalue problems. An important property of the ﬁnite element matrices is sparsity. The element matrices are sparse, meaning that most of the elements are known to be zero. Using a grid with 108 nodes yields a linear system of the same dimension, and storing a full matrix clearly cannot (and should not!) be done. The matrix elements Aij where x [i] and x [j] are indices not belonging to the same element are known to be zero. Hence, the number of non-zeroes in the element matrix is of order N , yielding very sparse matrices as the fraction of non-zeroes is actually of order 1/N . We must mention, 104 5.1 – Introduction to Diﬀpack however, that this is a somewhat simpliﬁed picture. Higher order elements yields more couplings between the nodes (as more nodes belong to the same element) and hence increase the number of non-zeroes. Fig. 5.1 shows a typical moderately-sized ﬁnite element grid with linear triangular elements and the corresponding sparsity pattern. The ﬁgure shows a square picture where each point corresponds to a matrix element. Black dots are non-zeroes and white dots are zeroes. Dense Matrices. Dense N × M matrices are the most general matrices. Every entry is assumed to be a priori a possible non-zero. Hence, a full array of N · M (complex or real) numbers must be allocated in an implementation. The ith component of the matrix-vector product w = Av is given by wi = n Aij vj , j=1 and as w has M components, this amounts to N M multiplications and additions, and hence O(N M ) ﬂoating point operations are required for a matrix-vector product in the real case. For complex matrices, the multiplications and additions are of course complex, roughly quadrupling the number of multiplications. Dense matrices are implemented in the class Mat(NUMT). Diagonal Matrices. On the other extreme of dense matrices we ﬁnd the diagonal matrices, in which only the diagonal elements are non-zeroes. The lumped mass matrix is an example of a diagonal matrix. For a square N -dimensional matrix exactly N (real or complex) numbers must be stored. The matrix vector product reduces to wi = Aii vi , and hence only O(N ) operations are needed for a matrix-vector product. Again, complex matrices require about four times more work than the real matrices. Diagonal matrices are supported through the MatDiag(NUMT) class. Tridiagonal Matrices and Banded Matrices. Tridiagonal matrices are allowed to have non-zeroes at the ﬁrst super- and subdiagonal as well as the main diagonal, i.e., only Ai,j , j = i − 1, i, i + 1 are allowed to be nonzero. A square N -dimensional tridiagonal matrix is stored as a 3 × N dense matrix. It is easily seen that a matrix-vector product requires O(3N ) operations. (Again, complex matrices require a bit more.) Tridiagonal matrices are special cases of banded matrices. A banded matrix allows non-zeroes at the k ﬁrst sub- and superdiagonals. The total number 2k + 1 of non-zero diagonals is called the bandwidth. It is stored as a (2k + 1) × N dense matrix and clearly O((2k + 1)N ) operations are needed for a matrix-vector product. General Sparse Matrices. A matrix is called sparse if in some sense the number of (possible) non-zeroes is low. A typical example is diagonal and tridiagonal matrices, in which the number of non-zero elements is of order N . A matrix-vector product then requires only O(N ) operations as is easily seen. Fig. 5.1 shows the banded structure of an element matrix and the fact that many of the zeroes inside the band is zero, resulting in very many unnecessarily stored elements if we use MatBand(NUMT). Indeed, the fraction of non-zeroes in the band decreases rapidly for larger grids (i.e., larger element matrices), and hence we need support for matrices with an even more general structure. In general sparse matrices we store only the non-zero elements. The class MatSparse(NUMT) implements the so-called compressed sparse row storage scheme (CSR 105 Numerical Methods for Linear Algebra (1,1) (N,N) Figure 5.1: Sparsity pattern matrix for a ﬁnite element grid (inset). Non-zeroes are shown as black dots. The matrix dimension is 1148. for short). It is the most compact way of storing the matrix and allows very fast computations of matrix-vector products. The N × M sparse matrix is stored by means of three one-dimensional vectors a ∈ Cn , c ∈ Nn and r ∈ NN +1 , where n is the total number of non-zero entries and N is the number of rows. The elements of a are the non-zero matrix entries ordered row-wise from left to right, i.e., in reading order. The entries of c are the column indices for these, i.e., a(s) resides in column c(s). Element number i of r stores the index s of the ﬁrst entry of row i. In addition, r(N + 1) = M + 1 by deﬁnition. Hence, the number of non-zero entries in row i is r(i + 1) − r(i). For example, consider the matrix 1 0 2 0 −3 4 A= 5 0 0 . 0 0 6 Here, n = 6 and a = (1, 2, −3, 4, 5, 6) are the entries. The column-indices are c = (1, 3, 2, 3, 1, 3) and the row pointers are r = (1, 3, 5, 6, 7). 106 5.2 – Review of Methods for Linear Systems of Equations Solvers for Linear Systems of Equations. As we have seen, large sparse matrices arise from ﬁnite element discretizations and ﬁnite diﬀerence approximations. Hence, solving large systems of linear equations with a sparse coeﬃcient matrix is fundamental. As we shall see, the corresponding eigenvalue problems, i.e., ﬁnding the eigenvalues and eigenvectors of the discrete Hamiltonian, also requires solutions to linear systems in order to be solved. For linear systems Diﬀpack implements many solvers, from Gaussian elimination to Krylov subspace methods with preconditioning. In the next section we will review the most important methods that are implemented. Again, we refer to Ref. [34] for details. Here we merely mention that a separate hierarchy of classes are implemented in Diﬀpack (i.e., LinEqSolver and its descendants) that allows the user to solve linear systems with all available methods. Eigenvalue Problems in Diﬀpack. Unfortunately there is no direct support for solving eigenvalue problems in Diﬀpack as of yet. To achieve this one must either implement the proper methods or in some way connect Diﬀpack matrix classes with external solvers. As eigenvalue computation is fairly complicated business we would prefer to use ready-made libraries. In an external report written for Simula Research Laboratory (see Refs. [5, 44]) I have investigated the possibility of using the popular ARPACK eigenvalue package with matrices from Diﬀpack. A simple implementation was made and the eigenvalue solvers in this thesis are an extension of the implementation found in this work. 5.2 Review of Methods for Linear Systems of Equations In this section we consider square linear systems of equations, i.e., ﬁnd x such that Ax = b, where A is an N × N matrix and x and b are N -dimensional (complex) vectors. For large systems the time spent on solving the system might be much higher than constructing the system, for example via assembly of the element matrices. It is then vital to have access to a variety of numerical methods for solving linear systems. The optimal choice for solving the linear systems is highly problem dependent, and one should not rely solely on for example Gaussian elimination. It is then important that the programming environment makes is easy to switch between the methods. There are three basic properties that we need to consider for each method: For what matrices they apply, the computational cost (both number of operations and storage requirements) of the method and how the method actually works. Se Refs. [34, 40] for details on the methods presented here. 5.2.1 Gaussian Elimination and Its Special Cases Strictly speaking, Gaussian elimination is only one of the methods that usually is referred to by that name. The pure Gaussian elimination is rarely implemented; instead one uses the LU decomposition method, for example. LU Decomposition. The LU decomposition method assumes that we can rewrite our matrix A as A = LU, where L is lower triangular and U is upper triangular. Solving a lower or upper triangular system is trivial and is done by forward substitution and backsubstitution, 107 Numerical Methods for Linear Algebra respectively. Let us write LU x = L(U x) = b. The equation Ly = b is easily solved by forward substitution, i.e., y1 = b1 , α11 i−1 1 yi = bi − αij yj , αii j=1 and i = 2, 3, . . . , N. Here, αij are the lower triangular elements of L. Then we solve the equation U x = y with forward substitution, i.e., yN xN = , βN N and xi = N 1 yi − βij xj , βii j=i+1 i = N − 1, N − 2, . . . , 1. Here, βij are the upper triangular elements of L. As we see, this produces the solution of Ax = b in O(N 2 ) operations, given L and U . The LU decomposition is usually performed with an algorithm called Crout’s algorithm. (The algorithm also shows that is always is possible to do LU decomposition, by construction.) The algorithm needs O(N 3 ) operations for a dense matrix. Clearly, with N ∼ 107 the LU decomposition is not feasible in general, but for some special cases the amount of work needed reduces drastically. Notice that once the LU decomposition is performed, any right hand side b can be solved with forward and backsubstitution. If we need to solve several right hand sides this should be taken advantage of. In Diﬀpack, Gaussian elimination with LU decomposition is implemented in the GaussElim class, a subclass of LinEqSolver. Gaussian elimination with LU decomposition is mathematically fool-proof. If A is non-singular the solution x is given by the above formulas. Numerically we however cannot avoid round-oﬀ errors. One way to minimize such errors is to perform a process called pivoting in the LU decomposition. In fact, Gaussian elimination methods are numerically unstable if pivoting is not carried out. Round-oﬀ errors are still a problem if the matrix A is ill-conditioned, meaning that its determinant is almost zero. The condition number κ is deﬁned as the square root of the ratio of the magnitudes of the largest and the smallest eigenvalue, i.e., / |λmax | . κ= |λmin | If the condition number is inﬁnite the determinant is zero as one of the eigenvalues is zero. Hence, the matrix is singular. If κ−1 approaches the machine precision, i.e., the smallest representable number in the computer, it is singular as far as the numerical methods concerned. Hence we can get unpredictable results when using such ill-conditioned coeﬃcient matrices. Tridiagonal Systems and Banded Systems. The situation is greatly simpliﬁed if A is a tridiagonal matrix. Recall that tridiagonal matrices have non-zero elements only along the diagonal and the ﬁrst sub- and superdiagonal. Hence, it has approximately 3N nonzero elements. For such systems LU decomposition and forward and backsubstitution takes only O(N ) operations, which is optimal. The algorithm can be coded in 108 5.2 – Review of Methods for Linear Systems of Equations merely a handful of lines, see for example Ref. [40]. The storage requirements are only a vector of length N for temporary storage, contrary to an array of length N 2 needed for the full LU decomposition. Banded systems are more general, in that the matrix A has non-zero elements along k sub- and superdiagonals. Banded matrices are frequently output from element assembly routines, although general sparse matrices perform better. (One complication is however that Gaussian elimination is not implemented for complex sparse matrices, see the paragraph below on sparse systems.) The computational cost of banded Gaussian elimination depends on the bandwidth, but if n N it is much faster than full Gaussian elimination. It is noteworthy that the tridiagonal Gaussian elimination need almost no extra storage space. Nor does Crout’s algorithm for dense matrices need extra space if we can live with the original dense matrix being destroyed. Banded Gaussian elimination however creates additional non-zeroes called ﬁll-ins at locations outside the band in A. If the bandwidth is comparable to N the number of ﬁll-ins are considerable. Tridiagonal and banded Gauss elimination is also implemented in GaussElim. Sparse Systems. Sparse matrices contain mostly zeroes. Often the number of nonzeroes is of order N . The structure of a sparse matrix may vary greatly, see for example Fig. 2.7.1 of Ref. [40]. The structure may for example lie somewhere between a dense matrix and a banded matrix. The tridiagonal LU decomposition with forward and backsubstitution can be performed in only O(N ) operations. This is due to clever application of the algorithm and bookkeeping of the zeroes in the system. The structure of the sparse matrix plays a fundamental role here, and for general sparse matrices one must in some way analyze the sparsity pattern in order to implement an eﬀective algorithm. Keeping the number of ﬁll-ins reasonable and at the same time minimizing the number of operations needed is no simple task. [Should have more on Diﬀpack’s approach to sparse systems here. Also mention that no Gaussian elimination is implemented for complex sparse systems.] 5.2.2 Classical Iterative Methods Gaussian elimination is a direct method, i.e., an algorithm for ﬁnding x with an exactly known complexity. We can precisely count the number of operations needed for an exact solution to be found. Iterative methods take another approach. As the name suggests one creates a sequence of trial vectors xk that hopefully converges to the real solution x. Each iteration should be in some sense cheap in order make this meaningful. Classical iterative methods split A into A = M − N and hence M x = N x + b, suggesting an iterative process given by M xk+1 = N xk + b. This process is the same at each iteration, which is characteristic for classical methods. We see that the system M x = b must be in some sense cheap to solve if the iteration process should have any practical use. If we deﬁne the residual r k as r k := b − Axk , (note the similarity to the ﬁnite element formulation) we obtain xk+1 = xk + M −1 r k . 109 Numerical Methods for Linear Algebra Deﬁning the matrix G = M −1 N and the vector c = M −1 b we obtain xk+1 = Gxk + c, (5.3) and this makes us able to analyze the stability and convergence properties of the iteration. Through the following chain of identities, xk+1 − x = Gxk + c − x = Gxk + M −1 (b − M x) = Gxk + M −1 (b − Ax − N x) = Gxk − Gx = G(xk − x), we obtain xk − x = Gk (x0 − x). In other words the error ek = x − xk converges to zero if and only if lim Gk = 0. k→∞ If we can diagonalize G, then G = (maxλ∈σ(G) |λ|) =: (G), where σ(G) is the set of eigenvalues and is called the spectrum of G. The quantity (G) is called the spectral radius. Hence, lim ek = e0 lim Gk = e0 lim (G)k , k→∞ k→∞ k→∞ and the iteration process converges if and only if the spectral radius (G) < 1. The asymptotic rate of convergence is deﬁned as R∞ (G) := − ln (G), and to reduce the error with a factor one needs k = − ln /R∞ (G) iterations. Unfortunately, the convergence rate of the diﬀerent iteration methods are highly problem dependent. It is not obvious what method is the best. In Ref. [34] an analysis of a few cases and model problems are quoted and the iterative methods are interpreted in terms of a model problem. The text is highly recommended for further reading. In the following, notice that when A is a sparse matrix the iterations become exceedingly simple to perform as each iteration only costs O(n) operations, where n is the number of non-zeroes in the matrix. The classical iterative solvers are implemented in subclasses of BasicItSolver (which again is a subclass of IterativeSolver derived from LinEqSolver). Simple Richardson Iteration and Jacobi Iteration. For the Richardson iteration (or just simple iteration) one chooses M = I. The system M x = b then becomes trivial. This yields the iteration xk+1 = xk + r k = xk − Axk + b. For the Jacobi iteration we choose M = D, where D is the diagonal of A, i.e., Dii = Aii and Dij = 0 for i = j. Again, M x = b is very cheap. This yields for the iteration xk+1 = D−1 ((D − A)xk + b) = xk − D−1 Axk + D−1 b. Writing out the equation for xk+1 yields i xk+1 i 110 N 1 bi − = xki + Aij xkj , Aii j=i+1 i = 1, 2, . . . , N. 5.2 – Review of Methods for Linear Systems of Equations Relaxed Jacobi Iteration. If we by x∗ denote the Jacobi iteration, use instead xk+1 = ωx∗ + (1 − ω)xk , i.e., a weighted mean between the old and new Jacobi iteration. It may turn out that using 0 < ω < 1 leads to faster convergence, but this is highly problem dependent. Gauss-Seidel Iteration. As noted above in section 5.2.1, equations involving upper or lower triangular matrices are easy to solve by forward or backsubstitution. Hence, we could choose M to be the upper or lower triangular part of A, not to be confused with L or U in the LU decomposition. If we write M = D + L + U where D is the diagonal from the Jacobi iteration and L and U is upper or lower-triangular, respectively, we may choose M = D + L which deﬁnes the Gauss-Seidel iteration. Hence, (D + L)xk+1 = −U xk + b. Writing out the equations yields i−1 N 1 bi − = Aij xk+1 − Aij xkj , xk+1 i j Aii j=1 j=i+1 i = 1, 2, . . . , N. in xk+1 for Hence, it is similar to the Jacobi iteration, but we reuse the values of xk+1 i i i<i. SOR and SSOR Iteration. In successive over-relaxation (SOR) we use Gauss-Seidel iteration with relaxation, viz., xk+1 = ωx∗ + (1 − ω)xk , where x∗ is xk+1 from the Gauss-Seidel iteration. The relaxation parameter ω might be chosen to be greater than unity, and it is in fact a good choice for Poisson-type problems, see Ref. [34]. We obtain M = ω −1 D + L, N = (ω −1 − 1)D − U. In symmetric successice over-relaxation (SSOR) one combines SOR with a backwards sweep, i.e., one ﬁrst uses the lower triangular part in a “sweep” and then the upper triangular part (i.e., performing a Gauss-Seidel iteration with M = D + U ). The matrix M is given by −1 1 1 1 1 D+L D D+U . M= 2−ω ω ω ω Reading from left to right (which is the order of appliance for M −1 ) we see that ﬁrst the lower triangular part is used and then the upper triangular part. Both are relaxed with ω as parameter. 5.2.3 Krylov Iteration Methods An alternative approach to the presentation of the methods in this section can be found in Ref. [40]. Here we adopt the presentation in Ref. [34]. Krylov iteration methods are iterative methods originally proposed as direct methods for solving linear systems, but as such they are more expensive than Gaussian elimination. Nevertheless it turns out that the methods produce a good approximation to the exact solution in k N iterations. They do not ﬁt into the framework of classical iterative methods as we cannot devise a “constant” matrix G as in Eqn. (5.3). 111 Numerical Methods for Linear Algebra Again we seek a sequence of approximations x0 , x1 , x2 et.c. that hopefully converge to the real solution x. The idea is to devise a subspace Vk ⊂ CN with a basis qj , j = 1, . . . , k such that xk := xk−1 + δxk , δxk = k αj qj , j=1 i.e., δxk is sought in Vk . We must determine the coeﬃcients αj and an algorithm for creating subspaces Vk at each iteration. The subspaces employed are so-called Krylov subspaces. The iterative methods in this section is therefore referred to as Krylov iteration methods or Krylov subspace methods. The Krylov subspace is deﬁned as Vk (A, u) := span {u, Au, A2 u, . . . , Ak−1 u}. Note that Vk need not be of dimension k. For more information on Krylov subspaces, see for example Ref. [41]. Our chosen Krylov subspace is Vk = Vk (A, r 0 ), i.e., the span of iterates of the initial residual with A. Recall that r k := b − Axk = r k−1 − Aδxk . To deﬁne the coeﬃcients αj at each iteration k, two choices are popular, and these may be viewed as a Galerkin method or least-squares method, respectively. For the Galerkin method we require that the residual r k is orthogonal to Vk , i.e., that (r k , qj ) = 0, This leads to k j = 1, . . . , k. αj (Aqj , qi ) = (r k−1 , qi ), i = 1, . . . k. j=1 This is clearly a linear system of (at most) dimension k. This of course implies that at iteration N we need to solve a system at the size of A. On the other hand, if dim VN = N we must have r N = 0 so that the exact solution is found. In this way the method is a direct method, at least if r 0 and A are such that dim VN = N . The Galerkin method is referred to as the conjugate gradient method. For the least-squares method, we minimize the norm of the residual with respect to αi , i.e., ∂ (r k , r k ) = 0, i = 1, . . . k. ∂αi Assuming that A is a real matrix and that b is a real vector, we obtain ∂r k = −2αi (Aqi , r k ) = 0. ∂αi Hence, k αj (Aqj , Aqi ) = (r k−1 , Aqi ), i = 1, . . . k. j=1 We will not go into the details of ﬁnding the new basis vector qk for Vk in the two methods, see instead Ref. [34]. Choosing some orthogonality condition and assuming some properties of A will however simplify the iteration process and the algorithm for qk . We state the types of systems for which the Krylov iteration methods described here are eﬀective. The Galerkin method needs a symmetric (Hermitian for complex A), positive or negative deﬁnite matrix, i.e., that all the eigenvalues of A are either 112 5.3 – Review of Methods for Eigenvalue Problems positive or negative. The least-squares can be used for every kind of real matrix but may require more temporary storage. For sparse matrices each iteration needs O(n) operations, where n is the number of non-zero entries. If the method converges in k N operations, we have a method with speed comparable to the tridiagonal Gaussian elimination. A complication is the fact that we need a start vector x0 to set oﬀ the iteration. For time dependent problems, choosing an initial vector x0 (and hence an initial residual r 0 ) is easy; we simply use the solution at an earlier time step. The residual then starts out small and the method may converge quickly. As for the classical iterative methods, the convergence rate of the Krylov iteration methods are highly problem dependent. In addition to the two mentioned above, there are also many other Krylov iteration methods implemented in Diﬀpack, such as SYMMLQ for non-deﬁnite symmetric (Hermitian) systems (such as our discrete Hamiltonian from both FEM and FDM) and BICGSTAB for non-symmetric systems. See the Diﬀpack documentation in Ref. [45], under the KrylovIt class which contains links to subclasses for the diﬀerent solvers. 5.3 Review of Methods for Eigenvalue Problems In contrast to solvers for linear systems of equations, eigenvalue solvers are exclusively of an iterative nature, at least for large systems. Iterative methods based on Krylov subspaces are also very popular for large systems. Ref. [41] is a thorough exposition on numerical methods for large eigenvalue problems and is highly recommended reading. For moderately-sized systems, Ref. [40] describes the widely-used methods of QR factorization, the Givens and Househölder methods et.c. for Hermitian standard problems. In addition, the documentation to ARPACK++ in Ref. [47] is very informative with respect to the Arnoldi method used in this thesis. In this section we are concerned with the generalized eigenvalue problem, i.e., Ax = λBx, where A and B are square matrices of dimension N , x is a vector and λ is a scalar. The classical methods used for moderately-sized matrices are for standard eigenvalue problems, i.e., for problems where B = I. We describe these ﬁrst and then move on to the Krylov subspace methods such as the Arnoldi method of which the Lanczos method is a special case. Is B is symmetric and positive deﬁnite it turns out that we may manipulate the eigenvalue problem to make it a standard eigenvalue problem. This will be addressed in section 6.3 Algorithms for eigenvalue and eigenvector computations are often very complicated, often due to complicated convergence properties, stability control and orthogonalization procedures. It is therefore usually not a good idea to implement the methods from scratch but instead use an existing library known to be stable, fast and that suits ones needs. On the other hand, a knowledge of the complexity and areas of application of the algorithms is necessary in order to choose the right method and also to invoke it properly. Hence, the below exposition is not meant to be complete. One should consult the references if more details are needed. 5.3.1 Methods For Standard Hermitian Eigenvalue Problems The methods described in this section works for Hermitian matrices (and hence real symmetric matrices). They do not handle generalized problems. The material in this section is taken from Ref. [40], and this reference is well worth spending some time on. 113 Numerical Methods for Linear Algebra Jacobi Transformations. The method described here is ﬁne for small matrices, but as the dimension increases, so do the number of iterations. One should then instead use the Househölder or Givens transformations described below. Diagonalizing A is equivalent to ﬁnding a similarity transformation such that D = V −1 AV is diagonal. Then column vector i of V is an eigenvector of A with eigenvalue Dii . If A is Hermitian, i.e., A† = A, the eigenvectors are all orthogonal which implies that V −1 = V † , i.e., V is unitary. The idea in the Jacobi method is to use a sequence of orthogonal (i.e., unitary) transformations P i to zero out oﬀ-diagonal elements in our matrix A, gradually reducing it to diagonal form. The diagonalizing operator is then given by V = P 1P 2P 3 . . . P n. i The transformations P i are referred to as Jacobi rotations. It is deﬁned by Ppp = i i i i Pqq = cos θ, Pqp = −Ppq = sin θ and Pij = δij otherwise. The angle θ is chosen such that Apq and Aqp become zero under the similarity transform A −→ A = P † AP. (The choice of p and q is of course dependent on i.) Unfortunately, performing a succession V of Jacobi rotations destroy the previous “annihilations” of non-zeroes, but when chosen appropriately the sequence yields convergence to D, albeit in an inﬁnite number of steps. Each rotation requires O(N ) operations for a dense matrix. The number of rotations required to sweep through the whole matrix is about N (N − 1)/2, i.e., the number of elements in the upper (or lower) triangular part. Typically one needs several sweeps to achieve convergence to machine precision; typical matrices require 6 to 10 sweeps. This totals O(20N 3 ) operations if 10 sweeps are required, according to Ref. [40]. This is of course prohibitive for dense matrices of large order. If the matrix is sparse we may obtain better convergence properties, but the rotations introduces non-zeroes at new positions in the matrix. Househölder and Givens Reductions. Using the Jacobi method for reducing A to diagonal form takes an inﬁnite number of steps. Reducing it to tridiagonal form can however be done in a ﬁnite number of steps, and tridiagonal systems may eﬃciently be diagonalized by iterative methods such as the QR algorithm described below. The Givens method chooses a sequence of (modiﬁed) Jacobi rotations that in a ﬁnite number of steps reduces any Hermitian matrix to tridiagonal form. The Househölder method is a little bit more eﬃcient in achieving the same goal and generally preferred. See Ref. [40] for details on the Givens reduction and the Househölder reduction. The QR algorithm. The QR method eﬀectively diagonalizes a tridiagonal Hermitian matrix. In simple problems the coeﬃcient matrix may be tridiagonal to begin with, but usually one obtains a tridiagonal problem by means of the Househölder reduction. The method is an iterative method that diagonalizes A through a series of unitary transforms. As the name suggests it is based on the well-known QR factorization, i.e., that any N × M matrix X can be rewritten as X = QR, where Q is an N × N unitary matrix and R is N × M and upper triangular. Indeed, it is simply a way of rewriting the standard Gram-Schmidt process for creating an 114 5.3 – Review of Methods for Eigenvalue Problems orthonormal basis (the columns of Q) for the column space of X. The matrix R provides the change of coordinates. Equivalently, we may decompose X into X = QL, where L is lower triangular. The corresponding method is called the QL decomposition. It can be shown that the QR (and QL) decomposition preserves both Hermiticity and tridiagonality of the matrix X. Given our tridiagonal matrix A = A0 we construct an iterative process as follows: Decompose Ai = Qi Ri and deﬁne Ai+1 := Ri Qi = Q†i Ai Qi , where the last equality follows from unitarity of Qi . Hence, each iteration is a change of coordinates to an orthonormal basis for the columns of Ai . As is easily seen, one may choose either of the QL or QR decompositions to deﬁne the algorithm. Under reasonable mild conditions on the eigenvalues of A this series can be shown to converge to a diagonal matrix and hence the diagonal contains the eigenvalues. To improve convergence (and to lessen the assumptions on A) one may introduce shifts at each iteration by instead factorize Ai − σi I where σi is a scalar. If A is tridiagonal each iteration can be implemented in only O(N ) operations, yielding a very eﬃcient iteration process, also for large systems. However, large systems are not that often tridiagonal and hence other methods is needed for such matrices, as the Givens reduction may be too complicated. 5.3.2 Iterative Methods for Large Sparse Systems When the dimension of the matrices in our eigenvalue problem becomes very large the Househölder method described above fails badly as it requires too many operations to complete. Instead we must turn to pure iterative methods that work with A directly. Common to the methods are the fact that each iteration uses (mostly) matrix-vector products with A (or some simple modiﬁcation of A) as coeﬃcients. If each product can be calculated fast the iteration process is in principle fast as well. Ref. [41] is a comprehensive exposition into the methods discussed in this section. Ref. [39] also contains some details concerning the power method and the simple subspace iteration. The ARPACK user guide in Ref. [47] also describes the implicitly restarted Arnoldi method in detail. The Power Method and Inverse Power Method. The simplest method for ﬁnding an eigenvalue of a standard eigenvalue problem is the power method. Given a start vector x0 one simply deﬁnes the iteration xk = 1 Axk−1 , α(k) where α(k) = maxj |xkj |, i.e., the magnitude of the largest component of xk . It is easy to see that if there is only one eigenvalue of largest magnitude and if this eigenvalue is not degenerated, the sequence will converge to the corresponding eigenvector. See Ref. [41] section 1.1 for a proof. The method converges faster the larger the ratio of the largest and next-to-largest eigenvalue magnitude is. The power iteration method is the basis for the inverse power iteration method. Notice that the matrix A − σI has the same eigenvectors as A but that the spectrum is shifted by −σ, i.e., if λ is an eigenvalue of A then λ − σ is an eigenvalue of A − σI. Furthermore, if A − σI is non-singular, the matrix B = (A − σI)−1 has the eigenvalue (λ − σ)−1 . Hence, if we want an eigenvalue of A around σ we may iterate B instead in the power method as the eigenvalues of B corresponding to eigenvalues of A close to σ will dominate the process. The eigenvectors of A, A − σI and (A − σI)−1 are all the same. The iteration with B may be done by performing an LU decomposition and successively solve the linear 115 Numerical Methods for Linear Algebra system Bx∗ = xk and scale x∗ with the appropriate factor to obtain xk+1 . A process called deﬂation may be used in order to compute the next eigenvalue when the one of largest modulus is found. One may manipulate the matrix such that the largest eigenvalue is displaced to a lower value. Then a new iteration process will ﬁnd the eigenvalue of highest magnitude. The power method is appropriate if one seeks only a few eigenvalues and when a lot of information on A is known. More sophisticated methods are used if several eigenvalues and eigenvectors are needed, as in this thesis. Hence, similar to Gaussian elimination it is rarely used except for the case where only a very few eigenvalues are needed. Simple Subspace Iteration. The power method and the inverse power method generates information at each iteration which is one-dimensional, i.e., only one vector is generated at each step. It would be nice if we instead choose m vectors, iterated these and used the extra information to obtain faster or more results, in some way. This is the idea of the simple subspace iteration method. We start out with an N ×m matrix X 0 whose column vectors span an m-dimensional subspace Vm ⊂ CN . If we use the power method on each column vector by iterating X k+1 = A · X k · D, (k) where D is an m × m diagonal matrix with renormalization factors, i.e., Dii = 1/αi , each column vector of X k will converge to the eigenvector with eigenvalue of highest magnitude. In other words, the columns will gradually lose their linear independence. The idea is then to re-establish linear independence once in a while. The column vectors of X k will then converge to several eigenvectors if this is done properly. The re-orthogonalization is simply a Gram-Schmidt process, creating a set of orthonormal vectors from a set of linear independent vectors. Every once in a while one computes the QR factorization of X k and uses Q as starting point for further iterations, i.e., the orthogonal basis for the subspace spanned by the column vectors of X k . For further details on this algorithm see for example Ref. [41]. Krylov Subspace Methods. The methods based on Krylov subspaces are deﬁnitely the most complicated algorithms both to implement and use. They are however very fast, converge quickly and are applicable to very general matrices. The algorithm used in ARPACK is called the implicitly restarted Arnoldi method (IRAM) and is perhaps the most powerful Krylov method around. All the problems in this text are Hermitian, and traditionally most eigenvalue methods are developed for such systems. The reason is double: Most physics applications give rise to Hermitian problems and Hermitian problems are the easiest to solve. The IRAM ﬁnds eigenvalues and eigenvectors of real and complex matrices, both symmetric and non-symmetric and also handles generalized problems easily. It is also used in the commercial computational software Matlab for solving large and sparse eigensystems. Similar to the Krylov subspace methods in section 5.2.3 for solving systems of linear equations the eigenvalue methods employ Krylov subspaces Vk . Notice the similarity with the deﬁnition of the Krylov subspace Vk (A, x) and the simple iteration method. One basic idea in the Krylov subspace method is to exploit more of the information generated by a simple iteration sequence. This is described in detail in Ref. [47]. Some of the information is used in the subspace iteration but there is much more to gain. Actually, the IRAM is deﬁned in terms of a Galerkin condition and hence there is a close relationship between the ﬁnite element methods, conjugate gradient-like methods for linear systems of equations and the IRAM. 116 5.3 – Review of Methods for Eigenvalue Problems Diving further into the IRAM is out of the scope of this text. The implementation in ARPACK is well-tested and robust and we will use it without hestitating. The important things to know is what ARPACK can compute, what information ARPACK needs to perform the iteration process, how much each iteration costs and how much storage space we must set aside. As for what ARPACK can compute, it ﬁnds eigenvalues and eigenvectors in diﬀerent parts of the spectrum. It may ﬁnd the largest eigenvalues, the smallest, those with largest real or imaginary parts, centered around a shift σ and so on. We are for the most part interested in the lowest eigenvalues of the Hamiltonian which is the default. Fortunately, the only information needed to solve a standard problem is simple matrix-vector products, used in subspace iterations inside the IRAM. Internally ARPACK sets aside an amount of memory proportional to the space required by about N k real (or complex) numbers, i.e., an N × m matrix. The computational cost of each iteration varies slightly, but it is proportional to the number k of eigenvalues (or eigenvalues) sought and the number of operations needed for a matrix-vector product. The number of iterations needed varies however. In fact, seeking more eigenvalues does not necessarily slow down the process as it may take fewer iterations before convergence. This is due to the extra information in the Krylov subspace generated from more search vectors. If we introduce shifting to the algorithm, i.e., seek eigenvalues around σ, linear systems must be solved with coeﬃcient matrix A − σI. For a generalized problems we need to solve linear systems with B and A − σB as coeﬃcient matrix. In other respects the method is performed similar to the standard problem. We will keep an empirical approach to these matters and use ARPACK as a black box. In addition to Ref. [47], see the Simula report available from Ref. [5]. 117 Chapter 6 Quantum Mechanical Eigenvalue Problems In this chapter we will return to the time independent Schrödinger equation, i.e., the eigenvalue equation for the Hamiltonian: HΦ = EΦ, (6.1) where E is a scalar. We will study discretized versions of the equation using ﬁnite diﬀerence and ﬁnite element methods. Assume that Φn and En are the (orthonormal) eigenvectors and eigenvalues of H; we assume a discrete set of eigenvalues for simplicity and we always order the eigenvalues in increasing order as is the convention in quantum mechanical formalism. The time development of an initial state Ψ(0) = n cn (0)Φn is given by Ψ(t) = U(t, 0)Ψ(0) = e−iEn t/ cn Φn . n In other words, the coeﬃcients evolve as cn (t) = e−iEn t/ cn (0). Their change in time is only a phase change; the magnitude of cn is conserved. Introducing a spatial discretization leads to a natural discrete form of the eigenvalue problem, viz., Hh u = Eu, where H is an N × N Hermitian matrix and u is an N -dimensional complex vector. The eigenvalues and eigenvectors of Hh will reﬂect the eigenvalues and eigenvectors of the full Hamiltonian H. Intuitively, the better approximation Hh is to H, the better the correspondence will be. We may also view Hh as the Hamiltonian for an altogether diﬀerent physical system with ﬁnitely many degrees of freedom. The approximative character of Hh to H makes us expect that the behavior of this new system reﬂects the behavior of the original one; a line of thinking often applied in physics to capture essential features of a very complicated model. As for the time development of an initial state given as a linear combination of the eigenvectors of Hh , it is of course given by the ODE (4.37). Indeed, u(t) = N e−iEn t/ cn (0)un . n=1 119 Quantum Mechanical Eigenvalue Problems As discussed in chapter 5 we typically search for some k N eigenvalues and eigenvectors. On the other hand, when starting with a suﬃciently regular initial condition, the coeﬃcients cn are neglible whenever n > k. In this way we may actually solve the time dependent Schrödinger equation for stationary problems. For time dependent problems the time dependent part is typically some perturbation that is turned on and oﬀ. Before and after the perturbation we have a stationary problem, and the spectrum of the initial and ﬁnal state with respect to the unperturbed Hamiltonian contains important information. For example, if we study a Hydrogen atom perturbed with a laser beam and start out in the ground state, the spectrum after the perturbation shows us the probability of exciting the ground state to higher states with a laser. Hence, the eigenvalue problem of the Hamiltonian is of great relevance also for time dependent problems. In this chapter we will focus on time independent problems, solving numerically some analytically solvable problems. We will also treat a simple one-dimensional problem, ﬁnding the numerical solution analytically. 6.1 Model Problems To investigate the properties of the ﬁnite element method we will investigate some model problems. These problems are analytically solvable and provides excellent means for testing ﬁnite element discretizations with diﬀerent grids; in particular we will employ square grids and grids that approximate a circular region. 6.1.1 Particle-In-Box First, we consider the particle-in-box problem, in which our particle is free to move inside the domain whose boundary deﬁnes the shape of the box. The Hamiltonian is H =− 2 2 ∇ + V (x ), 2µ where the potential V is zero inside the domain Ω and inﬁnite everywhere else. This leads to the boundary condition Ψ(x ) = 0, ∀x ∈ ∂Ω, and to the time independent Schrödinger equation 2µE , x ∈ Ω. 2 For some particular shapes of Ω this problem can be solved analytically. We will consider two-dimensional geometries, namely a square and a circular domain Ω. −∇2 Ψ = λΨ, λ= Square Domain. First consider the square domain Ω = [0, 1] × [0, 1]. We may use separation of variables, i.e., Ψ(x, y) = u(x)v(y), which leads to the identical equations and −u (x) = λx u(x), − v (y) = λy v(y), u(0) = u(1) = 0, v(0) = v(1) = 0. The solution is readily obtained, viz., un (x) = C sin(nπx), where the quantum number n = 1, 2, . . . labels the eigenfunctions and where C is an irrelevant normalization constant. Hence, Ψnm (x, y) = C 2 sin(nπx) sin(mπy) and λnm = π 2 (n2 + m2 ). 120 6.1 – Model Problems Note that whenever n = m the eigenvalue is not degenerated. Otherwise, we have λnm = λmn , i.e., twice degenerated eigenvalues. (There could also be some accidental degeneracy due to solutions of the equation n21 + m21 = n22 + m22 .) Circular Domain. Second, consider a circular domain, viz., 0 1 Ω = (x, y) : x2 + y 2 ≤ 1 . This problem is more complicated to solve and involves Bessel functions and Neumann functions, see Ref. [24]. We will sketch some of the main steps in the solution. For a complete account, see Ref. [48]. Due to rotational symmetry it is wise do employ polar coordinates, i.e., x = r cos φ, and y = r sin φ. The Laplacian in polar coordinates is given by ∇2 = 1 ∂2 ∂2 1 ∂ + 2 2, + 2 ∂r r ∂r r ∂φ as is readily obtainable by use of the chain rule for diﬀerentiation. We use separation of variables and write Ψ(r, φ) = eimφ R(r). This yields for R(r) the eigenvalue equation (called the radial equation1 ) m2 1 d d2 + 2 R(r) = λR(r). − 2− dr r dr r As for the number m it must be an integer for the wave function to be periodic and diﬀerentiable. Furthermore, the energy λ must be positive (due to H being positive deﬁnite) and we write λ = k2 . By means of changing variable to ρ = rk and multiplying the radial equation by ρ2 we obtain ρ2 R + ρR + (ρ2 − m2 )R = 0, which is called Bessel’s equation. Its solutions are R(ρ) = C1 Jm (ρ) + C2 Nm (ρ), where Jm are called Bessel functions of the ﬁrst kind, and where Nm are called Neumann functions of the ﬁrst kind. As Nm (0) = −∞ they give unphysical solutions, hence R(ρ) = CJm (ρ). To impose the boundary conditions, note that Jm has inﬁnitely many zeroes rms . These are typically tabulated, and Table 6.1 shows some of the numerical values for various integral m. Note that J−m = (−1)m Jm , and this yields degeneracy of the eigenvalues λ whenever |m| > 0. We obtain an eigenfunction R(r) whenever kms is a zero of Jm (ρ). Thus, we label the energies λms . The energies are given by 2 λms = kms and are tabulated in Table 6.1. Fig. 6.1 shows the probability densities of the ground state and the state m = 2,s = 3, i.e., |Ψ0,1 |2 and |Ψ2,3 |2 . 1 In general, for rotationally symmetric potentials V (r) the operator on the left hand side has the form −d2 /dr 2 − r −1 d/dr + m2 /r 2 + V (r). 121 Quantum Mechanical Eigenvalue Problems Figure 6.1: The two states m = 0, s = 1 (left) and m = 2, s = 3 (right) of the circular particle-in-box. k0s k1s k2s k3s λ0s λ1s λ2s λ3s s=1 2.404825558 3.831705970 5.135622302 6.380161896 5.783185964 14.68197064 26.37461643 40.70646582 s=2 5.520078110 7.015586670 8.417244140 9.761023130 30.47126234 49.21845632 70.84999891 95.27757254 s=3 8.653727913 10.17346814 11.61984117 13.01520072 74.88700679 103.4994540 135.0207088 169.3954498 s=4 11.79153444 13.32369194 14.79595178 16.22346616 139.0402844 177.5207669 218.9201891 263.2008542 Table 6.1: Some zeroes kms of the Bessel functions Jm and the corresponding energies 2 λms = kms 6.1.2 Harmonic Oscillator The one-dimensional harmonic oscillator was solved in section 2.2. We now consider the two-dimensional isotropic harmonic oscillator, i.e., HΨ = EΨ, where H=− 2 2 1 ∇ + mω 2 (x2 + y 2 ). 2µ 2 Multiplying with 2µ/2 and deﬁning γ 2 /4 = µ2 ω 2 /2 and λ = 2µE/2 yields γ2 2 2 2 −∇ + (x + y ) Ψ(x, y) = λΨ(x, y) 4 We use separation of variables and write Ψ(x, y) = u(x)v(y). The corresponding equations read γ2 2 x u(x) 4 2 γ − v (y) + y 2 v(y) 4 −u (x) + and = λx u(x) = λx v(y). These equations are simply two one-dimensional harmonic oscillators. From section 2.2 we have En = ω(n+1/2) for the eigenvalues of a one-dimensional oscillator. The total energy of the two-dimensional oscillator is thus λnm = γ(n + m + 1), n, m = 0, 1, 2, . . . where we have used λ = 2µE/2 . Hence, the eigenvalues are precisely γν where ν is any positive integer. The multiplicity of each eigenvalue is easily seen to be ν. 122 6.1 – Model Problems Note that due to rotational symmetry we could write Ψ(r, φ) = eimφ R(r) as with the circular particle-in-box. However, the radial equation for the two-dimensional harmonic oscillator is more diﬃcult to solve. When solving the harmonic oscillator numerically, we will choose a circular and ﬁnite domain. This will in eﬀect set the potential to inﬁnity at a ﬁnite distance from the origin, a source of error in our eigenvalues and energies. However, if we choose the radius of the disk suﬃciently large we should be able to reproduce many of the lowest-lying energies accurately. 6.1.3 Two-Dimensional Hydrogen Atom The two-dimensional hydrogen atom was discussed in section 3.2.2. We did not however discuss the time independent Schrödinger equation for this problem. In polar coordinates (r, φ) and in the symmetric gauge the Hamiltonian is given by H = −∇2 − iγ 2 γ2 ∂ − + r2 , ∂φ r 4 (6.2) where the units of length, energy and the magnetic ﬁeld is given by Table 3.1. Weak Field Limit. In Ref. [29] the limits in which the ﬁeld γ is weak and strong, respectively, are treated perturbatively. The limit γ = 0 yields a pure Coulombic Hamiltonian, viz., 2 H w = −∇2 − . r Similarly to the harmonic oscillator and the particle-in-box, we employ separation of variables, i.e., Ψ(r, φ) = eimφ R(r). The radial equation then reads m2 d2 1 d 2 + 2 − − 2− (6.3) Rn (r) = nm Rn (r). dr r dr r r The energies are given in Ref. [29] as w nm = − 1 , (n − 1/2)2 m = −n + 1, −n + 2 . . . , n − 2, n − 1. Hence, we have a 2n − 1-fold degeneracy of the eigenvalues. We omit the derivations of the energies and the radial functions as they are lengthy and not particularly interesting. The ground state energy in normal units is E0 ≈ 13.603 eV · w 10 = −54.41 eV. Thus, the electron in the two-dimensional hydrogen atom is more bound than in the three-dimensional hydrogen atom, in which E0 ≈ −13.603 eV. Strong Field Limit. If we consider the strong ﬁeld limit the Coulomb term is considered as vanishing,2 viz., γ2 ∂ + r2 . H s = −∇2 − iγ ∂φ 4 In other words, the electron is considered free except for the inﬂuence of the magnetic ﬁeld. The energies are given by sN M = 2γ(N + 1/2). 2 Although we have a singularity in the potential at r = 0 the expectation value 1/r is dominated by γ 2 r 2 for large γ. This justiﬁes the perturbative treatment. 123 Quantum Mechanical Eigenvalue Problems The azimuthal quantum number m is given by the quantum numbers M and N through m = N − M .3 The number N numbers the so-called Landau-levels. We see that the Landau levels are degenerate, and in fact they are inﬁnitely degenerate. In particular the ground state energy (i.e., energy of the lowest Landau level) is 0M = 1 and the corresponding wave function is a Harmonic oscillator ground state multiplied with an arbitrary function of x + iy, i.e., Ψ0 = f (x + iy) · e−r 2 /4 , where f is analytic. A basis for the ground state eigenspace is then ψ0,m = (x + 2 iy)m e−r /4 with m = 0, 1, 2, . . .. This is the solution found in the symmetric gauge, see Ref. [7]. We may change the Hamiltonian by performing a gauge transformation of the vector potential. An example of a diﬀerent but physically equivalent gauge is A = γ(−y, 0, 0), whose corresponding magnetic ﬁeld is easily seen to be identical to that if Asymm , i.e., B = γ k̂. The solution to a gauge transformed problem is equivalent to the original solution and given by an explicit unitary transform, see section 1.6. The magnetic system with the alternative gauge is solved in Ref. [48]. The inﬁnite degeneracy of the ground state is easy to fathom intuitively. There is no sense of localization in the problem deﬁnition; the electron is free and the magnetic 2 2 ﬁeld is constant throughout space. If e−x /2 was a ground state, so must e−(x −x0 ) /2 (although these states are not orthogonal). Every choice of x0 gives rise to a diﬀerent ground state and the set of these states are easily seen to span an inﬁnite dimensional space. If the domain is on the other hand a disk with radius r0 (such as in our simulations below) the degeneracy is instead r2 g = 0. 2 See Refs. [7, 48]. In the symmetric gauge, the states ψ0,m are states that are easy to interpret with the correspondence principle. One can show that for large m the probability density |ψ0,m | is concentrated in a circle with radius rm = 2m/γ. Classically an electron in a magnetic ﬁeld moves in circular paths with a radius that increases with the kinetic energy. The kinetic energy of the quantum particle also increases with m. Furthermore, the classical path’s radius is given by √ 2cT 2µT c v2 = = , r= a qvB qB which, if we assume T = 2 m2 /r 2 quantum mechanically, becomes precisely 2m r= γ in our case. 6.2 The Finite Element Formulation In this section we will describe how we arrive at the discretized eigenvalue problem for the discrete Hamiltonian Hh with the ﬁnite element method. 3 The quantum number N is not to be confused with the dimension of the matrices in the numerical problem. It should be clear from the context which N we refer to. 124 6.2 – The Finite Element Formulation Assume that we are given a set of m ﬁnite element basis functions Ni (x ). Hence, the subspace V ⊂ H has dimension m. These functions must be deﬁned appropriately when given a grid G of m nodes representing an (approximate) subdivision of our domain Ω. As usual with the ﬁnite element method we assume that the exact solution Ψ is well approximated with an element in the ﬁnite element space V , viz., Ψ(x ) ≈ Ψ̂(x ) = m uj Nj (x ). j=1 Inserting Ψ̂(x ) for Ψ in the time independent Schrödinger equation, multiplying with Ni (x ) and integrating over the domain Ω yields uj Ni (x )[HNj (x )] = E uj Ni (x )Nj (x ). (6.4) j Ω j Ω If we write u for the vector whose jth component is uj , we arrive at the generalized eigenvalue equation Au = λM u, (6.5) where M is the mass matrix and A is the element matrix obtained by integrating by parts any ∇2 term in H. As an example, consider the two-dimensional Hydrogen atom Hamiltonian, viz., 2 γ2 ∂ − + r2 ∂φ r 4 ∂ ∂ γ2 2 2 +x = −∇ − iγ −y + (x2 + y 2 ). − ∂x ∂y 4 x2 + y 2 H = −∇2 − iγ (6.6) The ﬁrst term becomes the stiﬀness matrix K, and the second term yields a matrix L whose elements are ∂ ∂ +x Ni (x ) −y Nj (x ). Lij = −iγ ∂x ∂y Ω The Coulomb term yields a matrix C given by 2 Cij = Ni (x ) Nj (x ), 2 x + y2 Ω and the last harmonic oscillator term yields a matrix O, viz., γ2 Oij = Ni (x )(x2 + y 2 )Nj (x ), 4 Ω The total element matrix is then A = K + L + C + O. Note that no boundary conditions have been imposed as of yet. Hence A contains unknown boundary terms from the integration by parts. These terms will however be eliminated. How do we impose the boundary conditions in an eigenvalue problem such as this? The homogenous boundary conditions state that uk = 0 whenever x [k] ∈ G is a boundary node of our grid. If we diagonalize the pair (A, M ) as it stands in Eqn. (6.5) we have no guarantee whatsoever that a boundary component uk of an eigenvector is zero. Furthermore, doing what is usually done in ﬁnite element applications, i.e., letting Akk = 1 and Akj = 0 whenever j = k gives no meaning in this case as we do not deal with equations on the form Au = b. (In addition to modifying A we would set bk equal to the boundary condition; in our case zero.) 125 Quantum Mechanical Eigenvalue Problems Imposing homogenous conditions in an eigenvalue problem fortunately turns out to be very simple in principle. It amounts to erasing row k and column k from both matrices A and M , and at the same time erase uk from u. In other words, we reduce the dimension of our problem with 1 for each boundary node and remove every reference to uk in the equations. As matrices may be represented in many diﬀerent ways in the computer, e.g., dense matrices (Mat in Diﬀpack), banded matrices (MatBand) or sparse matrices (MatSparse), implementing this principle must be done for every kind of matrix used in the ﬁnite element context; a task that involves non-trivial algorithms and speed considerations. Let us ﬁrst state the boundary condition imposing process in a more abstract way to make it clearer. Let P be the projection matrix onto a subspace W of Cm V . We want W to be the subspace that corresponds to the interior of Ω, i.e., P : Cm −→ Cm−n , where n is the number of boundary nodes in G, and P is hence an (m − n) × m matrix. Indeed, it is the identity matrix Im with the row number k removed for all boundary nodes x [k] . The matrix P then maps u ∈ Cm to a vector v ∈ Cm−n with the boundary node values removed. The above reduction on dimensionality for all boundary nodes can then be written as Ãv = λM̃ v, (6.7) where v = P u and where X̃ = P XP T for any m × m matrix X. Note that X̃ equals X with the rows and columns corresponding to boundary nodes removed. If x [k] is a boundary node, then uk does not appear anywhere in the equations. On the other hand, if uk was a priori known to be zero, then the equations would look like Eqn. (6.7). Notice that the mentioned boundary terms in the matrix A resided precisely in row and column k, which are now removed from the problem. As for the implementation of the matrices A and M , i.e., the integrals of Eqn. (6.4), numerical integration must be used in general. The mass matrix M is evaluated exactly if we use Gaussian quadrature of suﬃciently high order.4 In fact, Diﬀpack provides class methods that creates the mass matrix automatically. As in the above example, the element matrix A may contain more complicated terms. Each potential term must be considered in particular. For example, both the harmonic oscillator term O and the angular momentum term L contain a simple polynomial expression that is very well tackled by Gaussian integration. On the other hand, the Coulomb term C contains an 1/r term which may or may not be well integrated with Gaussian integration as it has a singularity. In all our applications we will use default Gaussian integration. 6.3 Reformulation of the Generalized Problem Solving a generalized eigenvalue problem is much more diﬃcult than solving a standard problem. The numerical methods presented in chapter 5 were mostly only applicable to standard problems with Hermitian matrices. We observe that the mass matrix M is symmetric and positive deﬁnite. Let Vh be our ﬁnite element space and let uh ∈ Vh be arbitrary and nonzero. Let Ni , i = 1, 2, . . . , m be our ﬁnite element basis. Hence, Mij := (Ni , Nj ) and M is a real, symmetric matrix (if we assume Ni to be real). For the norm of uh we obtain ui Ni , uj Nj ) = u∗i uj (Ni , Nj ) = u† M u, 0 < (uh , uh ) = ( i j ij 4 Gaussian integration evaluates the integral of a polynomial of a certain degree exactly. See Ref. [34] for a description of common Gaussian integration rules that are implemented in Diﬀpack. 126 6.3 – Reformulation of the Generalized Problem and hence M is positive deﬁnite. Here u is the CN vector whose components are uj .5 Any positive deﬁnite matrix is invertible. Hence, our generalized eigenvalue problem is equivalent to M −1 Hx = λx. Alas, the matrix M −1 H is not Hermitian, rendering the algorithms from chapter 5 useless. Luckily, the Cholesky decomposition comes to our rescue. It is well-known that every symmetric and positive deﬁnite (real) matrix M can be rewritten as M = L · LT , where L is a lower triangular matrix. This decomposition is called the Cholesky decomposition. We now state a theorem. Theorem 18 Given the generalized eigenvalue problem Hu = λM u, where H is Hermitian and M is symmetric, real and positive deﬁnite. Let (u, λ) be an eigenpair. Let M = LLT be the Cholesky decomposition of M . Then the matrix C = L−1 H(L−1 )T is Hermitian and has the eigenpair (LT u, λ). Proof: Hermiticity is easily seen to hold, viz., C † = (L−1 H(L−1 )T )† = [(L−1 )T ]† H † (L−1 )† = L−1 H(L−1 )T . Multiplying Hu = λM u with L−1 from the left yields L−1 Hu = λLT u, and writing v = LT u gives L−1 H(L−1 )T v = λv, and we are ﬁnished. From this theorem it is easy to see that solving the eigenvalue problem Cv = λv gives the correct eigenvalues. Furthermore, obtaining u = (LT )−1 v is eﬃciently done with backsubstitution. One potential problem is the fact that C is a dense matrix and hence matrix-vector products, if they may be computed at all, is an O(N 2 ) process. However, to compute v = Cv = [L−1 H(LT )−1 ]v we may follow these steps: 1. Solve the equation LT x = v by backsubstitution; an O(N 2 ) process in worst case but much faster if LT is banded. 2. Compute y = Hx by matrix multiplication. 3. Solve Lv = x by forward substitution. It is easy to see that if H is a sparse matrix and M is sparse and stored in a banded format, this process is eﬃcient. We mention that the matrix L of the Cholesky decomposition of a banded matrix is again banded with the same bandwidth. Hence, a general sparse structure of M should 5I apologize for the ambiguous notation here. 127 Quantum Mechanical Eigenvalue Problems not be chosen but rather a banded structure, allowing the Cholesky decomposition to be made en place. Unfortunately, there was not time to implement this strategy in the HydroEigen solver for the two-dimensional hydrogen atom. Even though it does not represent any addition of ﬂexibility except for the ability to solve sparse generalized eigenvalue problems, it would represent a considerable speed-up in the simulations. It is on the other hand an obvious future project to implement the Cholesky factorized mass matrix for ﬁnite element eigenvalue problems. 6.4 An Analysis of Particle-In-Box in One Dimension As an introduction to the work with discretized eigenvalue problems we will analyze the particle-in-box in one dimension, using both ﬁnite diﬀerence methods and ﬁnite element methods. The exact eigenfunctions and eigenvalues can be obtained, and will serve as an indicator of the behavior of both methods. Recall the eigenvalue equation for a particle in a box, viz., −u (x) = λu(x), x ∈ [0, 1], with the boundary conditions u(0) = u(1) = 0. This problem has the eigenvectors and corresponding eigenvalues given by uk (x) = sin(kπx), λk = (kπ)2 , as stated in section 6.1.1. For the discretized version, we use a uniformly spaced grid with N + 1 points, hence the grid spacing is h = N −1 and the grid points are given by xj = hj, j = 0, 1, . . . , N . If we use the standard diﬀerence scheme [−δx δx v(x) = λv(x)]j , we obtain a ﬁnite-dimensional eigenvalue problem, viz., Av = λv, (6.8) where v is the components of the discrete solution, i.e., vj = v(xj ), j = 1, . . . , N − 1. (The components v0 and vN are identically zero due to the boundary conditions.) The matrix A is an (N − 1) × (N − 1) symmetric and positive deﬁnite matrix.6 Hence, we expect N − 1 positive and real eigenvalues for this problem. Let us assume that the discrete eigenfunctions are given as vk (x) = sin(kπx), i.e., we guess that the eigenfunctions have the same form as in the continuous problem. They clearly fulﬁll the boundary conditions and the components are given by vj = sin(kπhj). (We omit the subscript k to economize.) We will not consider the matrix A explicitly, but instead write out the diﬀerence equations, viz., − 6 See 128 1 (vj−1 − 2vj + vj+1 ) = λvj , h2 for example Ref. [33]. j = 1, . . . , N − 1. 6.4 – An Analysis of Particle-In-Box in One Dimension Using the trigonometric identity sin(x + y) + sin(x − y) = 2 cos y sin x, (6.9) we obtain vj−1 + vj+1 = 2 cos(kπh) sin(kπhj) = 2 cos(kπh)vj , and hence, −δx δx vj = 1 (2 − 2 cos(kπh))vj . h2 Using 1 − cos(x) = 2 sin2 (x/2), we arrive at −δx δx v(xj ) = λk v(xj ), with 4 kπh ), sin2 ( 2 h 2 and indeed the assumed form of v(x) is valid. The integer k now numbers the eigenvalues, but there are not inﬁnitely many as A has at most N − 1 eigenvalues. It is easy to see that vN ≡ 0, so it is not a proper eigenfunction, and furthermore vN +k does not yield further eigenvectors, this due to periodicity of vk (x). But for k = 1, . . . , N − 1 we have distinct eigenvalues. In summary, the eigenfunctions are identical to the N −1 ﬁrst ones of the continuous problem, but the eigenvalues are not the same. We may estimate the deviation by using a Taylor expansion for sin2 (x), viz., λk = 1 sin2 (x) = x2 − x4 + O(x6 ). 3 This yields 1 λk = k2 π 2 1 − π 2 k2 h2 + O(k4 h4 ) . 12 Now we turn to a simple ﬁnite element approach, employing linear and uniformly sized elements with the node points deﬁned by xj as in the ﬁnite diﬀerence case. We will obtain a generalized eigenvalue problem reading Kv = λM v, (6.10) where K and M are the stiﬀness matrix and mass matrix, respectively. These are (N − 1) × (N − 1) matrices given by Ki,i = 2 , h 1 Ki,i±1 = − , h Kij = 0 otherwise, and 4h h , Mi,i±1 = , Mij = 0 otherwise. 6 6 Note that K = hA, i.e., the stiﬀness matrix is essentially the ﬁnite diﬀerence operator used in the above analysis. In this case we also expect N − 1 real eigenvalues. If we try the same discrete eigenfunctions vk (x) as above, the left hand side becomes Mi,i = [Kv]j = 4 kπh sin2 ( )vj . h 2 If we write out the jth component of M v, we get [M v]j = h (vj−1 + 4vj + vj+1 ) . 6 129 Quantum Mechanical Eigenvalue Problems 3 Eigenvalues/N2π2 2.5 FEM 2 Exact 1.5 1 FDM 0.5 0 0.2 0.4 0.6 0.8 1 πkh/2 Figure 6.2: Plot of eigenvalues We use the trigonometric identity (6.9) and obtain [M v]j = h (2 + cos kπh)vj . 3 With the identity 1 + cos(x) = 2 cos2 (x/2) we obtain h 2 kπh [M v]j = 1 + 2 cos vj . 3 2 Hence, vk (x) is an eigenfunction of M also. We may convert the generalized eigenvalue problem to a standard one by writing M −1 Kv = λv, and hence we obtain λk = sin2 kπh 12 2 . 2 h 1 + 2 cos2 kπh 2 Using the Taylor expansion for λ, we get 1 2 2 2 2 2 4 4 λk = k π 1 + π k h + O(k h ) . 12 Figure 6.2 shows a plot of the normalized eigenvalues for the ﬁnite diﬀerence, ﬁnite element and exact calculations, respectively. Clearly, the ﬁnite diﬀerence eigenvalues underestimate the exact ones, while the ﬁnite element eigenvalues overestimate the eigenvalues. Both numerical methods yield good approximations for the lowest eigenvalues, but the ﬁnite element calculations are clearly qualitatively more correct than the ﬁnite diﬀerence calculations for the higher eigenvalues. Several questions and interesting topics arise already at this point. The ﬁnite diﬀerence method is equivalent to lumping the mass matrix, i.e., replace M by the is the sum of the elements in row i of M . diagonal matrix M whose element Mi,i Seemingly, the ﬁnite element method yielded qualitatively better results, but is this 130 6.5 – The Implementation true in general? We shall see that the answer is not aﬃrmative: Lumping the mass matrix may improve the convergence of the eigenvalues for some systems. Actually, if lumping the mass matrix turns out to be fortunate one must not immediately conclude that the ﬁnite element method is worthless in the case at hand, as for example the geometry may play a signiﬁcant role in the precision. A combination of lumping the mass matrix with using higher-order elements and exotic geometries may turn out to be very powerful. Speaking of which: How fast does the eigenvalues of the discrete problem converge to the exact values? In the above application the convergence was O(h2 ) in both ﬁnite diﬀerence and ﬁnite element approximations, and in fact the leading terms in the error were identical in the two cases. If we use higher-order elements, will the convergence be correspondingly higher? This could clearly give the ﬁnite element method an advantage. We will address these questions when doing numerical experiments below. Actually, the solution method used in this section may be extended to two (or more) dimensions by using separation of variables analogously to the continuous problem. If we write nm = v n (xk )v m (yl ), Ukl where the superscript indicates the quantum numbers in the diﬀerent directions we obtain nm nm (−δx δx − δy δy )Ukl = (λn + λm )Ukl , where λnm = 4 h2 nπh mπh ) + sin2 ( ) , sin2 ( 2 2 is the numerical eigenvalues of the two-dimensional ﬁnite diﬀerence formulation. As for the mass matrix, one may ﬁnd similar results, making it possible to obtain the exact eigenvalues also in the ﬁnite element case with linear elements. Nevertheless, we will use the computer program described in section 6.5 to numerically diagonalize our particle in box. This in order to test our implementation which may be used with more complicated problems. 6.5 The Implementation When programming with Diﬀpack one usually implements a class derived from one of the preexisting solver classes. In this case we derive class HydroEigen from class FEM, the base class for ﬁnite element solvers. HydroEigen is instantiated in the main() function and the simulation is then run. The class deﬁnition reimplements various virtual functions in class FEM which performs certain standardized tasks, such as initialization, evaluation of integrals and so on. In this way the programmer of a ﬁnite element solver does not need to bother with ﬁnite element algorithms or the sequence of actual functions from class FEM that is called, but instead focus on the formulation of the problem. Listings for class HydroEigen can be found in appendix B. The source code is available from Ref. [5] as well. The main() function is not listed in conjunction with the eigenvalue solver. When we turn to time dependent implementations in chapter 7, we derive class TimeSolver from class HydroEigen to make a more general program. The main() function listed instantiates class TimeSolver instead. The eigenvalue problems that are implemented in our class is the two-dimensional hydrogen atom with an applied magnetic ﬁeld. Furthermore, terms in the Hamiltonian may be turned on and oﬀ at will so that other problems such as the free particle or the harmonic oscillator may be solved as well. 131 Quantum Mechanical Eigenvalue Problems We will inevitably touch upon features of Diﬀpack that are beyond the scope of this text to describe further. Refs. [34, 45] contains a thorough description of most features used. However, the purpose of each feature should be clear from the context. The code is well commented and is recommended for additional reading. The program was compiled in the Debian Linux operating system with GNU gcc 2.95.4. We mention some important command line parameters for the application here. These are standard for Diﬀpack programs. The name of the executable is assumed to be SchroedSolve.x. (The makeﬁles are not listed in the appendix, but it can be obtained from Ref. [5].) If one passes --GUI as parameter, the menu system will be displayed in a graphical user interface. Otherwise, a command based interface in the console is used. If --casename cname is passed, the simulation casename will be set to cname . The casename is the base name of ﬁles produced by the simulator such as the log ﬁle (cname.log), the simres database holding ﬁelds, grids, curves et.c. (.cname.simres) and so on. The casename feature is particularly useful when performing a sequence of simulations with varying parameters, e.g., the magnetic ﬁeld, the grid size and initial condition. Finally, we have the --verbose 1 command line parameter that displays various runtime statistics, such as the time spent on solving linear systems of equations. During execution Diﬀpack also provides a lot of error and warning messages in case of unforeseen events, such as memory overﬂow, erroneous grid parameters and so on. 6.5.1 Class methods of class HydroEigen Here we describe the most important class methods. Hopefully this will draw a clear picture of how the class works. But ﬁrst we list some important class members that have a central role: Handle(GridFE) grid; This is a handle (i.e., a smart pointer) to a ﬁnite element grid. Handle(DegFreeFE) dof; This is a handle to a Diﬀpack object that holds the mapping q(e, r) from local node numbers and element numbers to global node numbers. It determines the correspondence between ﬁeld components Uj and the nodes x [k] in the grid. Handle(LinEqAdmFE) lineq; This object is a wrapper around linear systems of equations and solver methods. The solvers are also implemented as classes and are connected to this object via the menu system described below. Handle(Matrix(NUMT)) M; This is a handle to the mass matrix. Another handle K points to the element matrix of the Hamiltonian. The following description of the member functions is not exhaustive. It only describes the most important functions and lines of thinking. As always in large programming projects a lot of small and big problems needs to be solved, and to describe them all in this text would be both lengthy and unnecessary. The code is however well documented, such that algorithms for such things as removing a row and a column in a sparse matrix should be easy to read and understand. define(). An important feature of Diﬀpack is the menu system. Every parameter to a solver should be accessible via the menu system, and with proper usage experimenting with for example solver methods for linear equations become very easy. The menu system may be run in command-mode in which diﬀerent items are set with commands from standard in like ‘set gamma=1.0’. A list of available commands is listed with the ‘help’ command. If a Diﬀpack class comes with a (virtual) define() method, it usually adds some knobs and handles to the menu system. The define() method of class HydroEigen 132 6.5 – The Implementation adds parameters that may be used to adjust the deﬁnition of the problem, desired matrix encoding scheme and so on. The entries deﬁned are as follows: – gridfile File to read grid from or a preprocessor command that is used to create the grid. – nev Number of eigenvalues to search for. – nucleus Boolean (with value true or false) that turns on or oﬀ the Coulomb attraction in the Hamiltonian. – epsilon When evaluating the Coulomb term in the Hamiltonian, 1/r must be calculated. If r is too small this may lead to overﬂow and loss of accuracy in the calculations. Therefore, if r < epsilon, we use 1/epsilon instead in the integrand. – gamma Strength of magnetic ﬁeld. If this is zero together with false for nucleus, the problem will turn into particle-in-box. – angmom A boolean that turns on or oﬀ the term in the Hamiltonian (6.2) proportional to the angular momentum i∂/∂φ. Setting the parameter to false together with the nucleus parameter turns the problem into a harmonic oscillator. – nitg Number of integration points if trapezoidal integration is preferred over Gaussian integration. A zero value turns on Gaussian integration. See comments in listing for details on the trapezoidal rule used. – lump A boolean variable indicating whether or not lumping the mass matrix should be done. – warp The grading parameter w described on page 144. The grading makes the grid points and elements concentrate around the origin if w > 1. This feature may be used to improve the accuracy of the numerical integration around the origin. – scale A scalar that is used to uniformly scale the grid after warping. For example, a value of 2 will double the size of the grid in each direction. – renum A boolean variable indicating whether or not an optimization of the q(e, r) mapping is to be done in order to minimize the bandwidth of the element matrices. If banded matrices are used this option should be set to true. For sparse matrices it has no practical signiﬁcance except for that it spends some time. (A lot, actually, if the grid is large.) – savemat A boolean that indicates if one wishes to save the element matrices after they are assembled. Could be useful if one wishes to compare the eﬃciency of the program with, e.g., Matlab or Maple. Only works for sparse matrices and they are stored in Matlab compliant .m-ﬁles. – use arpack A boolean that indicates if ARPACK is to be invoked for solving the eigenvalue problem. Usually this is the case, but if one simply wishes to test other features (such as timing the assembly process) it could be handy to be able to turn it oﬀ. – gauge A string literal with value symmetric or non-symmetric indicating the gauge to be used for the magnetic ﬁeld. 133 Quantum Mechanical Eigenvalue Problems – store evecs If set to a value outside [0,nev], all the eigenvectors are stored. They are chosen for storing in order of increasing eigenvalue. Storing eigenvectors take up a lot of drive space, so for large problems it could be useful to store, e.g., only the ground state. – store prob density A boolean that indicates whether or not the probability density ﬁeld is to be stored alongside the eigenvectors. The number of ﬁelds stored is given by store evecs. The define() method also attaches submenus accessing parameters for linear solvers and matrix storage schemes. From the menu system, the submenu controlling matrices and linear solvers are accessed with the command ‘sub LinEqAdm’ or ‘sub L’. Two submenus Matrix prm and LinEqSolver prm can be accessed from here, making available settings for matrix storage and solver methods, respectively. The most important commands are ‘set matrix type=classname ’ and ‘set basic method=method ’, selecting matrix storage scheme and solver method, respectively. scan(). This method initializes the HydroEigen solver class according to the settings from the menu system. It allocates memory for various arrays and objects, creates (or reads from ﬁle) the grid, opens a log ﬁle to which various statistics and results is written and so on. In short, an instance of the class is ready to solve an eigenvalue problem when scan() is ﬁnished. solveProblem(). This method is the most central function in the program. It creates the matrix A (called K in the code) and the matrix M , imposes boundary conditions, instantiates EigenSolver and solves the eigenvalue problem. The eigenvalues and eigenvectors are then written to ﬁle by HydroEigen::report(). enforceHomogenousBCs(). Boundary conditions are implemented diﬀerently in eigenvalue problems than in regular ﬁnite element simulators. The problem deﬁning matrices have to be modiﬁed by erasing rows and columns corresponding to nodes on the boundary of the grid. This member function accomplishes this with both sparse matrices and banded matrices. It uses other member functions such as eraseRowAndCol() to do this. getElapsedTime() and reportElapsedTime(). These functions perform simple timing tasks. getElapsedTime() simply returns the number of seconds since the ﬁrst call to this function. reportElapsedTime() writes this to the log ﬁle as well. report(). After solveProblem() has ﬁnished its main task, report() is called to write the eigenvectors (i.e., the discrete eigenfunctions) to a ﬁle. In addition, the eigenvalues and various other information is written to the log ﬁle. This information may be used in visualization programs to show the approximate eigenfunctions found by the program. Each eigenvector is labeled with its eigenvalue for reference. integrands(). This method is perhaps the core of every ﬁnite element formulation. It evaluates the integrand in the numerical integration routines used in the element matrix assembly. Integrals in Diﬀpack are exclusively done numerically with Gauss integration of varying order. The integrands() method evaluates the integrand (multiplied with the integration weight and the Jacobian of the coordinate change mapping between local and global coordinates) at a given point (passed in a class FiniteElement parameter object) and updates the element matrix and vector also passed as parameters. calcExpectation r(). It would be useful if the simulator application could produce information on expectation values of various observables. Given two FieldFE objects 134 6.5 – The Implementation corresponding to two discrete functions u and v, this method calculates (u, rv). The method is implemented by integrating numerically over the elements in the grid similar to the assembly of the element matrix A. Even though only this and one other such method is implemented here (namely calcInnerProd()), it is easy to write new ones. In fact, the time dependent solver implements a more general expectation value method using a general matrix. This allows for computation of for example the total energy without integration over the elements as this is already done in the assembly process. 6.5.2 Comments Some comments are given here, summing up some information gained during implementation, simulation and in the aftermath after having implemented the time dependent solver (in chapter 7). The Matrices. The matrices stored in the handles M and K correspond to the mass matrix M and the matrix H, respectively. The matrices may be created in either MatBand or MatSparse format, the latter deﬁnitely being the most favorable as the matrices become increasingly sparse with increasing dimension of the problem. Banded matrices take up a lot more space and require much more operations during for example matrix-vector products. Furthermore, if a sparse matrix is used there is no need for optimizing the grid. In Fig. 5.1 this optimization is done. Without optimization the non-zeroes are scattered throughout the whole matrix, making a banded storage scheme not at all attractive. Other matrix formats should not be used as their support is not implemented in for example enforceHomogenousBCs(). The mass matrix can be lumped or not lumped. In the former case the full matrix is not used in the diagonalization process, but instead A (i.e., K) is multiplied with its inverse so that the eigenvalue problem becomes a standard problem. This speeds up the simulation considerably. Linear Solvers. The linear solver method is chosen by the following commands: sub LinEqAdm sub LinEqSolver prm set basic method = method The name method can be one of many choices, e.g., GaussElim, GMRES and ConjGrad. With complex sparse matrices Gauss elimination cannot be used because it is not implemented in Diﬀpack. The conjugate gradient method ConjGrad supposedly works best for positive deﬁnite systems, but it seemingly works well for other systems as well. The method used in the simulations in this thesis was GMRES, however. The method takes longer than Gaussian elimination for small systems, bot when the number of nodes increases, the method becomes much faster. Optimization of the grid does not seem to aﬀect the eﬃciency. (This is intuitively so because only matrix-vector operations are used in the algorithm.) The Invocation of ARPACK. The class EigenSolver instance used for diagonalization has support for many options that are not used at all. For example one might want to search for eigenvalues of intermediate magnitude instead of the lowest magnitudes. If one wishes to study for example the structure of highly excited states in a classically chaotic system, this may be the case. It is only minor modiﬁcations that is needed to incorporate this in the class. EigenSolver is also somewhat limited with respect to what features of ARPACK is actually used, such as monitoring of convergence, reading the Schur vectors and so 135 Quantum Mechanical Eigenvalue Problems L h 0 0 L Figure 6.3: A square grid with D = 6. A few linear and quadratic element shapes have been drawn to indicate their distribution. on. In a more complex implementation these features could be taken advantage of in HydroEigen as well. 6.6 Numerical Experiments The HydroEigen class may be used to solve the particle-in-box, the harmonic oscillator and the hydrogen atom. In this section we will go through some numerical experiments with the program. There are several parameters that may aﬀect the precision of the eigenvalues, such as mesh width, element type and order and the numerical geometry. In addition, the presence of the mass matrix M in the problem may or may not improve the quality of a ﬁnite element approximation in comparison with a ﬁnite diﬀerence approximation. 6.6.1 Particle-In-Box Square Domain When studying the particle-in-box in a square domain, the most natural choice for grid is a uniform square grid. The analytic eigenfunctions have the form sin(kx πx) sin(ky πy) and do not concentrate in some regions of the square. Hence, the grid should be uniform. Fig. 6.3 shows a square grid with sides L. Each side is subdivided into D intervals of length h = L/D, where h is the mesh width, and the grid has (D + 1)2 nodes. When using linear elements, each elements requires 2 × 2 = 4 nodes, and when using quadratic elements 3 × 3 = 9 nodes are required. If we then choose D as an even number we may use both linear and quadratic elements on the same subdivision and hence the same mesh width. To study the quality of the ﬁnite element method (and the ﬁnite diﬀerence method in this case) we study the relative error of the eigenvalues as function of h and the element order p, i.e., λnum − 1. δ(h, p) = λ The qualitative behavior of the relative error, e.g., if it is increasing slowly or rapidly will also indicate the quality of the methods. Furthermore, comparing a simulation where the mass matrix is lumped and a simulation with the full mass matrix may indicate if lumping improves the quality of the eigenvalues. This particular problem of a particle-in-box was solved analytically in one dimension and also partially in two dimensions in section 6.4. These results should be set in 136 6.6 – Numerical Experiments n k 20 19.77982922 49.69408652 79.60834382 100.3720148 1 2,3 4 5 40 19.74935767 49.43433728 79.11931689 99.1128192 ∞ 19.73920881 49.34802202 78.95683523 98.69604404 60 19.74371889 49.38636767 79.02901644 98.88109094 Table 6.2: Results from numerical simulations of particle-in-box with linear elements ␦k(n) 0.1 0.09 0.08 0.07 n=20 0.06 0.05 0.04 0.03 n=30 0.02 n=40 n=50 n=60 k 0.01 0 10 20 30 40 Figure 6.4: Relative error of eigenvalues, linear elements conjunction with the present discussion. Linear Elements. A sequence of simulations was run with n = 20, 30, 40, 50 and 60 subdivisions of the sides in the grid. Table 6.2 shows the ﬁrst few numerical eigenvalues compared to the analytic eigenvalues for some grid sizes. Fig. 6.4 shows the relative error for each simulation. Note that many eigenvalues are pairwise equal, also in the discrete formulation, and this shows up as many pairwise equal relative errors. If we assume that δ(h) ∼ Chν = Cn−ν we obtain ln δ = C − ν(ln n), i.e., a straight line if we plot ln δ against ln n. The constant should depend on the eigenvalue number k. Fig. 6.5 shows such plots for a few eigenvalues λnum . As seen k from the ﬁgure, we obtain perfectly linear plots; hence the assumed form of δ ﬁts well. Notice that increasing eigenvalues yield increasing Cs, reﬂecting that higher eigenvalues tend to have higher error, as seen in Fig. 6.4 as well. By inspection, ν = −2 for all the sample plots. Hence, λnum = λ + O(h2 ) is a reasonable guess. Quadratic Elements. Performing the same experiments but with quadratic elements yields very similar results. Simulations with n = 20, 30, 40 and 50 were done. Fig. 6.6 shows the relative error for each eigenvalue when using quadratic elements, while Fig. 6.7 shows ln δ as function of ln n. Clearly, δ = O(h4 ), which is a much better convergence rate than for linear elements. 137 Quantum Mechanical Eigenvalue Problems ln ␦ –4 –5 k=20 k=5 –6 k=4 k=2 –7 –8 3 3.2 3.4 3.6 3.8 4 k=1 ln n Figure 6.5: ln |δ| versus ln n. The graphs are straight lines with slope −2. ␦k(n) 0.03 0.025 0.02 0.015 n=20 0.01 0.005 n=30 n=40 n=50 0 10 20 30 k 40 Figure 6.6: Relative error of eigenvalues, quadratic elements The same number of nodes and hence the same dimensions of the matrices yields much faster convergence with quadratic elements. What we have to pay is an increase in the bandwidth of the matrices, as quadratic elements contain more nodes, and hence couplings between nodes farther away from each other than in linear elements. We notice that δ > 0 for all performed experiments, i.e., the ﬁnite element method seems so overestimate the eigenvalues. Lumping the Mass Matrix. It is a well-known fact that when solving the one-dimensional wave equation utt = uxx the standard ﬁnite diﬀerence method δt δt u = δx δx u yields the exact solution. This diﬀerence scheme is actually equivalent to using linear elements and lumping the mass matrix; hence an improvement of the numerical results is a consequence of lumping in this case. Lumping the mass matrix when using linear elements creates an eigenvalue problem equivalent to using the ﬁnite diﬀerence method, as shown in section 6.4. Furthermore, lumping the mass matrix makes our eigenvalue problem a standard eigenvalue problem which is easier and quicker to solve numerically as we do not need to solve linear systems of equations. Hence, it is of importance whether or not lumping improves or degrades the eigenvalues and eigenvectors. A few numerical experiments shows that δ = O(h4 ) also for the lumped eigenvalue problem with quadratic elements, see Fig. 6.7. In addition, δ < 0, i.e., the lumped system tends to underestimate the eigenvalues. 138 6.6 – Numerical Experiments ln ␦ –6 –8 k=20 –10 k=4 –12 k=1 –14 3 3.2 3.4 3.6 ln n 3.8 Figure 6.7: ln |δ| versus ln n. The graphs are straight lines with slope −4. The dashed lines are for the lumped eigenvalue problem. 0.04 ⱍ␦k(k=20)ⱍ lumped 0.02 regular k 0 10 20 30 40 50 Figure 6.8: Comparison of the relative error |δ| in the eigenvalues for lumped system and regular system. A grid with n = 20 was used. . The qualitative behavior of the relative error is however diﬀerent in the lumped system when compared to the regular eigenvalue problem. The regular problem has a relative error that ﬂuctuates a great deal with increasing eigenvalue number. The lumped version shows a smoother behavior, see Fig. 6.8 in which a comparison is made. As we see, the relative error grows faster with increasing eigenvalue number than in the original problem. This eﬀect can also be seen in Fig. 6.2, section 6.4. As we see, lumping might be an option when only the lowest eigenvalues are required. Furthermore, with a lumped mass matrix the eigenvalues are in principle much easier to compute, an important fact. As the behavior of the eigenvalues are qualitatively very diﬀerent when the mass matrix is lumped, it should not come as a surprise if the method is superior for some Hamiltonians. Indeed, when we study the twodimensional hydrogen atom we see that lumping the mass matrix drastically improves the behavior of the method. Circular Domain When discretizing a circular domain we use triangulation. This is supported by Diﬀpack, see Ref. [34]. Parameters to the circular grid are its radius r, the number of line segments on the boundary and the total number of desired elements. The only element type supported is linear triangular elements. As the node locations and element shapes are not known a priori we will have to estimate the mesh width h. If we by Ã denote 139 Quantum Mechanical Eigenvalue Problems the average area of the elements, we have Ã ∝ h2 , i.e., Ã = πr 2 ∝ h2 , nel where nel is the number of elements. Thus, r h∝ √ . nel In section 3.5 of Ref. [34] the suggested number of elements to produce a uniform grid with optimal triangle angles are nel = (n − 1)2 , 2π where n is the number of line segments used to discretize the border of the circular region. (In the grid generation utility, nel must be supplied, but it is treated as a “wish,”, i.e., the program will try to create this number of elements.) Using the formula for nel , we obtain r h∝ , n−1 hence using evenly spaced n will produce meshes comparable to the ones used for the square domain. We will use n = 50, 70, 90, 110 and 130. Fig. 6.12 shows two grids used in the simulation. We will do numerical simulations similar to the square domain case. We will do a comparison between the lumped system and the regular system in addition to ﬁnding the relative error order. Fig. 6.9 shows the relative error δ(k) for the diﬀerent grid sizes. Fig. 6.10 shows the same for the lumped system. The relative error is negative for the lumped system, hence, eigenvalues are underestimated in this case. For the regular system, eigenvalues are exclusively overestimated. Fig. 6.11 shows ln |δ| as function of ln n for selected eigenvalues for both the regular and lumped system. Clearly, ν = −2 in the case of linear, triangular elements (both lumped and regular) similar to the rectangular domain case. I.e., λnum = λ + O(h2 ). The irregularity of the curve shapes are probably due to the deﬁnition of h. The grids are not related in a simple way as with the rectangular grid used above. Clearly, the lumped system performs better than the regular for this system as the relative error is lower for all eigenvalues. 6.6.2 Two-Dimensional Hydrogen Atom The Hamiltonian for the two-dimensional hydrogen atom with an applied magnetic ﬁeld in polar coordinates is H = −∇2 − ∂ γ2 2 − iγ + r2 , r ∂φ 4 where we have used the symmetric gauge in which the magnetic vector potential is given by γ A = (−y, x, 0). 2 The parameter γ is the strength of the magnetic ﬁeld which is aligned along the z-axis perpendicular to the plane of motion of the electron. The time independent Schrödinger equation reads HΨnm = nm Ψnm 140 6.6 – Numerical Experiments ␦k(n) 0.35 0.3 n=50 0.25 0.2 0.15 n=70 0.1. n=90 0.05 0 10 20 30 40 n=110 n=130 k Figure 6.9: Relative error for each eigenvalue for the diﬀerent grids in the regular system. -␦k(n) 0.2 n=50 0.18 0.16 0.14 0.12 0.1 n=70 0.08 0.06 n=90 0.04 n=110 0.02 k 0 10 20 30 40 Figure 6.10: Relative error for each eigenvalue for the diﬀerent grids in the lumped system. . 141 Quantum Mechanical Eigenvalue Problems ln ␦k(n) regular ±2 lumped ±3 k=1 ±4 ±5 k=20 ±6 ±7 3.8 4 4.2 4.4 4.6 4.8 5 ln n Figure 6.11: ln |δ| as function of ln n for a few eigenvalues λnum . Reference lines with k slope −2 are shown. . where the eigenvalues are given by nm = − 1 , (n + 1/2)2 n = 1, 2, . . . , m = −n + 1, −n + 2, . . . , n − 2, n − 1. We see that each eigenvalue has degeneracy 2n − 1. The corresponding eigenfunctions are obtained by separation of variables, viz., Ψnm = eimφ Rnm (r), where Rnm (r) fulﬁlls a diﬀerential equation we cannot solve in closed form. The numerical values of the ﬁrst few eigenvalues are 4 4 4 ≈ −0.4444, 3m = − = −0.16, 4m = − ≈ −0.08163. 9 25 49 Table 6.3 shows the standard deviation σnm = r 2 , quoted from Ref. [29]. The width of the wave function tends to grow rapidly with higher energy. Our domain of discretization must be large enough in order to capture the essential features of Ψnm for as many states as we wish. On the other hand we have a singularity at r = 0 and the potential varies rapidly here. In order to reproduce the Hamiltonian faithfully near the origin we must have an appropriately ﬁne mesh. As σ00 = 3/8 is a small number it is not surprising if the region r < 1 needs a quite ﬁne mesh. Numerically we do not use the natural numbering of the eigenvalues, simply because we do not know a priori if they are degenerate or not. We simply choose to sort them in increasing order and we label them with an integer k. The eigenvectors found for the ﬁnite element matrix corresponds to an approximate eigenfunction through the mapping Uj Nj (x ), uh = 10 = −4, 2m = − j and therefore we will use the term eigenvector and eigenfunction interchangeably in this section. 142 6.6 – Numerical Experiments 1 0 -1 -1 0 1 -1 0 1 1 0 -1 Figure 6.12: Grids for n = 50 (above) and n = 130 (below). . 143 Quantum Mechanical Eigenvalue Problems n=1 n=2 n=3 n=4 |m| = 0 3 ≈ 0.612 8 14 58 ≈ 3.825 103 18 ≈ 10.155 385 78 ≈ 19.644 |m| = 1 |m| = 2 11 14 ≈ 3.354 93 34 ≈ 9.682 367 21 ≈ 19.170 Table 6.3: Standard deviation 65 85 ≈ 8.101 312 83 ≈ 17.764 |m| = 3 220 12 ≈ 14.849 r 2 for diﬀerent eigenstates. The simulations for the two-dimensional hydrogen atom is very similar to the particle-in-box-simulations. However, in this system we do not have a bounded geometry, hence we must truncate it. The obvious ﬁrst-choice is to create a rotationally symmetric domain, i.e., a disk with radius r0 . However, this limits the element type to linear triangles due to constraints in Diﬀpack’s grid preprocessors. We will use triangulated disk grids for our simulations in this section. The grid generation utility from the particle-in-box simulations had to be replaced, due to apparent instabilities of the algorithm. Nevertheless, the grids are parameterized by the radius r0 and the typical element size h. To sum up, we have nel = Ã = 1 (n − 1)2 2π πr 2 = h2 , nel where n is the number of line segments used to approximate the boundary of the disk. To keep the number of elements ﬁxed but increase the accuracy near the origin we may introduce a grading to our grid. The grading is a transformation of the nodes in the grid given by [k] w−1 x x [k] . x [k] → r0 The eﬀect of this mapping is to change the length of each point x into x w and then rescale such that the size r0 of the grid is conserved. With w > 1 this will create a grid with smaller triangles near the origin. The parameter must be chosen with some care, however, as a value which is too large will stretch some triangles and produce undesiredly small angles. (This may introduce round-oﬀ errors in the computations.) Notice that grading with this procedure keeps h ﬁxed as the average element area is preserved.7 Importance of grading. To illustrate the importance of the mesh width near the origin we present simulations with r0 = 20, h = 0.2 and with diﬀerent grading parameters, viz., w ∈ {1.0, 1.5, 1.8, 2.0, 2.5}. The grid had nno = 1275 nodes (of which 1174 remained after incorporating boundary conditions) and nel = 2447 triangular elements. Fig. 6.13 shows the ﬁrst few eigenvalues for each simulation compared to the analytic eigenvalues. We have used a lumped mass matrix in each simulation. It is worthwhile to mention that the simulation time increased with increasing w, even though the dimension of the system was the same in each simulation. The simulation with w = 2.0 took 13 times as long time to ﬁnish as the homogenous grid, i.e., w = 1.0, while w = 1.8 took 6.5 times as long. 7 More sophisticated methods for grading and/or reﬁning the grid can be used as well, such as specifying each element’s desired in a second pass of grid generation. This is out of scope for this thesis. See for example Ref. [46]. 144 6.6 – Numerical Experiments 0.5 linear behaviour 0 -0.5 εn(k)m(k) -1 faithful reproduction of εnm -1.5 -2 -2.5 -3 analytic w=1.0 w=1.5 w=1.8 w=2.0 -3.5 -4 0 5 10 15 20 k 25 30 35 40 Figure 6.13: Eigenvalues for grids with diﬀerent grading factors w. Parameters for grid: r0 = 20, h = 0.2. The Numerical Eigenfunctions. Fig. 6.13 shows the numerical eigenvalues together with the analytic eigenvalues. Clearly, up to a certain k, the qualitative behavior is reproduced. At some point the numerical eigenvalues seems to grow linearly instead of the expected 1/n(k)2 behavior. Intuitively, the eigenfunctions of such high states must analytically go beyond the limits of the disk, i.e., σ > r0 , and this is the reason for the wrong behavior of the eigenvalues. For high energies the particle-in-box features of the eﬀective potential (i.e., Coulomb attraction plus inﬁnite box potential) dominate. Fig. 6.14 shows plots of a few eigenstates (the real parts) for the grid with parameters (r = 120, h = 2). By looking at the imaginary parts (not shown here) they are easily seen to be proportional to the real parts, hence the eigenfunctions found by ARPACK can be written as eiα Ψ(x ), with α ∈ R and Ψ a real function. When solving the eigenvalue problem analytically one typically uses separation of variables, i.e., Ψnm = eimφ Rnm (r). The radial function R(r) is a solution of Eqn. (6.3), and it depends on both the quantum number n and the magnitude |m|. For a given n we have an eigenspace of dimension 2n − 1 for which we may ﬁnd an orthonormal basis, and indeed the functions eimφ Rnm (r) are orthogonal. The IRAM also ﬁnds orthogonal eigenvectors, but this time discrete eigenvectors. Incidentally, it ﬁnds something very similar to what we ﬁnd when using separation of variables in this case. Consider the three eigenfunctions that clearly correspond to 2 ≈ −0.4444. We see that the ﬁrst one is approximately independent of φ, hence it corresponds to Ψ20 . The two next are clearly a radial function modulated with a periodic function approximately equal to cos φ and sin φ. We obtain these by taking linear combinations of eiφ and e−iφ , viz., 2 cos φR(r) = (eiφ + e−iφ )R(r), 2i sin φ = (eiφ − e−iφ )R(r). In other words, ∝ Ψ2,−1 + Ψ2,1 Ψnum 3 145 Quantum Mechanical Eigenvalue Problems ⑀=-3.383182 ⑀=-0.407101 ⑀=-0.425269 ⑀=-0.426326 ⑀=-0.146063 ⑀=-0.155577 ⑀=-0.155459 ⑀=-0.151320 ⑀=-0.150936 ⑀=-0.068412 ⑀=-0.068736 ⑀=-0.050594 Figure 6.14: Eigenstates found with r0 = 20, h = 0.2, w = 2.0. Gray tones indicate low values (white) and high values (black). Lowest-lying states are upper left. 146 6.7 – Strong Field Limit and ∝ Ψ2,−1 − Ψ2,1 . Ψnum 4 In the same manner we proceed with the ﬁve graphs with ≈ −0.15, i.e., they corresponds to n = 3. Again, one of these is independent of φ and corresponds to m = 0. Two functions are proportional to sin φ and cos φ, and the last two are proportional to cos 2φ and sin 2φ, corresponding to m = 2. In addition we see that the radial functions R30 , R31 and R32 are diﬀerent, as expected from the radial equation’s dependence on |m|. We must emphasize that this structure of the eigenfunctions are partially a coincidence, probably due to the approximate rotational symmetry of the grid. There is no other reason why ARPACK should not have chosen a diﬀerent orthogonal basis for the eigenspace corresponding to n . Furthermore, the eigenvalues are only approximately degenerate in the numerical case. Increasing the Grid Size. It was mentioned that the eigenvalues tend to grow linearly when k > 10. We can ﬁnd an explanation in terms of the eigenfunction plots. For the states with the lowest eigenvalues the dominating features are near the origin. For higher-lying states we see that the features approach the rim of the disk. Therefore the particle-in-box part of the numerical model starts to show up in both the eigenvalues and eigenfunctions. Already for n = 4 we have σnm ∼ r0 and hence we are losing features of the functions. To study this behavior in more detail simulations with increasing disk sizes were carried out. The grids had radii given by r0 ∈ {20, 30, 40, 50}, and they all had h = 0.2 and w = 1.5. As seen in Fig. 6.16 the eigenvalues get progressively better as we increase the grid size, also for the lowest-lying states. Physically, this corresponds to the slow dying out of the Coulomb potential. Fig. 6.15 shows the 10th eigenstate for r = 50. If we compare this to the corresponding plot in Fig. 6.14 we see that more of the features are captured by this large grid. 6.7 Strong Field Limit We have studied the eigenvalue problem for the weak-ﬁeld limit, i.e., γ = 0. We now turn to the limit in which the Coulomb interaction is treated as a perturbation, i.e., we turn it oﬀ and set γ = 1 in order to study the numerical properties of the eigenvalue problem. If we ﬁnd a common grid in which both the weak-ﬁeld limit (i.e., the pure Coulomb system) and the strong-ﬁeld limit (i.e., the pure magnetic system with γ = 1) yield results that agree with analytic results, it is reasonable to believe that also the combined system (i.e., Coulomb attraction and magnetic ﬁeld) will be solved with accuracy. Our goal in this thesis is not to achieve high-precision results in this respect. In order to achieve this we must perform simulations that are too heavy for the resources available at this stage in the work. Instead we aim at a good understanding of the behavior of the numerical problem for diﬀerent grid parameters and physical parameters. When more resources are available we can solve the eigenvalue problem with higher accuracy. The Hamiltonian for the pure magnetic system reads H = (−i∇ + Asymm )2 = −∇2 − iγ ∂ γ2 + r2 . ∂φ 4 147 Quantum Mechanical Eigenvalue Problems ⑀=-0.080439 50 40 30 20 10 0 -10 -20 -30 -40 -50 -50 -40 -30 -20 -10 0 10 20 30 40 50 Figure 6.15: Sample eigenstate of a simulation with r0 = 50, h = 0.2, w = 1.5. Grid size from Fig. 6.14 shown for reference. 0.5 0 0.9 -0.5 0.8 0.7 -1 0.6 ε n(k)m(k) 0.5 -1.5 0.4 0.3 0.2 -2 0.1 0 -2.5 -0.1 30 40 50 60 70 80 100 analytic r=20 r=30 r=40 r=50 r=60 -3 -3.5 -4 90 0 5 10 15 k 20 25 30 Figure 6.16: The eigenvalues for successively larger grids. Here, h = 0.2, w = 1.5. The inset graph shows the continuation of the larger. 148 6.7 – Strong Field Limit 7 6 5 εk 4 3 2 r=30, h=0.2, w=1.5 r=40, h=0.2, w=1.5 1 0 r=30, h=0.1, w=1.5 0 10 20 30 40 50 k 60 70 80 90 100 Figure 6.17: Eigenvalues for thee simulations of the magnetic Hamiltonian The eigenvalues (called the Landau levels) are inﬁnitely degenerate if the domain is R3 . The symmetric gauge is employed here. Recall that any other gauge will give diﬀerent eigenfunctions but the same eigenvalues. Recall that the degeneracy of each landau level if the domain is a disk with radius r0 such as in our simulations is g= r02 . 2 (6.11) There are two important facts to be aware of. First, we do not know which basis of eigenvectors our numerical method will choose for us. (As we shall see it is a quite interesting basis.) Second, we do not know if the numerical methods are indeed gaugeinvariant such as the original problem. If we diagonalize in a diﬀerent gauge, will we ﬁnd the same eigenvalues? We will exclusively use the symmetric gauge in our simulations. First we consider two simulations with r0 = 30, h = 0.2, w = 1.0 and 1.5, and compare the results. The eigenvalues are shown in Fig. 6.17. Clearly, the graded grid yielded better eigenvalues than without grading. Next, we do a simulation with r0 = 30, h = 0.1, w = 1.5, (6.12) i.e., we reduce the mesh width. The eigenvalues are shown in Fig. 6.17. Clearly, reducing the mesh width has a dramatic impact on the approximate degeneracy of the lowest Landau level. We also see the eﬀect on the energy levels that correspond to 2N + 1. Analytically, the eigenvalue = 1 is multiply degenerate. Numerically, the eigenvalues starts out close to 1 but rapidly increase. Clearly, a smaller mesh width enhances the convergence of the lowest eigenvalues. It is intuitively clear that all g = r02 /2 states for every Landau level will not be found with a ﬁnite mesh width, as the number of states equals the number of internal grid points in the mesh which is perhaps much less than g to start with. It also becomes clear that decreasing the mesh width should increase the approximate degeneracy of the lowest Landau level in particular. 149 Quantum Mechanical Eigenvalue Problems 40 30 30 20 20 10 10 0 0 -10 -10 -20 -20 -30 -30 -40 -40 -30 -20 -10 0 ⌿*⌿ ⑀17=2.094214 Re(⌿) 40 10 20 30 40 -40 -40 -30 -20 -10 0 10 20 30 40 Figure 6.18: The 17th eigenfunction of the Landau Hamiltonian in the symmetric gauge. Light colors are low values and black colors are high values. We notice that large fractions of the numerical eigenvalues tend to cluster around the analytical eigenvalues 2N +1, creating a (crude) approximation to a ladder function. In fact, as h → 0 one expects that the function k converges to (k) = 2k/g + 1, representing the exact eigenvalues although we have no proof of this. Recall that for the hydrogen atom ARPACK tended to ﬁnd eigenfunctions that were closely related to the eigenfunctions found analytically by separation of variables. This is also the case in the Landau system and in a both interesting and amusing way. Fig. 6.18 shows the state Ψ16 from the simulation with h = 0.1. It belongs to the lowest Landau level as seen from Fig. 6.17. First of all the quantum number m = 16 is readily identiﬁed if we assume Ψ0M =√eimφ R0M (r). Second, notice the localization of the probability density around r16 = 2 · 16 ≈ 5.6. This state is qualitatively representable for the eigenfunctions found for the lowest Landau level. The angular quantum number m is readily identiﬁed and the probability is concentrated in rings. As the momentum is proportional to the gradient of Ψ, it is easily seen that in this case it is tangent to the ring at the centre. (At the edges of the ring the gradient points in a slightly diﬀerent direction.) Hence the probability density corresponds to that for a particle moving in a circular path. To sum up, ARPACK ﬁnds a basis that is “identical” to the analytical basis, and in addition we can see the correspondence principle in work from the numerical results! 6.8 Intermediate Magnetic Fields The ﬁnal experiment is a series of simulations with a constant grid but with varying magnetic ﬁeld γ. Both the Landau level simulations in the previous section and the pure hydrogen atom simulations yielded qualitatively correct eigenvalues for the lowest-lying states. As we have pointed out it is therefore natural to expect that with an intermediate magnetic ﬁeld γ ∈ [0, 1] we will also obtain qualitatively correct eigenvalues and eigenfunctions. More information on the intermediate magnetic ﬁeld regime can be found in Refs. [29, 30]. We shall not perform a thorough analysis. The simulation data is too coarse to ﬁnd numerical values that are very interesting to present. The simulation produces 150 6.9 – Discussion and Further Applications 1 0.5 0 energy levels -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ␥ Figure 6.19: The energy levels of the hydrogen atom for intermediate magnetic ﬁelds however qualitatively very interesting results. We will reuse the grid from Eqn. (6.12) for our simulations. We use a magnetic ﬁeld with values γ ∈ {0, 0.05, 0.1, . . . 0.95, 1.0}. Fig. 6.19 shows the eight lowest energy levels as function of the magnetic ﬁeld. To produce the ﬁgure the eigenvalues from each simulation were sorted before they were plotted. We easily recognize the degenerate energy levels at γ = 0, and as the ﬁeld is turned on, the degeneracy is broken, producing fork-like structures in the graph. We can also easily recognize crossings of the eigenvalues. As argued in Ref. [29] there must be inﬁnitely many crossings. Furthermore, the eigenvalues accumulate at the Landau levels 2γ(N + 1/2). Intuitively this is so because the Landau levels are inﬁnitely degenerate, needing inﬁnitely many hydrogen eigenvalues that are ﬁnitely degenerate to converge. For example, we see a crossing around γ = 0.15 of the 2,−1 and 3,−2 hydrogen levels. Here, we interpret the curves in terms of the states Ψnm found with separation of variables, and use the well-known fact that it is indeed these states that wander to the split energy levels, see Ref. [29] for the perturbative treatment of the magnetic ﬁeld. We also see hints of crossings with curves not shown in the graph, such as the abrupt change of slope of the 2,0 hydrogen state around γ = 0.6. 6.9 Discussion and Further Applications Was This a Good Idea? We have spent quite a lot of time ﬁnding eigenfunctions and eigenvalues of the two-dimensional hydrogen atom with numerical methods that are quite complicated. We could have found much better eigenvalues and eigenfunctions by concentrating on the radial equation instead of the full Hamiltonian. However, when introducing more complicated Hamiltonians we can no longer use the separation of variables, and the full Hamiltonian must be considered anyway. For example, consider 151 Quantum Mechanical Eigenvalue Problems a two-dimensional non-isotropic harmonic oscillator potential with an applied magnetic ﬁeld. This system has no rotational symmetry, and hence techniques such as separation of variables are no longer applicable. Furthermore, we have gained a thorough insight into the eigenvalue problem in the ﬁnite element context. Besides its value as a catalyst for insight into the numerical methods, information can be extrapolated to time dependent problems as the quality of eigenvalues have a profound eﬀect on the quality of the time dependent simulations. Time Evolution of Time-Independent Systems. If the system under consideration is independent of time, i.e., ∂H ≡ 0, ∂t we know from section 1.4 that the solution to the time-dependent Schrödinger equation is easy to calculate: 1. Diagonalize Hh , i.e., ﬁnd the k (where k N is a possibility) lowest eigenvalues and their corresponding eigenvectors, viz., Hh Φ n = E h Φ n , n = 1, 2, . . . k. 2. Pick an initial condition Ψ(0) assumed to be suﬃciently smooth, i.e., Ψ(0) ≈ k cn (0)Φn . n=1 The coeﬃcients are calculated by (Φn , Ψ(0)) cn (0) = . (Φn , Φn ) 3. To ﬁnd Ψ(t), form Ψ(t) ≈ k e−iEn t/ cn (0)Φn . n=1 This can be used to calculate the time development of the initial state to the ﬁnal state. If the initial state is equal to the superposition of the k ﬁrst eigenstates the evolution is perfect. The cost of diagonalizing Hh may however be much more than using for example the leap-frog scheme or the theta rule. Then again, these stepping methods introduce numerical errors in the phases of the various components. If the wave function is desired at very many instants the cost of time-stepping may be cheaper or more expensive than simply forming the superposition in point 3. Forming the linear combination of eigenvectors requires O(kN ) operations. A time stepping procedure typically includes a forward or backsubstitution of an LU factorized matrix (costing O(N 2 ) operations) or using an iterative method for solving the linear systems at hand (with a priori unknown price.) Consequences for the Time Dependent Problems. In addition to providing a means for analyzing probabilities for excitations in time dependent problems, knowledge of the eigenvalues and eigenvectors of the numerical Hamiltonian can tell us a lot about the development of an arbitrary chosen initial state. Such a state can be written as a linear combination of the eigenvectors. Let us assume that the Hamiltonian is independent of time. In that case, it was explained in section 1.4 that solving the time dependent Schrödinger equation was equivalent to diagonalizing the Hamiltonian. 152 6.9 – Discussion and Further Applications Assuming that we can solve the spatially discretized Schrödinger equation exactly, i.e., that we can solve ẏ = −iHh y, we know that its solution operator is given by U(t) = e−itHh . Expanding y in the eigenvectors φn of Hh , i.e., the discrete eigenvectors of H, viz., y(0) = c n φn , n then yields y(t) = U(t)y(0) = cn e−itn φn . n For the particle in box we know that linear elements reproduce the exact eigenvectors. It is reasonable to believe that with quadratic (or higher-order) elements the eigenvectors are still (or very close to) exact. Hence, for the time development it is easy to see that the better the eigenvalues we have obtained, the better the time development of a wave packet will be. If our ODE solving algorithm is exact, or close to exact, it is clear that the eigenvalues of Hh will enter the numerical solution operator of the ODE. For example, the Crank-Nicholson scheme has a solution operator U∆ explicitly given by the eigenvectors and eigenvalues of Hh ; see section 1.4. The better eigenvalues we obtain, the better will the solution of the time dependent Schrödinger equation look as well. This gives us a means for predicting whether or not a spatial discretization of some system is reasonable or not. Thus, if we ﬁnd that the eigenvalues of, say, the twodimensional hydrogen atom is way-oﬀ the analytic eigenvalues, we must think twice before we start solving time dependent problems with this discretization. Doing Large-Scale Simulations. The simulations in this chapter were done on various computers, but in general they were ordinary workstations that happened to be available at the time with moderate CPU speed and amounts of memory. For example, the hydrogen simulations were performed on a machine running Debian Linux with 1 GB of memory and a 2.4 GHz Intel Pentium 4 processor. The simulation times ranged from a few minutes to two hours, and the simulation time of course increases with the dimension of the matrix. The increase is not optimal (i.e., linear) but neither is it quadratic. (Detailed results are not sought in this thesis.) For realistic simulations however, we will typically use a cluster with dozens of CPUs and lots of memory. Both Diﬀpack and ARPACK has parallelization support and this is (not only in principle) possible to take advantage of, even though it requires some details in the implementation area that are not yet sorted out. Simulating for example the eigenvalues of a hydrogen atom with an arbitrary magnetic ﬁeld with a grid with for example 106 grid points is then easily done, yielding results superior to those presented in this text. Of course, the various details of memory and storage requirements, the expected accuracy from such simulations and so on must be investigated before embarking on such a mission. Quantum Chaos. Quantum chaos is the study of quantum mechanical systems corresponding to classical systems whose trajectories are chaotic, e.g., a double pendulum, three-body systems, particle in a stadium-shaped box, et.c. In section 15.6 of Ref. [9] eigenstates of the latter system is depicted, and perhaps the most characteristic feature is the so-called scarring of the highly-excited states. The classically periodic paths, 153 Quantum Mechanical Eigenvalue Problems which are unstable in the sense of Lyapunov, “shine through” the otherwise noisy structure of the eigenstates. This is obviously connected with the correspondence principle, but the fundamental mechanisms are not fully understood at the time of writing. When we studied the strong-ﬁeld limit of the two-dimensional hydrogen atom with an applied magnetic ﬁeld we saw the correspondence principle in work as the classically periodic trajectories were seen in the quantum mechanical eigenstates. The idea is then to use the techniques presented here for arbitrary systems to produce scarred eigenstates of high energy. In addition one may do statistical computations on the eigenvalues, e.g., as function of mesh width. To what extent the ﬁnite element method has been applied to such systems I have not investigated, but it is on the other hand clear that rich insight into both the understanding of the ﬁnite element method and the spectrum and eigenfunctions of the ﬁnite element matrices is possible to gain. Gauge Invariance. We know that quantum physics is invariant with respect to gauge transformations as described in section 1.6. What we have not studied up to this point is whether or not the numerical methods are gauge invariant or not. By gauge invariance of the numerical methods in the eigenvalue context we mean that the eigenvalues are left unchanged under a gauge transformation and that the eigenfunctions are given by a unitary change of basis similar to that of the continuous problem. Our HydroEigen simulator class actually implements a non-symmetric gauge in addition to the symmetric gauge used in the simulations in this chapter. Due to time limitations no simulations worth mentioning has been done for this thesis. Gauge invariance of numerical methods in general is a very interesting subject to do further studies on. First of all, it has a profound impact on our understanding of numerical methods. Second, it will automatically yield valuable conﬁrmative or dismissive information on the quality of the results of a numerical experiment. If the methods are known to be gauge invariant up to a term proportional to for example O(∆t4 ), we know that it is much less than the error from a simulation with the leap-frog scheme or the split-operator method used in section 4.3. Improvement of the Program and the Methods. Even though we have used the ﬁnite element method quite generally, the complete picture has not been drawn. So-called adaptive ﬁnite element methods may be used to successively improve the grid based on local a posteriori error estimates. Such estimates must be derived for each PDE in question and allows for estimating the error in for example the norm of the error over each element. If the error is above some threshold one reﬁnes the grid around the element and performs a new simulation. In Ref. [49] such a technique for the Schrödinger equation is presented. Adaptive ﬁnite element methods are treated to some extent in Ref. [34]. One can imagine that adaptive FEM can be used to ﬁnd a suitable grid for the lowest-lying states of the physical problem, and then use this grid in a time-dependent simulation. As for the program, many improvements can be done. Besides cosmetic changes, a more robust handling of matrix types would be valuable. As for now, only MatSparse and MatBand are supported, and this only in a limited manner. Not all storage options inside the matrices are handled properly. A complete and robust eigenvalue program for the Hamiltonian could also serve as a starting point for more general eigenvalue problem implementations in Diﬀpack, as the built-in support for this is virtually nonexistent at the time of writing. 154 Chapter 7 Solving the Time Dependent Schrödinger Equation 7.1 Physical System We will implement the solver for a single charged particle in an attractive Coulomb potential with an applied time-dependent magnetic ﬁeld. We will choose a simple model for an ultra-short laser pulse with amplitude γ0 , viz., πt 2 γ(t) = γ0 sin cos[2πω(t − T /2 + δ)]. (7.1) T Here, ω is the frequency of the laser and δ is a phase shift. The envelope rises from zero to γ0 at t = T /2 and falls back again to zero at t = T , i.e., at the end of the simulation. Such laser pulses are actually possible to create experimentally, see Ref. [50]. Usually one considers very long pulses in which ω 1/T . With this model perturbative methods may yield very accurate answers, but when ωT ∼ 1 we are outside the regime for such treatment. The physical system is also possible to create in a laboratory, and variants include electrons trapped in anharmonic oscillator potentials, ions in Paul traps, et.c., see Ref. [51]. Therefore it poses an interesting starting point for doing simulations. Furthermore, simpler systems such as the free particle can be created as special cases in the program by turning on and oﬀ the various parameters available. In this thesis we will only consider this magnetic ﬁeld as a toy model with which we test our simulator. We will implement the possibility of either using an eigenfunction of the Hamiltonian (typically with γ = 0) or a Gaussian wave packet as initial condition. This is described further in the next section. We have to make a comment on the Hamiltonian. The vector potential A in both the symmetric and non-symmetric gauges is obtained via the dipole-approximation, i.e., one assumes that the electromagnetic ﬁelds depend only on time. This is reasonable if one imagines a very distant source of radiation. However, as was pointed out on page 54, we are actually neglecting Maxwell’s equations. A spatially independent magnetic (or electric ﬁeld) must be a constant ﬁeld in order to be consistent. We should have an extra term proportional to the scalar potential in the Hamiltonian. The electric ﬁeld is given by 1 t dτ A(τ ), E= c but this is easily seen to be spatially dependent! It is also easy to see that the physical electric ﬁeld becomes very diﬀerent in the two gauges. 155 Solving the Time Dependent Schrödinger Equation We will ignore the diﬃculties in this chapter, and simply note that introducing rigor to this topic is an interesting extension of the discussion. 7.2 The Implementation The implementation of the time dependent solver (class TimeSolver) is derived from class HydroEigen. There are several advantages of reusing the code from the eigenvalue solver. First, many features of the problems are common, e.g., the Hamiltonian and the ﬁnite element assembly process. This reduces the need for code. Indeed, the source code for the time dependent solver is a great deal shorter than the HydroEigen class deﬁnition even though one hardly can say that the problem is less complex for that reason. Second, the ﬁnal application may be used to solve both the eigenvalue problem and the time dependent problem. It is also easier to keep the interfaces common for the two simulation types, which is highly desirable. Unfortunately some loss of eﬃciency is inevitable with this approach. 7.2.1 Time Stepping Although we have described the numerical methods in details elsewhere, we here present in explicit terms the updating algorithms for the numerical wave functions in the leapfrog and theta-rule methods. The Leap-Frog Scheme. form reads The updating rule for the leap-frog scheme (4.27) in matrix u+1 = u−1 − 2i∆tM −1 H u , where H is the ﬁnite element Hamiltonian used in the eigenvalue solver and where M is the mass matrix; a positive deﬁnite and symmetric matrix. The strategy is to build a linear system M v = b at each time step, where b = H u . The solution at the next time step then becomes u+1 = u−1 − 2i∆tv. The linear system becomes very easy to solve explicitly if we lump the mass matrix. This is taken advantage of in the implementation with a special hand-coded diagonal solver. Given the initial condition u0 (which may be a Gaussian wave packet or an eigenstate of the Hamiltonian) we need a special rule to ﬁnd u1 . This special rule is simply a single iteration with the theta-rule. The implementational details of building the linear system is somewhat complicated due to the structure of Diﬀpack’s solution algorithms and the wish for reusing the code for calculating the Hamiltonian from class HydroEigen. The resulting code should be quite eﬀective, though. The overhead of extra function calls does not dominate the time spent in the new element assembly routine TimeSolver::makeSystem(). The Theta-Rule. In matrix form the updating rule (4.19) for the theta-rule reads u+1 = [M + iθ∆tH +1 ]−1 [M − i(1 − θ)∆tH ]u . If we deﬁne b = [M − i(1 − θ)∆tH ]u , our new wave function becomes the solution to the linear system Au = b, with A = M + iθ∆tH +1 . 156 7.2 – The Implementation The theta-rule is somewhat trickier than the leap-frog scheme to implement in an eﬃcient manner. This is due to the fact that the Hamiltonian at diﬀerent time levels is needed on the left hand side and the right hand side of the linear system. We will always use θ = 1/2 as we know this is the best choice. For this reason we refer to the Crank-Nicholson method instead of the theta-rule in the following. Solving Linear Systems. For the time dependent solver the total time spent at the assembly process can be considerable. The linear systems must also be solved in an eﬃcient manner as we deal with matrices of potentially very large sizes. It is obvious for two reasons that we must avoid the use of banded matrices. First, banded matrices store a lot of zeroes for very sparse systems, as can be seen in Fig. 5.1. Second, the linear solvers are very slow with banded matrices due to numerous superﬂuous multiplications and additions with zeroes. For our simulations we will therefore exclusively use iterative methods and sparse matrices. For the simulations in this chapter we stick to the Generalized Minimal Residual method (GMRES in Diﬀpack, see Ref. [34]). It performs well on both the types of linear equations we deal with. Keeping Track of the Solution. For the purpose of analyzing data from the simulation, the log ﬁle is written as a Matlab script. The script deﬁnes arrays and ﬁlls them with simulation data, such as the time (the t array), the square norm of the solution (the u norm array) and so on. The arrays are located in a struct variable casename data. Some of the variables and arrays that are deﬁned are: – casename: The case name for the simulation. – n rep: The number of reports. Equals the number of rows in each array. – t: The time array. – gamma: The magnetic ﬁeld array. – dt: The time step. – theta: The parameter for the theta rule. – gamma0, omega and delta: The parameters for the magnetic ﬁeld. – gaussian params: A string with the parameters for the Gaussian initial condition. – u norm: Array with the norm of the solution. – u energy: Array with the energy of the solution, i.e., the expectation value of the Hamiltonian. – u pos: Array with the expectation value of the position. The ﬁrst column is the x-coordinate and the second column is the y-coordinate. – system time: An array with the time for each assembly of the linear system. – solve time: The time for each solution of a linear system. (Not reported if the leap-frog method is used with a lumped matrix.) – solve niter: The number of iterations taken for the solution to converge. – solve total and system total: The total time for the solution of linear systems and the assembly of the systems, respectively. 157 Solving the Time Dependent Schrödinger Equation 7.2.2 Description of the Member Functions Here we describe the most important new methods of class TimeSolver. Many member functions are inherited from class HydroEigen and therefore not described. define(). The menu system is extended with several parameters. The menu items from class HydroEigen such as the parameters for turning on and oﬀ terms in the Hamiltonian are kept. All the new parameters except for simulation in time are added inside a submenu to distinguish them from the eigenvalue problem parameters. The submenu can be reached with the command ‘sub time’. – simulation in time This boolean parameter selects the time dependent solver if its value is true and solves an eigenvalue problem otherwise. – time method This parameter is used to select the time integration method. The only legal values are the strings theta-rule and leap-frog. – T This is the time at which the simulation is to be stopped. In other words the simulation is performed for t ∈ [0, T]. – dt This is the time step. Thus, about T/dt steps are taken in the simulation. – n rep This parameter is an integer signifying how many reports are desired throughout the simulation. Thus, at the end of every time interval of length T/n rep ﬁelds and various physical quantities are written to the ﬁeld database and the log ﬁle. A report is automatically issued before any time integration is done, making the total number of reports n rep + 1. – theta This is the parameter for the theta-rule. – gamma0, delta and omega These are the parameters for the time dependent magnetic ﬁeld in Eqn. (7.1). – ic type Two kinds of initial conditions are implemented. If ic type is set to gaussian, a Gaussian wave packet will be used. If ic type is equal to field, a ﬁeld is read from a database and used as initial condition. – field database This is the casename of a previously stored set of eigenvectors from an eigenvalue simulation. The ﬁelds are read with the method loadFields() and overwrites any grid deﬁned in gridfile, see section 6.5. If the value of the parameter is none, no ﬁeld database will be loaded. – ic field no This integer is the index of the ﬁeld to use as initial condition. If the index is larger than the number of ﬁelds in the field database, the ground state, i.e., ﬁeld number one, is automatically used. – gaussian This is a string with parameters for the Gaussian initial condition. It is a string with switches such as -x followed by a real number. The default value of gaussian is ‘-x -10 -y 0 -sx 2 -sy 2 -kx 0 -ky 1’. The switches -x and -y sets the mean position, the switches -sx and -sy sets the standard deviations in the x and y directions, and ﬁnally -kx and -ky sets the mean momentum. fillEssBC(). This method imposes the essential boundary conditions, i.e., the homogenous Dirichlet conditions. It loops through the grid nodes and gives the DegFreeFE *dof object information on where to prescribe zeroes. This object is then responsible for altering the element matrix before any linear system of equations is solved. 158 7.2 – The Implementation setIC(). This method sets the initial condition according to the choices in the menu. The two ﬁelds u and u prev are set equal to a Gaussian or an eigenvector from the field database. The two ﬁelds represent the current solution (u) and the previous solution (u prev). In the theta-rule the u prev ﬁeld is not really necessary to use because we only need the current solution to ﬁnd the next. scan(). The scan() function is updated heavily, with support for reading previously stored ﬁelds from a simres database. It is important to notice that if such a database is loaded, the grid deﬁned with the gridfile menu entry is overruled with the grid stored in the database. The scan() function also allocates memory for various objects, such as scratch vectors needed in the time integration. In other respects the function is similar to the scan() function in class HydroEigen. gammaFunc(). as parameter. This method updates the magnetic ﬁeld according to the time passed calcExpectationFromMatrix(). This function calculates the matrix element of a Matrix(NUMT) object with respect to two given ﬁelds. This is particularly useful if one wants to calculate the expectation value of observables available as matrices, e.g., the Hamiltonian.1 calcExpectation pos(). It is useful to be able to calculate the mean position of the wave function during the course of a simulation. This is the purpose of this method which returns a Ptv(real) object holding x and y. loadFields(). This function reads ﬁelds stored in a simres database. The ﬁelds are assumed to have been stored by HydroEigen as the method selects only ﬁelds with an integer as label. If ﬁelds are actually read, the GridFE object in the simulator is overwritten with the grid from the ﬁelds. integrands(). The new integrands() method calculates the integrand used in the assembly of the left and right hand side of the linear equations. It can also compute the Hamiltonian; this is necessary if one wants to solve an eigenvalue problem instead of a time dependent problem. We will not go into details on the integrands() method; the code should contain more than enough comments. We mention though that the use of the local-to-global mapping of nodes q(e, r) is central when calculating the right hand side terms. This mapping can be found in the (public) member VecSimple(int) loc2glob u of class ElmMatVec. This class contains an n × n matrix Mat(NUMT) A and an n-dimensional vector Vec(NUMT) b, where n is the number of nodes in the current element. These are the local element matrix and element vector, respectively, and in the assembly process each term Ae and be of Eqn. (4.16) is given in terms of these smaller matrices and the q(e, r) mapping, in a way generalizing the one-dimensional example from section 4.4.2. solveProblem(). This is a central class member function. It ﬁrst checks whether a time dependent simulation actually is wanted (and branches to HydroEigen::solveProblem() if not) and then loops through the desired time levels. 1 The matrix used is the Hamiltonian modiﬁed due to essential boundary conditions and then symmetrized, see Ref. [34]. If this matrix has the same expectation value as the original matrix when we know that the BCs are fulﬁlled is unclear at the moment, but when the wave function is essentially zero near the boundary we may assume that it is very close. 159 Solving the Time Dependent Schrödinger Equation reportAtThisTimeStep(). During the simulation it is desirable that the simulator provides some quantitative results, such as the norm of the solution, the mean position, the energy and so on. This is provided by the reportAtThisTimeStep() method. It combines dumping the solution u and the corresponding probability density ﬁeld to the simres database (named with casename ; see the description of class HydroEigen) with writing statistical data to stderr and to caseneme.log in Matlab format. 7.2.3 Comments More on the Initial Conditions. The wave packet used for initial conditions is implemented as a subclass GaussFunc of the Diﬀpack class FieldFunc, i.e., a functor that encapsulates functions that can be evaluated at a space-time point. For example, a FieldFE object such as the current solution u can be initialized to a Gaussian with this code: GaussFunc gaussian; // create functor gaussian.scan(menu.get("gaussian")); // init functor u.fill(gaussian); // fill u The mathematical expression used for the nodal values of the Gaussian is (x[j] − x0 )2 (y [j] − y0 )2 [j] Uj = exp − − + ik · x . 2σx2 2σy2 The parameters x0 , y0 , σx and σy correspond to the switches -x, -y, -sx and -sy of the initialization string, respectively. The momentum k corresponds to -kx and -ky. Numerical normalization is provided by the setIC() method and calcInnerProd(). As described above one can use a ﬁeld read from a ﬁeld database as initial condition as well as a Gaussian wave packet. In the eigenvalue solver class HydroEigen the ﬁelds are stored and labeled with the integer index k from the eigenvalue k of the ﬁeld. (The corresponding probability densities are labeled prob k .) The loadFields() method reads the ﬁelds with integer labels only and stores them in an array of Handle(FieldFE) objects. The ﬁeld objects are useful when one wants to compute the spectrum of a numerical state, i.e., the overlap with the eigenstates of the (stationary) Hamiltonian. This can be done easily with HydroEigen::calcInnerProduct(). Analytical Integration. It is claimed in this thesis that Diﬀpack solely relies on numerical integration to compute the systems of linear equations. This is not entirely true as one can override the FEM::calcElmMatVec() method with another one which actually computes the element vectors and matrices analytically. For the Coulomb interaction term this is actually possible, and we may avoid any numerical inaccuracy due to the limited order of the Gaussian quadrature. This is not investigated any further in this thesis, but might be interesting in a future project. Obtaining a Finite Diﬀerence Scheme. We may obtain the standard ﬁnite diﬀerence scheme (at least for some classes of Hamiltonians) with the ﬁnite element formulation and a special set of parameters. More speciﬁcally, we use a uniform grid, linear elements and simple nodal point numerical integration. Let us demonstrate this for a onedimensional example. Let h be the grid spacing. Let there be m nodes such that Ω = [0, (m − 1)h] and the grid points are given by xk = (k − 1)h. The nodal-point integration is deﬁned by f (x) dx ≈ h Ω 160 m k=1 f (xk ). 7.3 – Numerical Experiments The mass matrix becomes Mij = h Ni (xk )Nj (xk ) = h k δik δkj = hδij , k i.e., we obtain the lumped mass matrix. For an arbitrary operator V (x) we similarly obtain the matrix Vij = hδij V (xi ), i.e., a diagonal operator. Notice that the nodal point integration corresponds to zeroeth order Gaussian quadrature in local coordinates. Hence, it integrates constant functions analytically. The derivatives Ni (x) are constant over each element. Therefore, the stiﬀness matrix is integrated analytically, and yields the standard ﬁnite diﬀerence operator (multiplied by h), viz., 2 −1 . Kii = , Ki,i±1 = h h Thereby, any Hamiltonian on the form H =− ∂2 + V (x) ∂x2 is discretized with a scheme identical to the standard ﬁnite diﬀerence scheme. The argument generalizes to more dimensions, but we must be careful with the angular momentum operator and other ﬁrst order diﬀerential operators which do not ﬁt so easily in. Anyway we can consider the ﬁnite element method with linear elements and nodal point integration (and a lumped mass matrix) as corresponding to a (modiﬁed) ﬁnite diﬀerence scheme. This rule also in some sense generalizes the ﬁnite diﬀerence concept if we include non-uniform meshes and perhaps triangular elements as well. 7.3 Numerical Experiments In this section we do a collection of numerical experiments with various settings to ﬁgure out some properties of the numerical methods. We do not focus on physics right now, because it is more important to know whether our methods work or not and what methods are best to use. We will use a mixture of analytical reasoning and physical intuition when analyzing the results. The aspects we wish to study are: – The time spent on building and solving linear systems. When using ﬁnite element methods (and implicit ﬁnite diﬀerence methods) most of the computational eﬀort is put into solving linear systems of equations. For time dependent problems the time spent on building the systems is also an important factor. We try to determine the relationship between the size (i.e., the number of nodes) in the grid and the time spent on these operations for diﬀerent methods. – Comparison of the Crank-Nicholson scheme and the leap-frog scheme. The schemes may seem similar when it comes to computational cost, at least for ﬁnite element methods. There are however some diﬀerences, and we will try to study some of them. – Diﬀerent element types. The ﬁnite element method allows a wide range of element types, such as triangular elements, quadratic elements and so forth. We do a quick comparison between linear and quadratic ﬁnite elements. – A simulation of the full physical model. Finally we are ready to do a simulaton of the full system. We will describe the results qualitatively and compare simulations with diﬀerent methods. 161 Solving the Time Dependent Schrödinger Equation Hopefully, these experiments will provide us with valuable information on the performance of the numerical methods. Along the way we will also come across several interesting questions that may lay grounds for exciting future work, both physical and numerical. It is important to keep in mind that the experiments performed here are very simple and to not exploit the full capability of the ﬁnite element method, such as the extremely ﬂexible geometry of the grid. Analyzing numerical results are much simpler on highly structured grids. When one wants to perform “real” experiments, the ﬂexibility in the location and size of the elements should be exploited. 7.3.1 Building and Solving Linear Systems 2.5 1.5 polyfit(log(n),log(tsolve )=[2.0007 -8.9127] 2 1 1.5 0.5 log(tsolve ) log(tsystem ) polyfit(log(n),log(tsystem))=[2.0005 -8.0297] 1 0.5 0 -0.5 0 -1 -0.5 -1.5 -1 3.6 3.8 4 4.2 4.4 4.6 4.8 5 -2 3.6 5.2 3.8 4 4.2 log(n) 4.6 4.8 5 5.2 30 t solve niter n=40 n=80 n=120 n=160 9 8 25 tsolve and n iter 7 6 tsystem 4.4 log(n) 10 5 4 20 15 10 3 2 5 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 time step l 0.7 0.8 0.9 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 tl Figure 7.1: Lower left: the time for creating linear systems for each time step. Upper left: Log-log plot of the average time spent on building the system. Lower right: The time for solving the linear systems with n = 160 and the number of iterations needed. Upper right: Log-log plot of the average time spent on the solving. The most time consuming part of a ﬁnite element simulation is the solution of linear systems. These tend to be very large and sparse. Here we will investigate the increase in the time spent on solving linear equations as function of the number of nodes in the grid. Here and in the rest of the simulations on this section, we will employ a “ﬁnite diﬀerence grid,” i.e., a grid with uniform spacing in every direction and elements with a square shape. The time spent on building linear systems should be approximately the same from time step to time step. In the theta-rule we build two Hamiltonians and mass matrices (for the left and right hand side, respectively) at each time step, roughly doubling the time spent in the leap-frog scheme. The linear systems are solved with the GMRES method. The iteration process requires 162 7.3 – Numerical Experiments a start vector, and we use the previous solution u as initial vector when solving for u+1 . At least for the Crank-Nicholson scheme it is intuitively close to the new solution and therefore we expect that the iteration process converges fast. One may also expect that the time step aﬀects the convergence rate: The smaller the time step, the faster the convergence. This can turn out to be very important if true! Indeed, then the simulation time does not increase linearly with the total number of time steps, but slower than linear, i.e. O(∆t−1/p ), with p ≥ 1. We perform a series of simulations with constant physical problem deﬁnition but with increasing number of subdivisions n in each spatial direction. – We use the full Hamiltonian with the Coulomb term and the time-dependent magnetic ﬁeld with ω = δ = 0 and γ0 = 1 in order to capture eﬀects from every aspect of the full problem. – Our initial condition √ is√a Gaussian wave packet located at x0 = (−10, −10) with momentum k = ( 2, 2) and width σ = (4, 4). It is located safely away from the origin but moving towards it with velocity 2k . (Its speed is 2k = 4.) – We simulate for t ∈ [0, 1] with ∆t = 0.002. – We utilize a square domain Ω = [−20, 20] × [−20, 20] subdivided into n2 squares with sides 40/n. – We let n vary over a series of single simulations, viz., n = (40, 80, 120, 160). (7.2) – We utilize the leap-frog scheme in these simulations. We monitored the time tsystem (n, ) spent on building the linear systems and the time tsolve (n, ) spent on solving the linear equations, where is the time step. It is expected that the time spent on building the linear system is independent of because it is the exact same sequence of operations that is performed each time. This is also observed, see Fig. 7.1 in which we have graphed the time spent on building the linear systems for each simulation. All the simulations produced qualitatively the same timedependent solutions. (They are not very interesting so we do not show them here.) The time spent on solving the equations is also expected to not depend too much on the time step number , and this is also seen in Fig. 7.1. The time step was unfortunately chosen arbitrarily and believed to be within the stability condition for the leap-frog scheme with linear elements, while it actually violated the condition for the largest grids. The simulations for the two ﬁnest grids developed instabilities near the middle and end of the simulations. This is the reason why the corresponding graphs are shorter. The average (over ) time in seconds spent on building linear systems is t̄system = (0.5226, 2.0796, 4.7194, 8.3457). The order of the numbers is from low to high n. The average time spent on solving the systems is t̄solve = (0.2346, 0.7656, 1.7548, 3.9920). We conjecture that the average time t̄(n) = O(np ) for some p ≥ 1 for both cases. To check this we graph the logarithm of t̄ versus the logarithm of n, similar to what we did in section 6.6. Fig. 7.1 shows these plots. The data were run through Matlab’s regression function polyfit and the resulting polynomials are shown as well. (For t̄system the linear function is not plotted because it is indistinguishable from the data.) Clearly, we have t̄system, solve ∼ n2 = N 163 Solving the Time Dependent Schrödinger Equation 18 16 14 12 〈x(t)〉 10 8 6 LF 0.0015 LFL 0.0015 CN 0.0015 CN 0.01 CN 0.02 CN 0.1 expected 4 2 0 0 0.5 1 1.5 2 2.5 t 3 3.5 4 4.5 Figure 7.2: Expectation value x for the diﬀerent simulations where N is the dimension of the element matrix. In other words we have approximately a linear scaling of the computational cost with the leap-frog method. It is not unreasonable to guess that this holds also for the Crank-Nicholson scheme, because the matrices have exactly the same structure in the two cases. Contrast the O(N ) result of solving linear systems with the O(N 3 ) result for gaussian elimination of dense matrices. It would be interesting to check this scaling property with other element types, such as quadratic elements that generate denser matrices. As for Fig. 7.1 a few comments are in place. First, the plots with t on the horizontal axis (the lower plots) show small ﬂuctuations. This may be due to for example other processes running on the computer. The number of iterations niter is clearly proportional to tsolve , reﬂecting that each iteration requires a ﬁxed amount of numerical work. 7.3.2 Comparing the Crank-Nicholson and the Leap-Frog Schemes The Crank-Nicholson scheme and the leap-frog scheme might look similar at ﬁrst when it comes to numerical cost. If we do not lump the mass matrix in the leap-frog scheme, we need to solve a linear system of equations at each time step. Considering the stability criterion for each scheme, the leap-frog scheme might look useless when compared to the Crank-Nicholson scheme. As we saw in section 4.5.4, the unitarity of the scheme (for a time-independent Hamiltonian) is only secured if the time step was smaller than the inverse of the largest eigenvalue of H, viz., ∆t ≤ 1 max . The Crank-Nicholson scheme has no such restrictions in the time-independent case. However, there are two questions that must be answered. First, is the accuracy of the Crank-Nicholson scheme just as good as the leap-frog scheme if we take longer time steps in the former? Maybe we need to reduce the time step in order to achieve the 164 7.3 – Numerical Experiments same precision as the leap-frog scheme. Second, are the solutions to the linear equations utilizing M as the coeﬃcient matrix just as expensive as the ones used in the CrankNicholson scheme? Intuitively no, because M is positive deﬁnite and symmetric, while the coeﬃcient matrix in the Crank-Nicholson scheme is non-symmetric. Furthermore, lumping the mass matrix in the leap-frog scheme makes the linear systems trivial to solve, improving the eﬃciency drastically. (If we also use nodal-point integration and linear elements, we obtain a scheme equivalent to the standard ﬁnite diﬀerence scheme.) To determine some answers experimentally we perform a series of simulations with a time independent Hamiltonian with diﬀerent methods and time steps. We use the particle-in-box Hamiltonian and a Gaussian initial conditions, because we know (at least before the packet hits the boundary) the exact qualitative behavior of the wavepacket and expectation values of position. Indeed, the free wave packet has mean momentum k , so the mean position should be x = x0 + 2tk , where x0 is the initial mean position and where k is the momentum. The wave packet used for our simulations start out at x0 = (0, 0) with momentum k = (2, 0). The expected velocity is therefore v = 2k = (4, 0). The width of the wave packet is σ = (5, 5). The grid is a “ﬁnite diﬀerence grid” discretizing the domain Ω = [−16, 16]2 with n = 160 subdivisions along each side. We use linear elements. The stability criterion for the leap-frog scheme is h2 = 0.005 ∆t ≤ 8 for the ﬁnite diﬀerence method, and roughly ∆t ≤ h2 = 0.0016666 24 for the linear ﬁnite elements. (The latter is only an estimate because the highest energy in two dimensions in the ﬁnite element method is not exactly twice the one-dimensional energy.) Therefore we use ∆t = 0.0015 for the leap-frog method, hopefully staying on the right side of the stability criterion. In the simulations we let t ∈ [0, 4]. We perform six diﬀerent simulations with the following numerical parameters: 1. The leap-frog method with the full mass matrix, ∆t = 0.0015. This case is referred to as LF 0.0015 in the ﬁgures et.c. The total simulation time (in seconds) for this case was ttotal ≈ 15.5 · 103 . 2. The leap-frog method with a lumped mass matrix and nodal point integration, i.e., the ﬁnite diﬀerence method, and using the same time step as before. This is referred to as LFL 0.0015. The total simulation time was ttotal ≈ 10.1 · 103 . 3. The Crank-Nicholson method with ∆t = 0.0015. This is referred to as CN 0.0015. The total simulation time was ttotal ≈ 32 · 103 . 4. Same as the previous, but with time step ∆t = 0.01. This is referred to as CN 0.01. The total simulation time was (somewhat surprisingly) ttotal ≈ 2.3 · 103 . 5. Same as the previous, but with time step ∆t = 0.02 This is referred to as CN 0.02. The total simulation time (in seconds) for this case was ttotal ≈ 1.1 · 103 . 165 Solving the Time Dependent Schrödinger Equation -2 -3 log 10 (abs(||u||2 -1)) -4 -5 -6 -7 LF 0.0015 LFL 0.0015 CN 0.0015 CN 0.02 CN 0.01 CN 0.1 -8 -9 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 t Figure 7.3: Deviation from unitarity in the simulations 6. Finally, we use a time step ∆t = 0.1 in the Crank-Nicholson method. This we refer to as CN 0.1. The total simulation time was ttotal = 665.8. Animations of the simulations can be reached from Ref. [5]. These are only interesting inasmuch that qualitatively they are all the same, and they reproduce the expected qualitative behavior.2 The simulation times must be taken with a pinch of salt. The ﬁnite diﬀerence scheme LFL 0.0015 was not optimized in the implementation. On the contrary, building linear systems and solving them (even if they are trivial) represent much more work than needed in an implementation of the method. A proper implementation is much faster. The Crank-Nicholson method on the other hand needs solutions to linear systems, even in the ﬁnite diﬀerence case. In Fig. 7.2 the mean position x as function of t is plotted for each simulation. The expected behavior is plotted as a dashed straight line. We notice that the expected behavior is largely reproduced. The slope of the curves drop near the end of the simulations due to collision with the boundary. Contrary to a classical particle which bounces right oﬀ the wall, the quantum particle turns around in a smooth fashion. The diﬀerent simulations show small but important variations in the mean position x. Some of the simulations overestimate the velocity of the packet while others underestimate it. The simulations that overestimate the velocity are LF 0.0015, CN 0.0015, CN 0.01 and CN 0.02. The simulation LFL 0.0015, i.e. the ﬁnite diﬀerence method, yielded qualitatively the same result as CN 0.1, i.e., Crank-Nicholson with a very large time step. This indicates (bot it does not prove) that the ﬁnite diﬀerence leap-frog method is similar to Crank-Nicholson with (much) larger time step than the stability criterion. 2 Actually, behavior characteristic to quantum mechanical collisions show up in the animations, namely the packet’s interference with itself as it collides with the wall. See also the link to other animations and descriptions in Ref. [5] with further investigation of this phenomenon. 166 7.3 – Numerical Experiments 40 1 linear quad. 0.9 35 0.8 30 0.7 25 tsolve γ(t) 0.6 0.5 0.4 20 15 0.3 10 0.2 5 0.1 0 0 0.5 1 1.5 2 t 2.5 3 3.5 0 4 0 1.0005 0.5 1 1.5 2 t 2.5 3 3.5 4 150 linear quad. linear quad. 1 0.9995 100 niter ||u(t)||2 0.999 0.9985 0.998 50 0.9975 0.997 0.9965 0 0.5 1 1.5 2 t 2.5 3 3.5 4 0 0 0.5 1 1.5 2 t 2.5 3 3.5 4 Figure 7.4: Various simulation data as function of time: The magnetic ﬁeld (top left), the square norm (bottom left), the number of iterations per linear system (top right) and the time per linear system (bottom right) The leap-frog method for ﬁnite element methods (LF 0.0015) used roughly half the simulation time than the corresponding Crank-Nicholson simulation. We will not show it here, but for larger time steps, solving linear equations in the Crank-Nicholson method takes more time, indicating that the closeness of the previous solution plays some role on the performance of the time stepping. This is a subject for later studies. The Crank-Nicholson method preserves the norm of the solution exactly, but as we employ iterative solvers there might be some ﬂuctuations. The leap-frog scheme is not exactly unitary because we have violated the assumption we used when deriving the stability criterion by using the Crank-Nicholson integration for the ﬁrst time step.3 Fig. 7.3 shows the logarithm of the deviation from unity of the square norm of the numerical solution. Clearly, Crank-Nicholson preserves unitarity better, with variations around 10−6 . The leap-frog method shows clear ﬂuctuations, but never larger than 10−3 . This is acceptable if we normalize the wave function before calculating physical quantities such as expectation values. In summary, it seems that the Crank-Nicholson method can take bigger time steps while reproducing the same results as the leap-frog scheme. The explicit ﬁnite diﬀerence method is quick to implement, but one loses the advantages from the ﬁnite element method, such as geometry freedom, the mass matrix (which yields higher accuracy of the eigenvalues) and so on. 7.3.3 Comparing Linear and Quadratic Elements. We have several parameters that aﬀect the ﬁnite element method, most important is perhaps the element order. Increasing the element order will increase the number of 3 The “hole” in the LF 0.0015 curve is due to missing data because of a mishap during the simulation. 167 Solving the Time Dependent Schrödinger Equation Figure 7.5: Stills from simulation with linear (left) and quadratic (right) elements. The time levels are (from bottom to top) t = 0.0000, t = 1.9825, t = 2.6825 and t = 4.0000. 168 7.3 – Numerical Experiments non-zeroes in the matrices of the problem, thereby increasing the time needed for each iteration in the linear solver. On the other hand, increasing the element order (while keeping the number of grid points ﬁxed) may aﬀect the accuracy of the solution. We will do a simulation with a varying magnetic ﬁeld with linear and quadratic elements, respectively, to check if they give very diﬀerent results. If they do, then one or both of the, experiences loss of numerical precision, because both simulations certainly cannot be correct. This will indicate whether or not the Schrödinger equation is sensitive to the order of the elements. There are of course many more aspects to study, such as the grading of the grid, the choice of geometry and so on. Again a square grid was used. This time the domain was given by Ω = [−40, 40] × [−40, 40], with n = 200 subdivisions along each side. Two simulations with linear elements and quadratic elements, respectively, were performed. Thus, we had 2002 = 40, 000 elements in the linear case and 1002 = 10, 000 elements in the quadratic case. The magnetic ﬁeld parameters were ω = δ = 0 and γ0 = 1 and the initial condition was a Gaussian with x0 = (0, 0), k = (0, 2), and σ = (10, 10). The Coulomb attraction was turned oﬀ; hence we had a free particle with an applied magnetic ﬁeld. The time interval was t ∈ [0, 4] with ∆t = 0.0025. We used the Crank-Nicholson scheme in both cases. The only diﬀering setting was the element type. Hence, if the simulations agree we must conclude that the Schrödinger equation appear insensitive to the element order (even though only two experiments is too little to make decisive conclusions.) If the simulations disagree, we must conclude that the element order do have some importance. The particular system and initial condition we tested turned out to show very interesting dynamics: The wave packet moves like a free particle as expected until the magnetic ﬁeld starts to become signiﬁcant. The wave packet then shrinks very quickly, concentrating into almost a single point. As the magnetic ﬁeld diminishes it spreads out again. When concentrated in an area covering only a few elements, all the information from the original wave packet is reduced to only a handful of numbers, i.e., the components of the numerical wave function. Hence, there is loss of information. This means that we are on thin ice when studying the wave function dynamics after this event. Animations showing the development of the probability density can be reached from Ref. [5]. Fig. 7.5 shows some still images of the probability density for the two simulations at key points in the development. Fig. 7.4 shows the development of the square norm of the wave function, the magnetic ﬁeld γ(t) and the time and number of iterations spent at solving the linear systems for each time step. The time levels at which we have plotted the probability density is showed as circles on the graphs. The qualitative behavior of the probability density is similar for linear and quadratic elements, at least before the “collapse” of the wave function into approximately a single point around t = 2.6825. A noteworthy diﬀerence is that the quadratic element simulation tend to develop high-frequency spatial oscillations near the edge of the wave packet. These can be seen at t = 1.9825. If this is a physical eﬀect or a numerical eﬀect is not easy to say, but certainly deserves an investigation. If this is a numerical eﬀect it indicates that (at least for this magnetic Hamiltonian) that higher order elements introduce noise. On the other hand, if it is a physical eﬀect, it indicates that using higher order elements captures features of the wave function that needs a reﬁning of the grid in the ﬁnite diﬀerence method. 169 Solving the Time Dependent Schrödinger Equation The wave functions collapse approximately at the same time and place as can be seen from the ﬁgure. After the collapse the wave functions do not bear resemblance to each other, except for spreading rapidly. The spreading is easily understood in terms of Heisenberg’s uncertainty principle. Highly localized wave packets have a high uncertainty in the momentum, making the packet spread fast. The fact that the wave functions are so diﬀerent means that the error must be large in either or both simulations after the collapse. Turning to the graphs in Fig. 7.4, the norm is seen to be approximately conserved for both linear and quadratic elements, but shows a tendency to drop somewhat when the magnetic ﬁeld is strong. The time spent on solving linear systems and the corresponding number of iterations show the opposite behavior. When the magnetic ﬁeld is strong several times more iterations are needed for solving the systems than γ = 0. The time of collapse of the wave function (where the wave function changes very rapidly) this does not seem to behave diﬀerently with respect to this. 7.3.4 A Simulation of the Full Problem As a ﬁnal study we will do a numerical experiment with the complete physical system. We will start with a Gaussian wave packet in a Coulomb ﬁeld and add an ultra-short laser pulse. We will not discuss the accuracy of this simulation, for that we need much more time and space. However, we will describe the results qualitatively and generate a motion picture of the resulting “trajectory” of the wave packet. We will do the simulations for three diﬀerent parameter sets for the time integration for comparison. The magnetic ﬁeld has parameters γ0 = 1, ω = 1.23, δ = 0.13 and is shown in Fig. 7.7. The initial condition has parameters x0 = (−5, −5), k = (2, 0), σ = (3, 3). The initial condition along with other stills from one of the simulations is shown in Fig. 7.6. We will use three representative time integration schemes. – We use the leap-frog method with ∆t = 0.001 and nodal point integration, i.e., we use the ﬁnite diﬀerence method. The time step is chosen based on h = 80/150 which gives ∆t < 0.03555. We assume that the Coulomb potential has no eigenvalue of magnitude greater than the maximum energy for the particle-in-box. This is reasonable, since the ground state has magnitude |0 | ≤ 4 and since the system “looks like” a particle in box for very highly excited states. Due to the changing magnetic ﬁeld we choose ∆t even lower, namely ∆t = 0.001. We refer to this simulation as LFL 0.001 in the ﬁgures. – For the second simulation we use the Crank-Nicholson method with larger time step, viz., ∆t = 0.02. We use linear elements and Gaussian quadrature and refer to the simulation as CN 0.02. 170 7.3 – Numerical Experiments Figure 7.6: Stills from simulation. Time levels are t = 0.00 (bottom left), t = 0.98 (bottom right), t = 1.98 (middle left), t = 2.98 (middle right), t = 3.48 (top left) and t = 3.98 (top right). – The third and last simulation also uses Crank-Nicholson but with quadratic elements. We refer to the simulation as CN 0.02 quad. Stills from the LFL 0.001 simulation are shown in Fig. 7.6. Animations of all three simulations can be reached from Ref. [5]. The diﬀerent simulations yielded qualitatively similar results for the probability density. Actually it is hard to spot any diﬀerence at all. Furthermore, we can not see any high-frequency oscillations in the quadratic elements case as we did in Fig. 7.5. In Fig. 7.7 we have graphed the magnetic ﬁeld γ, the square norm u2 of the wave function, the energy H and the expectation value x of the position as function of time. All the plots indicate that the Crank-Nicholson simulations behave very similar to each other. Furthermore, they are qualitatively diﬀerent from the leap-frog simulation in all plots but the energy plot. The square norm has large ﬂuctuations in the Crank-Nicholson scheme. The ﬂuctuations clearly follow the time development of the magnetic ﬁeld. Recall from section 171 Solving the Time Dependent Schrödinger Equation 1 ω=1.23, δ=0.13 0.8 0 0.2 -1 〈y〉 1 0.4 γ(t) 0.6 0 CN 0.02 LFL 0.001 CN 0.02 quad -2 -0.2 -3 -0.4 -4 -0.6 -5 -4 -2 0 -0.8 -1 2 4 6 8 〈 x〉 0 0.5 1 1.5 2 t 2.5 3 3.5 4 25 1.01 CN 0.02 LFL 0.001 CN 0.02 quad 1 20 0.99 〈 H〉 ||u(t)|| 2 15 10 0.98 0.97 0.96 5 CN 0.02 LFL 0.001 CN 0.02 quad 0.95 0 0.94 0 0.5 1 1.5 2 t 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 t Figure 7.7: Plots of dynamical quantities: The magnetic ﬁeld (top left), the mean position (top right), the mean energy (bottom left) and the square norm (bottom right). 4.5.4 that the norm should experience O(∆t3 ) ﬂuctuations depending on ∂H/∂t for both methods. This is clearly the case for all the simulations, and in fact the smaller ﬂuctuations for the leap-frog method must be due to the smaller time step. Even though we have ﬂuctuations in the norm, the expectation value H seems insensitive to these ﬂuctuations as all three simulations give the same result, both qualitatively and quantitatively.4 The energy ﬂuctuates and follows the “beat” of the magnetic ﬁeld. This indicates that violating the strict unitarity not necessarily destroys the physical content of the wave. In the graph of the expectation value x (and in the animations) the wave packet actually reverses its motion, indicating that the kinetic energy should have local minima. This is clearly seen (if we assume that the potential energy is neglible). The changing direction of the magnetic ﬁeld (as γ goes from positive to negative or vice versa) means that a changing torque is exerted on the particle. This classical line of thinking is of course possible due to the correspondence principle and indicate that our simulations behave according to the laws of quantum mechanics. The expectation value x of the position shows that the wave packet is being deﬂected by the Coulomb potential, similar to a classical system. An interference pattern is also observed connecting the wave packet with the origin with long “ﬁngers” of probability. At the end of the simulation we observe something interesting. Similar to the system in which the Coulomb potential was absent the wave packet starts to shrink. Thinking classically we would expect the opposite to happen, i.e., that the packet would spread out due to the uncertainty in the linear momentum. But in our simulations it seems that the magnetic ﬁeld’s eﬀect on the particle is to localize it. This localization feature 4 The 172 program output is not normalized, but this is done prior to plotting. 7.4 – Discussion is not present in the pure Coulomb system, see Ref. [26]. An additional animation showing a simulation in which γ = 0 is also found in Ref. [5]. We have obtained these qualitative results in diﬀerent systems with diﬀerent numerical integration schemes, indicating that there might be some interesting physics behind this behavior as well. Studying wave packet dynamics in magnetic ﬁelds should be an interesting future project, also because it poses a numerical challenge if the wave collapses into a very small area as in Fig. 7.5. From chapter 6 we know that the time development of the wave function (at least for a time independent Hamiltonian) is given directly in terms of the eigenvalues and eigenvectors of the system. If the discretization reproduces the eigenvalues then the time development of the exact system will be accurately reproduced. On the other hand, if the eigenvalues are wrong, the time development of the discrete system will not reﬂect the exact development. Even though our diﬀerent numerical methods all yielded very similar results we cannot say that it reﬂects the true time development. For that we need to know in some way whether the eigenvalues are correct. In the present case we could do a diagonalization of the Hamiltonian over the current grid for γ = 0. This would give further indications of a good numerical method and simulation. 7.4 Discussion Even though the experiments were limited in extent we have come across several important indications on the performance and behavior of the numerical methods. Can we draw any conclusions from our work? The most important beneﬁt of the Crank-Nicholson method over the leap-frog scheme is stability. The method is stable for much larger time steps than the leap-frog method. The time step required to reproduce the accuracy of the leap-frog scheme (with ﬁnite diﬀerences in space) is much larger than the stability criterion for the explicit scheme, making this beneﬁt even more important. On the other hand, the ease of implementing the leap-frog scheme with ﬁnite differences makes it very attractive. The linear systems in the ﬁnite element method is cheaper to solve with the leap-frog method, which is an important fact. As for the spatial discretization, ﬁnite diﬀerences are without doubt the fastest to implement and run. But we have seen that the ﬁnite element approach reproduces eigenvalues that are qualitatively more correct than the ﬁnite diﬀerence method. Furthermore, there might be something to gain from using higher-order elements. We saw that the time development of a wave function with quadratic elements was qualitatively diﬀerent than with linear elements. Then again, the systems of linear equations are harder to solve. This takes us to the next important fact we have learned. Choosing the linear solver appropriately is very important. The eﬃciency of the implicit time stepping improves drastically if we choose a somewhat more sophisticated linear solver (i.e., iterative methods) than the usual banded Gaussian elimination that is used in most two-dimensional approaches. In three dimensions it becomes vital. With the generalized minimal residual method the time spent on solving linear systems scaled linearly with the matrix dimension, and this is optimal. Fine-tuning the choice and parameters of the linear solver is an interesting future project. Our simulations exclusively used a Gaussian wave packet for initial condition even though we implemented the possibility of using the eigenvectors of the Hamiltonian over the current grid. This was mostly due to limitations in computing time when the submission deadline approached. If the time-dependent simulations done here were interesting, using for example the ground state of the hydrogen atom over a properly graded and ﬁne-tuned grid would be really cool, to put it that way. 173 Chapter 8 Conclusion This short chapter summarizes the results and ﬁndings of this cand. scient. project. Along the way valuable experience with the numerical methods has been gained, and some ideas for possible future work have emerged as well. The focus has not been as much on physics as on numerical methods, but this is not a limitation. On the contrary, I believe that the experience gained with the numerical methods may serve as an invaluable basis when complicated simulations and work is to be done. The Eigenvalue Problem. The eigenvalue problem of the ﬁnite element discretized Hamiltonian is closely related to the development of the time-discrete wave function, as we have seen. The eigenvalues dictate the evolution, but apart from this the eigenvalue problem has few consequences for the time dependent simulations. Of course, we still have the usefulness of being able to analyze the time dependent state with respect to the spectrum of the Hamiltonian, but this is not directly linked with the numerical methods. On the other hand, the eigenvalue problem by itself is very interesting to consider. Many questions one might have concerning a physical system can be answered in terms of the allowed energy levels. Take for example the hydrogen atom, in which the spectrum tells us the frequency of light that is needed to be able to excite or ionize the electron from its ground state. By exploiting the ﬂexible geometry speciﬁcation in the ﬁnite element method we are able to concentrate our numerical work in areas where the potential energy varies rapidly. We saw examples of this when trying to ﬁnd the spectrum of the two-dimensional hydrogen atom. With other methods such as ﬁnite diﬀerence methods, the choice of geometry is very limited. Because of the ﬂexible geometry in the ﬁnite element method, the possibility of studying quantum chaotic systems was mentioned in chapter 6. This implies the study of highly excited states, i.e., eigenvectors of the Hamiltonian far from the ground state. We saw that for the particle-in-box, the ﬁnite element method produced eigenvalues that qualitatively followed the analytic eigenvalues better than the ﬁnite diﬀerence method. If this is a general feature of ﬁnite element methods (or variational methods in general) they must be superior when studying such systems. Furthermore, if the energy of the wave function in a time dependent simulation is very high, we should use whatever method that most faithfully reproduces the higher eigenvalues. Hence, further studies of the ﬁnite element eigenvalue problem is very attractive, as it has implications that are far-reaching. It must be mentioned that our approach to the eigenvalue problem was rather naı̈ve and based on the idea that we could learn something about the time development. Very sophisticated methods for ﬁnding eigenstates of the Hamiltonian called adaptive ﬁnite element methods have been developed and applied to some classes of systems, even though they are not completely general. Doing a critical study of adaptive methods 175 Conclusion could prove interesting, perhaps also for the time development of the wave function. As starting point for the work on eigenvalue problems one could perform an extension of the one-dimensional analysis in section 6.4 to include higher order elements in several dimensions. This will then indicate lines of action with respect to timeintegration as well. An additional study is the energy levels for the two-dimensional hydrogen atom at intermediate magnetic ﬁelds. We discussed this system to some extent, ﬁnding interesting qualitative results. Extending this analysis to for example spatially varying magnetic ﬁelds could be interesting, as well as doing one-dimensional calculations on the radial equation. For the tree-dimensional atom this has been done with ﬁnite element methods as early as in the eighties, see Refs. [52, 53]. We have also mentioned gauge invariance of the eigenvalues and eigenvectors as an important physical and numerical concept. Establishing results concerning this may prove fruitful. This has never been properly done before. Time Integration Methods. We have just scratched the surface of the wide range of time integration methods that exist for classical Hamiltonian systems and the Schrödinger equation, see Ref. [36]. A concept that becomes more and more emphasized in numerical analysis of ordinary diﬀerential equations is geometric integration, i.e., integration methods that preserve certain qualitative features of the diﬀerential equations, thereby incorporating stability in a natural way. It is clear that a thorough insight into what methods exist and to what kinds of systems they have been applied to would form an invaluable basis for future work on solving the time dependent Schrödinger equation. Trying out these methods with diﬀerent spatial discretizations and physical systems is very interesting and should form interesting projects. As for the methods studied in this text, there are still some unanswered questions. Most important is perhaps the question of gauge invariance, which in the literature is not even asked. For the time dependent methods this must be viewed in conjunction with the spatial discretization, since both spatial and temporal degrees of freedom enter the problem. A gauge invariant integration scheme will provide more accurate results in a simulation, simply because the original formulation is gauge invariant and that the chosen gauge is arbitrary in principle. If we must “choose the correct gauge” in order to be able to trust our results in each case, we automatically have a lot more parameters to adjust before starting the numerical simulations, and we must be able to ﬁnd out what gauge is best in the ﬁrst place. In the extension of this analysis we may study what kinds of schemes that in general might prove gauge invariant, such as implicit schemes, explicit schemes and unitary schemes, if any. We stress that gauge invariance is an area of numerical analysis that never before has been studied properly. On the other hand the concept of covariance of numerical methods is a ﬁeld of research in progress. Linear Solvers and the Schrödinger Equation. A lot of simulation time was spent on solving linear systems. Linear systems pervade most computational scientist’s work, and having a thorough knowledge of what solver to use and when to use it really distinguishes an experienced practitioner from the crop. In this thesis we simply used a ﬁxed solver because it worked and spent reasonably small time at doing its job. The Schrödinger equation and the particular time integration method yield linear systems of a particular kind. The leap-frog scheme (and any other explicit scheme) have the mass matrix M as coeﬃcient matrix. The Crank-Nicholson scheme have a coeﬃcient matrix M + i∆tH/2. As we saw the solution of the latter systems required longer time than the former, indicating that knowing what method to use is important. Are there any way of deciding what solver is the best for the Schrödinger equation? What about modifying an existing solver to ﬁne-tune it for the Schrödinger equation? 176 Such questions also become increasingly important if higher-dimensional systems are to be solved, in which the matricial dimensions approach the limits of what is possible to handle with today’s computer technology. The technique of preconditioning the linear systems in order to improve the convergence rate for the Schrödinger equation is also a topic which could be interesting to analyze further, see Ref. [34]. As mentioned in the concluding remarks of chapter 6, both Diﬀpack and ARPACK has supports parallelization. Studying the parallelized version of the numerical methods will certainly make us able to do much more detailed simulations and give better physical results. Physical Applications. In this thesis we have focused on two-dimensional systems. They were chosen for several reasons, most prominent were perhaps their ease of visualization and their intermediate diﬃculty of implementation. We can use moderately large grids and obtain accurate results, something that would be diﬃcult in a threedimensional problem. Two-dimensional systems also arise in a variety of applications, such as solid-state physics and plasma physics. In Ref. [54] several atomic systems are considered, in particular Rydberg atoms (i.e., highly excited atoms) in electromagnetic ﬁelds. Several interesting conclusions and results are presented, several of which could prove interesting to reproduce and study in further details, such as the excitation of Rydberg atoms with laser pulses. With the ﬁnite element method and accurate integration schemes we may study these aspects in a systematical manner and provide further insight into an exciting (no pun intended) area of physics. Atomic physics describe systems that we actually can create and control in a laboratory, providing basic insight into quantum mechanical systems and also possible technological advances in the future. In the extension we may also study systems of ions trapped in arbitrary geometrically shaped potentials, such as anharmonic and anisotropic harmonic oscillators, quadrupole potentials, Paul traps and even quantum wires and nanotubes. See Ref. [51] for details. We could also study the Schrödinger equation on more exotic manifolds, such as a graph with several one-dimensional manifolds that meet at common points (see Ref. [55]), a curved two-dimensional manifold such as a sphere and so on. Pointinteraction Hamiltonians (i.e., Hamiltonians with δ-function potentials) could also be attacked in a direct manner with ﬁnite element methods in which the singularities are integrated out of the formulation, leaving a well-deﬁned matrix equation instead. Closely related to the point-interaction Hamiltonians are the systems from solidstate physics mentioned in section 3.2.1. These systems encourage in a natural way the use of the ﬁnite element methods. If the system is very regular we may ﬁnd the exact solution, but if the particles in the model are displaced in a random fashion the behavior of the system is largely unknown. On the other hand we know that irregularities in such systems are responsible for many well-known phenomena, such as super-conductance at low temperatures. The Schrödinger equation is linear, but in several areas of physics non-linear variants of the equation arise. Bose-Einstein condensates can be described by the GrossPitaevskii equation, see Ref. [54]. This equation has an extra term proportional to |Ψ(x)|2 and describes the eﬀective one-particle wave function for the interacting bosons. Solving the time-dependent Gross-Pitaevskii equation is a challenge and will certainly provide valuable insight into both the physics and the numerics of linear and non-linear systems. 177 Appendix A Mathematical Topics This appendix discusses some mathematical and technical points in the text in detail. For a thorough discussion on linear operators on Hilbert spaces and in quantum mechanics, see [56]. The physics textbook [20] contains mathematically oriented appendices easier to grasp at ﬁrst reading. A.1 A Note on Distributions The most famous example of a distribution is perhaps the Dirac distribution, also (improperly) called the δ-function. Originally, this was deﬁned as the “function” δ(x) satisfying δ(x) = 0, ∀x ∈ R, x = 0, and +∞ δ(x) dx = 1. −∞ If δ(x) is a function, then its value at x = 0 must be inﬁnte in order to make the integral diﬀerent from zero. The Dirac δ-function is an example of a distribution. Distributions play a fundamental role in the mathematical theory of ﬁnite elements as well as quantum mechanics. The rigorous deﬁnition is beyond the scope of this text, see Ref. [16] for an introduction. Roughly speaking they represent a way to take a weighted average of a function. From the “deﬁnition” of the Dirac distribution we must have δ(x)f (x) = f (0), and so δ : L2 → C, δ(f ) = (δ, f ) = f (0). Here, we must assume that f (0) is well-deﬁned, but it is not! Any element f in L2 is considered identical to elements g whose function values diﬀer at a set of (Lebesgue) measure zero. Thus, the δ-distribution is not well-deﬁned on the whole space L2 . A.2 A Note on Inﬁnite Dimensional Spaces in Quantum Mechanics Quantum mechanics use linear algebra in Hilbert spaces as framework for the theory. More speciﬁcally, the Hilbert spaces employed are inﬁnite-dimensional and separable. 179 Mathematical Topics Separable means that one may always ﬁnd a countable orthonormal basis. Every ﬁnitedimensional vector space is separable. As stated in section 1.3, the Hilbert space for one particle is H = L2 (R3 ) ⊗ C2s+1 , where s is the intrisic spin of the particle. The space L2 is the square-integrable functions on R3 into the complex numbers, viz., 2 3 3 L (R ) := {f : R → C : |f |2 < ∞}. Usually, physicists work with H as if it is a ﬁnite-dimensional space, in the sense that Hermitian operators always may be diagonalised (see below) and that operators are well deﬁned for every vector in H. This is a rough simpliﬁcation of the real situation. Physicists also work with the elements as if they have well-deﬁned values at each point in space. Because of this, quantum mechanical calculations are actually often merely formal calculations, or “juggling with symbols.” For example, consider the point-interaction Hamiltonian in one-dimension, viz., H =− ∂2 + µδ(x0 ), ∂x2 where µ is some coupling constant. This is not an operator on L2 ! First, the kinetic energy term is not deﬁned on the whole space, and δ is not an even an operator, much less a function. Consider the momentum operator in one dimension. We write this as P = −i ∂ , ∂x but strictly speaking this is an abuse of notation and not correct. There are many vectors (i.e., functions) in H that cannot be diﬀerentiated, i.e., the operator is not deﬁned for all vectors. (It is not even deﬁned on a closed subspace of H, since a sequence of diﬀerentiable functions not necessarily converges to something diﬀerentiable.) If we on the other hand consider the weak derivative of an element f in L2 , the operator is well-deﬁned. If f happens to be diﬀerentiable in the classical sense, the weak derivative is the same as the classical derivative. This is why physics calculations with the momentum operator usually pose no diﬃculties. A.3 Diagonalization of Hermitian Operators As discussed in chapter 1, quantum mechanical observables are represented as linear operators A on the Hilbert space H. All observables are Hermitian operators, that is A† = A. This is equivalent to the requirement: (Ψ, AΦ) = (AΨ, Φ), for all Ψ, Φ ∈ H. Note that we like to consider operators not deﬁned for all vectors, such as the momentum operator or the position operator. Many of these (including the momentum operator) do not have an orthonormal basis of eigenfunctions in L2 . Diagonalization of an operator A means ﬁnding an orthonormal basis of vectors φn and a set of scalars an such that Aφn = an φn , 180 n ∈ I, A.4 – The Fourier Transform where I is some set. Usually it is a countable set, i.e. a subset of the natural numbers, or it is uncountable as a subset of the real numbers. All ﬁnite-dimensional (complex) Hilbert spaces are isomorphic to Cn where n = dim(H). Thus linear operators become n × n matrices, and Hermitian operators in ﬁnite-dimensional spaces may in this way always be diagonalized, since ﬁnite-dimensional matrices may always be diagonalized. For iniﬁte-dimensional Hilbert spaces such as L2 , the situation is not so simple. Many operators may be diagonalized, but others can not. Take for example the operator P = −i ∂ , ∂x acting on L2 (R)-vectors. Writing out the eigenvalue equation (using the classical derivative!) yields ∂φ = pφ(x). −i ∂x This diﬀerential equation has the solutions φp (x) = Aeipx . These solutions are however not vectors of L2 (R), since ∞ eipx dx −∞ does not exist. As another example, consider the operator X; multiplication by x. The eigenvalue equation becomes xφ(x) = x φ(x), whose only sulution is φ(x) = Bδ(x − x ). This is not a function! Both operators considered above are fundamental in quantum mechanics. Even though their “eigenvectors” are not proper vectors of H, it is very useful to consider them as such. Physicists use the name eigenvectors for the above solutions, even though it is mathematically incorrect. Actually, they are distributions. Do the “eigendistributions” form a basis for H in some way? In the cases of P and X they do, since any square integrable function ψ(x) obeys ψ(x) = δ(x − x0 )ψ(x0 ) dx0 , and ψ(x) = eipx φ(p) dp, so that there exist a superposition (i.e., an integral) of the eigendistributions that equals ψ(x). The above equations are nothing more than the formal deﬁnitions of the Fourier transform and the Dirac δ-function, respectively. In most cases physicists encounter the eigendistributions yields a basis for H in this sense. A.4 The Fourier Transform Given a suﬃciently nice function f : R → R we may deﬁne the Fourier transform as: 1 (A.1) g(k ) = F[f ] := f (x )e−ik ·x dn x. (2π)n/2 181 Mathematical Topics This transforms the function f (x ) into another function g(k ), and the transformation may be inverted, viz., 1 f (x ) = F −1 [g] := (A.2) g(k )eik ·x dn k. (2π)n/2 In other words: A function f (x ) may be viewed as equivalent to g(k ) and conversely. Furthermore, f (x ) is the Fourier transform of g(−k ), as may easily be seen. There is a strong connection between the Fourier transforms and quantum mechanics. The inverse Fourier transform may be interpreted in the following way. The function f (x ), a function of space coordinates, is represented as a superposition of plane waves exp(ik · x ) of wavenumber k with coeﬃcients g(k ). It is the description of the spatial wave function in the basis of plane waves. A.5 Time Evolution for Time Dependent Hamiltonians Consider the time dependent Schrödinger equation, viz., i ∂Ψ = H(t)Ψ(t), ∂t where the Hamiltonian H (which is linear and Hermitian) may depend explicitly on time. We should really make some assumptions on what function space Ψ belongs to, but let us instead consider the calculations as formal. Assume that the solution Ψ(t) exists in an interval t ∈ [0, T ]. Let U (t) be the operator that takes the initial condition Ψ(0) into Ψ(t). As the Schrödinger equation is linear, U is a linear operator. This yields i ∂U Ψ(0) = H(t)U (t)Ψ(0). ∂t This holds for all Ψ(0) so that i ∂U = H(t)U (t) ∂t with U (0) = 1 is the diﬀerential equation governing the solution operator (or the propagator.) Integration yields U (t) = 1 − i t H(t1 )U (t1 ) dt1 . (A.3) 0 This suggests an iteration process called Picard iteration. The ﬁrst iteration is U (t) = 1 − i t H(t1 )[1 − i t1 H(t2 )U (t2 ) dt2 ] dt1 t t t1 =1−i H(t1 ) dt1 + (−i)2 H(t1 )H(t2 )U (t2 ) dt1 t2 . 0 0 0 0 0 The succesive iterations is given by replacing U (tn ) with Eqn. (A.3). Performing inﬁnitely many iterations then gives U (t) = ∞ (−i)n n=0 182 t 0 t1 dt1 dt2 . . . 0 tn−1 dtn H(t1 )H(t2 ) . . . H(tn ). 0 A.5 – Time Evolution for Time Dependent Hamiltonians If [H(t1 ), H(t2 )] = 0 we can rewrite this as U (t) = t t ∞ (−i)n t dt1 dt2 . . . dtn H(t1 )H(t2 ) . . . H(tn ). n! 0 0 0 n=0 Unfortunately, we cannot. The Hamiltonians at diﬀerent time levels may be arbitrarly diﬀerent from each other. We therefore introduce the time ordering operator T . For H(t1 ) and H(t2 ) it is deﬁned by T [H(t1 )H(t2 )] = H(max{t1 , t2 })H(min{t1 , t2 }). For an set of n time levels S = {ti , i = 1, . . . , n} it is deﬁned by 2 2 H(t) = H(max{S}) T H(t) , T t∈S t∈S where S = S − max{S}. We also deﬁne T [A + B] = T A + T B. With this deﬁnition, we have t t t ∞ (−i)n T dt1 dt2 . . . dtn H(t1 )H(t2 ) . . . H(tn ) U (t) = n! 0 0 0 n=0 t = T exp −i H(t) dt . 0 This is the formal expression for the propagator in the case of a time dependent Hamiltonian. As we see, the expression reduces to U (t) = exp [−itH] in the case of a time independent Hamiltonian. 183 Appendix B Program Listings B.1 DFT Solver For One-Dimensional Problem This is the Matlab source code for the simulations in chapter 4. The code is very simple and easy to adapt to new problems. B.1.1 fft schroed.m % % % % % % % % % % % % % % % % % % % % % % S i m p l e p r o g r a m for s o l v i n g the time d e p e n d e n t S c h r o e d i n g e r eqn . in one d i m e n s i o n with the s p e c t r a l m e t h o d . The PDE is g i v e n by i p s i _ t = H psi , psi ( t0 ) = phi . The h a m i l t o n i a n H is H = T + V = - del ^ 2 + V ( x ) The e x a c t s o l u t i o n is g i v e n by psi ( t ) = U (t , t0 ) phi , w h e r e the p r o p a g a t o r U is g i v e n as U (t , t0 ) = exp ( - i ( t - t0 ) H ) We e m p l o y a split - o p e r a t o r m e t h o d in time , in w h i c h U ( t0 + tau , t0 ) is a p p r o x i m a t e d by U = DED , D = exp ( - i tau /2 T ) and E = exp ( - i tau V ) . Note that T ( and h e n c e also D ) is d i a g o n a l in the F o u r i e r t r a n s f o r m e d r e p r e s e n t a t i o n , viz . , T g ( k ) = - k ^2 g ( k ) , g ( k ) = FT ( psi ( x ) ) . Thus A b e c o m e s a d i a g o n a l op . with d i a g o n a l e l e m e n t s D (k , k ) = exp ( - ik * k * tau /2 ) . V is d i a g o n a l in position - rep . , so E (x , x ) = exp ( - iV ( x ) * tau ) . % s o l v e e q u a t i o n on i n t e r v a l [ xmin , xmax ]. % Use N p o i n t s with s p a c i n g h . xmin = -30; xmax = 30; L = xmax - xmin ; N = 1024; h = L /( N -1) ; b = 2.5; % half w i d t h of b a r r i e r a = 28; % h e i g h t of b a r r i e r % c h o o s e time step and end - of - s i m u l a t i o n time . tau = 0.00025; t_final = 2.0; plot_at_time = [0 , 0.5 , 1 , 1.5 , 2 , t_final ]; plot_colors = [ ’g ’ , ’r ’ , ’g ’ , ’r ’ , ’g ’ , ’r ’ ]; % make a g a u s s i a n i n i t i a l cond . c e n t e r e d at x0 with i n i t i a l m o m e n t u m k0 . % note : k i n e t i c e n e r g y <T > = k0 ^2. % w i d t h of g a u s s i a n : sqrt ( s i g m a 2 ) x0 = -10.0; sigma2 = 4; k0 = 5; X = linspace ( xmin , xmax , N ) ; % x - c o o r d i n a t e s of grid p o i n t s . psi0 = exp ( -( X - x0 ) .*( X - x0 ) / sigma2 ) .* exp ( j * k0 * X ) ; % also c r e a t e a p o t e n t i a l V . V = zeros (1 , N ) ; for i =1: N if ( X (i ) >= - b ) & ( X ( i) <= b ) V (i ) = a ; end ; end ; rel_energy = k0 * k0 / a ; % r e l a t i v e e n e r g y of i n i t i a l wave p a c k e t % set ICs . % psi is the s o l u t i o n with DFT , p s i _ f d with FDM 185 Program Listings psi = psi0 ; psi_fd = psi0 ’; % d e f i n e f r e q u e n c i e s for A o p e r a t o r . freqs = [ linspace (0 ,N -1 , N ) ] * 2* pi / L ; freqs ( N /2+1: N) = freqs ( N /2+1: N ) - N *2* pi /L ; % d e f i n e d i a g o n a l o p e r a t o r s for split - o p e r a t o r s c h e m e D = exp ( - j * tau * freqs .* freqs /2) ; E = exp ( - j * tau * V ) ; % b u i l d the f i n i t e d i f f e r e n c e o p e r a t o r as a s p a r s e m a t r i x . e = ones (N ,1) ; FD = spdiags ([ e -2* e e ] , -1:1 , N , N ) ; FD (N ,1) =1; FD (1 , N ) =1; FD = - FD /( h * h) ; hold off newplot plot (X , V /a , ’b ’) ; plot_indices = floor ( plot_at_time / tau ) ; % p r o p a g a t e ... for step = 0: round ( t_final / tau ) n = find ( plot_indices == step ) ; if ~ isempty ( n ) hold on plot (X , rel_energy + real ( conj ( psi ) .* psi ) , plot_colors ( n ) ) ; plot (X , rel_energy + real ( conj ( psi_fd ) .* psi_fd ) , strcat ( ’: ’ , plot_colors ( n ) ) ) ; end ; % p r o p a g a t e the s p e c t r a l m e t h o d wave f u n c i o n psi . phi = fft ( psi ) ; psi = ifft ( D .* phi ) ; psi = E .* psi ; phi = fft ( psi ) ; psi = ifft ( D .* phi ) ; % p r o p a g a t e the f i n i t e d i f f e r e n c e m e t h o d wave f u n c t i o n p s i _ f d . tempFD1 = j * tau /2* FD * psi_fd ; tempFD2 = - tau * tau /4* FD *( FD * psi_fd ) ; temp = psi_fd + tempFD1 + tempFD2 ; psi_fd = ( E .*( temp ’) ) ’; tempFD1 = j * tau /2* FD * psi_fd ; tempFD2 = - tau * tau /4* FD *( FD * psi_fd ) ; psi_fd = psi_fd + tempFD1 + tempFD2 ; end ; xlabel ( ’x ’) ; ylabel ( ’ |\ psi (x , t ) |^2 ’) ; B.2 The HydroEigen class This is the class deﬁnition of HydroEigen. It is not used directly in the main program but rather through the subclass TimeSolver B.2.1 HydroEigen.h # ifndef HydroEigen_h_IS_I NC LU D ED # define HydroEigen_h_IS_I NC LU D ED # include < FEM .h > // i n c l u d e s F i e l d F E .h , G r i d F E . h # include < TimePrm .h > # include < DegFreeFE .h > # include < Arrays_real .h > // for M a t D i a g # include < SaveSimRes .h > # include < FieldFormat .h > # include < FieldSummary .h > # include < LinEqAdmFE .h > # include < MatBand_Complex .h > # include < IsOs .h > // // s i m p l e i n n e r p r o d u c t i n t e g r a n d . i n l i n e d . // class InnerProdIntegrand Ca lc : public IntegrandCalc { protected : Handle ( FieldFE ) u , v ; NUMT result ; public : InnerProdIntegran dC al c () { result = 0; } virtual void setFields ( FieldFE & u_ , FieldFE & v_ ) { u . rebind ( u_ ) ; v . rebind ( v_ ) ; } virtual NUMT getResult () { return result ; } virtual void integrandsG ( const FiniteElement & fe ) { real detJxW = fe . detJxW () ; NUMT uval = u - > valueFEM ( fe ) ; NUMT vval = v - > valueFEM ( fe ) ; 186 B.2 – The HydroEigen class result += conjugate ( uval ) * vval * detJxW ; // i n t e g r a t e and s t o r e in r e s u l t . } }; // // s i m p l e i n t e g r a n d for <r >. i n l i n e d . // class IntegrandOfExpect at io n_ r : public InnerProdIntegran dC al c { public : IntegrandOfExpect at i on _r () { } virtual void integrandsG ( const FiniteElement & fe ) { real detJxW = fe . detJxW () ; NUMT uval = u - > valueFEM ( fe ) ; NUMT vval = v - > valueFEM ( fe ) ; Ptv ( real ) x = fe . getGlobalEvalPt () ; real r = sqrt ( x (1) * x (1) + x (2) * x (2) ) ; result += conjugate ( uval ) * vval * r * detJxW ; // i n t e g r a t e and s t o r e in r e s u l t . } }; // // // // // // // ********************************************************************** C l a s s d e c l a r a t i o n of H y d r o E i g e n - - a c l a s s that s o l v e s the time i n d e p . öS c h r d i n g e r e q u a t i o n for the two - d i m e n s i o n a l h y d r o g e n atom with an a p p l i e d m a g n e t i c f i e l d in two d i f f e r e n t g a u g e s . ********************************************************************** class HydroEigen : public FEM { protected : // P r o p e r t i e s that c o n c e r n s g e o m e t r y of the s y s t e m // and the s t r u c t u r e of the l i n e a r e q u a t i o n s . Handle ( GridFE ) grid ; // h n a d l e of the grid Handle ( FieldFE ) u; // a f i e l d over the grid Handle ( DegFreeFE ) dof ; // m a p p i n g : f i e l d < - > e q u a t i o n s y s t e m // P r o p e r t i e s that c o n c e r n s matrices , v e c t o r s and f i e l d s // used . // Note : the L e n E q A d m F E obj is not a c t u a l l y used for eqns in this c l a s s . // H o w e v e r it is a c o n v e n i e n t m e a n s for b u i l d i n g the e l e m e n t // m a t r i x with m a k e S y s t e m (* dof , * l i n e q ) ; Vec ( NUMT ) linsol ; // a v e c t o r that is used for l i n e a r eqns Handle ( LinEqAdmFE ) lineq ; // k e e p s l i n e a r e q u a t i o n s Ax = b Handle ( Matrix ( NUMT ) ) M; // mass m a t r i x Handle ( Matrix ( NUMT ) ) K; // e l e m e n t m a t r i x Handle ( SaveSimRes ) database ; // for d u m p i n g f i e l d s to disk // P a r a m e t e r s // in scan () . real gamma ; bool nucleus ; bool angmom ; bool lump ; real epsilon ; that are read from the m e n u s y s t e m and i n i t i a l i z e d // // // // // s t r e n g t h of m a g n e t i c f i e l d c o u l o m b on / off ang - mom op . on / off lump mass m a t r i x or not . singularity tolerance String mat_type ; String gridfile ; String gauge ; // c h o s e n s t o r a g e s c h e m e for m a t r i c e s . // g r i d f i l e from menu s y s t e m // i n d i c a t e s g a u g e . " s y m m e t r i c " or " non - s y m m e t r i c " int nev ; bool store_evals ; int store_evecs ; bool savemat ; bool store_prob ; bool arpack_solve ; bool * erase_log ; // // // // // // // // n u m b e r ov e i g e n v a l u e s/ v e c t o r s to c o m p u t e . s t o r e the e i g e n v a l u e s or not . n u m b e r of e i g e n v e c t o r s to s t o r e . save m a t r i c e s or not . s t o r e prob . d e n s i t y or not . use a r p a c k to s o l v e s y s t e m . a r r a y that i n d i c a t e s u n k n o w n s to be e r a s e d . when i n c o r p o r a t i n g BCs . void * the_solver ; // a void p o i n t e r used to s t o r e the E i g e n S o l v e r o b j e c t . Os logfile ; String real_format ; // w r i t e s info to c a s e n a m e . log . // c o n t a i n s e . g . " % 1 0 . 1 0 g " public : // C o n s t r u c t o r s , d e s t r u c t o r s and " c o m p u l s o r y " m e t h o d s // for a FEM c l a s s . HydroEigen () ; virtual ~ HydroEigen () ; virtual void adm ( MenuSystem & menu ) ; virtual void define ( MenuSystem & menu , int level = MAIN ) ; virtual void scan () ; virtual void solveProblem () ; virtual void report () ; virtual void resultReport () ; // H e l p s k e e p i n g t r a c k of time s p e n t on c o m p u t a t i o n s. time_t getElapsedTime () ; void reportElapsedTime () ; // Add a s t r i n g to c a s e n a m e . log and s t d e r r . void addToLogFile ( const String & s ) ; // M e t h o d s that e r a s e rows and c o l u m n s from m a t r i c e s . void eraseRowAndCol ( MatSparse ( NUMT ) & A , int k ) ; void eraseRowAndCol ( MatDiag ( NUMT ) & A , int k ) ; void eraseRowAndCols ( MatBand ( NUMT ) & New , MatBand ( NUMT ) & Old ) ; // f i l l E r a s e L o g() m a r k s the d e g r e e s of f r e e d o m that c o r r e s p o n d s to // b o u n d a r y nodes , uses bool * e r a s e _ l o g d e c l a r e d a b o v e . 187 Program Listings void fillEraseLog () ; // e n f o r c e H o m o g e n o u s B C s e r a s e s the rows and c o l u m n s from the m a t r i c e s // i n d i c a t e d by bool * e r a s e _ l o g . void enforceHomogenousBC s ( Handle ( Matrix ( NUMT ) ) & K , Handle ( Matrix ( NUMT ) ) & M ) ; // e r a s e dofs s t o r e d in e r a s e _ l o g ( shrink matrices ) // S a v e s a m a t r i x in M a t l a b c o m p a t i b l e f o r m a t . ( D i f f p a c k ’s own is not r e l i a b l e ...) void saveMatrix ( MatSparse ( NUMT ) & A , const String & Aname , const String & fname ) ; // C r e a t e s the mass matrix , l u m p e d or not , in a p p r o p r i a t e s t o r a g e f o r m a t . void makeMassMatrix2 ( Handle ( Matrix ( NUMT ) ) & Dest , const Handle ( Matrix ( NUMT ) ) & WithPattern ) ; // M u l t i p l i e s A with i n v e r s e of d i a g o n a l m a t r i x D . void multInvMatDiag ( Handle ( Matrix ( NUMT )) D , Handle ( Matrix ( NUMT ) ) A ); // M e t h o d that c a l c u l a t e d e x p e c t a t i o n v a l u e s . NUMT calcExpectation_r ( FieldFE & u , FieldFE & v ) ; NUMT calcInnerProd ( FieldFE & u , FieldFE & v ) ; // C a l u c l a t e the ( a p p r o x i m a t e) p r o b a b i l i t y d e n s i t y . ( u_j - - > | u_j |^2) void calcProbabilityDen si ty ( const FieldFE & u , FieldFE & prob , bool redim = false ) ; protected : // C o m p u t e s the i n t e g r a n d in the FEM m a t r i x f o r m u l a t i o n. C a l l e d in FEM :: m a k e S y s t e m () . virtual void integrands ( ElmMatVec & elmat , const FiniteElement & fe ); }; # endif B.2.2 HydroEigen.cpp # include # include # include # include # include # include # include # include < HydroEigen .h > < ElmMatVec .h > < FiniteElement .h > < readOrMakeGrid .h > < SparseDS .h > < IsOs .h > < time .h > < MatSparse_Complex .h > # include # include # include # include < RenumUnknowns .h > < AMD .h > < GibbPooleStockm .h > < Puttonen .h > # include < IntegrateOverGridFE .h > # include " EigenSolver . h " // * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // // C l a s s d e f i n i t i o n of E i g e n S o l v e r. See also E i g e n S o l v e r. h // // * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // m i n i m a l c o n s t r u c t o r HydroEigen :: HydroEigen () { erase_log = NULL ; } // d e s t r u c t o r HydroEigen ::~ HydroEigen () { if ( erase_log ) delete [] erase_log ; } // adm void HydroEigen :: adm ( MenuSystem & menu ) { SimCase :: attach ( menu ) ; define ( menu ) ; // let user come up with some c h o i c e s . menu . prompt () ; // i n i t i a l i z e s o l v e r . scan () ; } // // Set up menu s y s t e m e x t e n s i o n s . // void HydroEigen :: define ( MenuSystem & menu , int level ) { menu . addItem ( level , " gridfile " , " file or prepro command " , " P = PreproStdGeom | DISK r =40 degrees =360 | e = ElmT3n2D nel =1000 resol =100 ") ; menu . addItem ( level , menu . addItem ( level , menu . addItem ( level , menu . addItem ( level , menu . addItem ( level , menu . addItem ( level , menu . addItem ( level , menu . addItem ( level , menu . addItem ( level , menu . addItem ( level , menu . addItem ( level , menu . addItem ( level , menu . addItem ( level , menu . addItem ( level , " nev " , " no of eigenvalues " , " 100 ") ; " gamma " , " magnetic field strength " , " 0.0 " ) ; " nucleus " , " turn on / off coulomb term " , " true " ) ; " epsilon " , " singularity tolerance " , " -1.0 " ) ; " angmom " , " turn on / off angmom term " , " false " ) ; " nitg " , " number of itegration points (0 uses GAUSS_POINTS ) " , " 0 " ) ; " lump " , " lump mass matrix or not " , " false " ) ; " warp " , " warp factor for grid " , " 1.0 " ) ; " scale " , " scale factor for grid " , " 1.0 " ) ; " renum " , " try to renumber nodes ? " , " true " ) ; " savemat " , " save matrices as Matlab files " , " true " ) ; " store prob density " , " store probability density fields " , " false " ) ; " use arpack " , " diagonalize with arpack or not " , " true " ) ; " gauge " , " gauge " , " symmetric " ) ; // or " non - s y m m e t r i c " menu . addItem ( level , " store_evecs " , " number of evecs to store " , " -1 ") ; // -1 m e a n s all ... menu . addItem ( level , " real format " , " real format for streams used " , " %10.10 g " ) ; // T h e s e c o m m a n d s m a k e s the d i f f e r e n t c l a s s e s 188 B.2 – The HydroEigen class // f e t c h d e f a u l t s e t t i n g s from the menu s y s t e m . // For example , the d e f a u l t s o l v e r type for l i n e a r // s y s t e m s of e q u a t i o n s h a n d l e d by L i n E q A d m ( FE ) // can be set in the menu , and t h e s e c h o i c e s are // p a s s e d on when c o n s t r u c t i n g s o l v e r o b j e c t s . SaveSimRes :: defineStatic ( menu , level +1) ; // L i n E q A d m F E :: d e f i n e S t a t i c( menu , l e v e l +1) ; LinEqAdm :: defineStatic ( menu , level +1) ; FEM :: defineStatic ( menu , level +1) ; } // // I n i t i a l i z e s i m u l a t o r // void HydroEigen :: scan () { MenuSystem & menu = SimCase :: getMenuSystem () ; // Get the m a t r i x type . mat_type = menu . get ( " matrix type " ) ; // If s a v e m a t == true , the m a t r i c e s of the ev - p r o b l e m will be // s a v e d in M a t l a b c o m p a t i b l e f i l e s . savemat = menu . get ( " savemat " ) . getBool () ; // If s t o r e _ p r o b == true , then the p r o b a b i l i t y d e n s i t y // for each e i g e n v e c t o r that is s t o r e d is also s t o r e d . store_prob = menu . get (" store prob density " ) . getBool () ; // If a r p a c k _ s o l v e == true , then A R P A C K will be c a l l e d // to a c t u a l l y d i a g o n a l i z e the p r o b l e m . U s u a l l y this is done . // but to c o m p a r e the d i a g o n a l i z a t i o n t i m e s with e . g . M a t l a b or // L A P A C K one m i g h t want to i n s t e a d save the m a t r i c e s and // skip A R P A C K d i a g o n a l i z a t i o n. arpack_solve = menu . get ( " use arpack ") . getBool () ; // g a u g e i n d i c a t e s what g a u g e to use for the p o t e n t i a l s . // U s u a l l y set to " s y m m e t r i c ". gauge = menu . get ( " gauge " ) ; if (!(( gauge == " symmetric " ) || ( gauge == " non - symmetric " ) ) ) { // u n k n o w n g a u g e s_e < < " Unknown gauge ! must be ’ symmetric ’ or ’ non - symmetric ’. Now set to ’ symmetric ’. " < < endl ; gauge = " symmetric " ; } // // Set up the l o g f i l e . // logfile . rebind ( *( new Os ( aform ( " % s . log " , casename . c_str () ) , NEWFILE ) ) ) ; // // Get the grid from the menu s y s t e m . // gridfile = menu . get ( " gridfile " ) ; grid . rebind ( new GridFE () ) ; readOrMakeGrid (* grid , gridfile ) ; // // If the grid is a disk , the f i r s t node is a l w a y s // at the edge . We s h o u l d r e a l l y loop t h r o u g h the n o d e s // to be sure . . . : ) // r is only used when w a r p i n g ( g r a d i n g ) the grid . // real r = grid - > getCoor (1) . norm () ; // get the r a d i u s of the disk , if it is a disk . // // // // if If r e n u m == true , we r e n u m b e r the n o d e s in the grid to o p t i m i z e the b a n d w i d t h of the m a t r i x . This is i m p o r t a n t (!) if we use b a n d e d matrices , but not that i m p o r t a n t if M a t S p a r s e is b e i n g used . ( menu . get ( " renum " ) . getBool () ) { // try to r e n u m b e r the n o d e s to o p t i m i z e ... s_e < < " Optimizing grid ... " < < endl ; Handle ( RenumUnknowns ) renum_thing ; // not s t a b l e ?? c r a s h e s for some disk g r i d s . :( // r e n u m _ t h i n g. r e b i n d ( new G i b b P o o l e S t o c k m() ) ; renum_thing . rebind ( new Puttonen () ); renum_thing - > renumberNodes ( grid () ); } // Get grid m o d i f i c a t i o n p a r a m e t e r s . // See t h e s i s text for d e t a i l s on the // g r a d i n g ( w a r p i n g ) p r o c e s s . real warp = menu . get ( " warp " ) . getReal () ; real scale = menu . get (" scale " ) . getReal () ; if ( warp != 1) { s_e < < " Warping grid with factor " < < warp < < " ... " < < endl ; for ( int i =1; i <= grid - > getNoNodes () ; i ++) { real d = pow ( grid - > getCoor ( i ) . norm () , warp -1) ; grid - > putCoor ( grid - > getCoor ( i ) * d* pow (r ,1 - warp ) , i ) ; } } if ( scale != 1) { s_e < < " Scaling grid with factor " < < scale < < " ... " < < endl ; for ( int i =1; i <= grid - > getNoNodes () ; i ++) { grid - > putCoor ( grid - > getCoor ( i ) * scale , i ) ; } } // S a v c e the grid to . temp . grid . // U s u a l l y not n e c e s s a r y as d u m p i n g f i e l d s // will c r e a t e c a s e n a m e . grid as well ... Os gridf ( " . temp . grid " , NEWFILE ) ; grid - > print ( gridf ) ; // w r i t e w a r p e d / o p t i m i z e d grid to file s_e < < " Saving the warped / optimized grid to . temp . grid ... " < < endl ; 189 Program Listings // Get n u m b e r of e i g e n v e c t o r s to find . String nev_string = menu . get ( " nev " ) ; nev = atoi ( nev_string . c_str () ) ; // n u c l e u s t u r n s on / off the C o u l o m b i n t e r a c t i o n // in the H a m i l t o n i a n. g a m m a is the m a g n e t i c f i e l d . // If lump == true , we lump the mass m a t r i x . // a n g m o m t u r n s on / off the term prop . to d / d phi in H . nucleus = menu . get ( " nucleus " ) . getBool () ; gamma = menu . get ( " gamma " ). getReal () ; lump = menu . get ( " lump " ) . getBool () ; angmom = menu . get ( " angmom " ) . getBool () ; // Get n u m b e r of e i g e n v e c t o r s to s t o r e . // A n e g a t i v e v a l u e a u t o m a t i c a l l y s a v e s all v e c t o r s . store_evecs = atoi ( menu . get ( " store_evecs " ) . c_str () ) ; if (( store_evecs < 0) || ( store_evecs > nev ) ) store_evecs = nev ; // Init the s i m r e s d a t a b a s e . // We s t o r e f i e l d s ( i . e . , e i g e n v e c t o r s) with // database - > dump ( f i e l d ) ; FEM :: scan ( menu ) ; database . rebind ( new SaveSimRes () ) ; database - > scan ( menu , grid - > getNoSpaceDim () ) ; // Init the f i e l d u and the dof . // dof is a c t u a l l y not used in the s i m u l a t o r , // n e e d e d in the D e g F r e e F E o b j e c t . u . rebind ( new FieldFE (* grid , " u " ) ) ; dof . rebind ( new DegFreeFE (* grid , 1) ) ; // 1 for 1 u n k n o w n per node // Init the l i n e a r s y s t e m . ( H o l d s m a t r i x in e i g e n v a l u e f o r m u l a t i o n) // FEM s y s t e m s are not s o l v e d in H y d r o E i g e n , but the s t r u c t u r e is n e e d e d . lineq . rebind ( new LinEqAdmFE () ) ; lineq - > scan ( menu ) ; linsol . redim ( grid - > getNoNodes () ) ; linsol . fill (0.0) ; lineq - > attach ( linsol ) ; // // // // Set up i n t e g r a t i o n r u l e s . This is s o m e w h a t t e n t a t i v e : A t r a p e z o i d a l rule is set up if nitg > = 2 . This rule s h o u l d ONLY be used with t r i a n g u l a r e l e m e n t s . s_e < < " Setting up integration rules ... " < < endl ; String n_string = menu . get ( " nitg " ) ; int n = atoi ( n_string . c_str () ) ; int N = n *( n +1) /2; if ( n >=2) { // set up t r a p e z o i d a l rule . // NB : ONLY W O R K S FOR T R I A N G U L A R E L E M E N T S ! s_e < < " Using " < < N < < " integration points w / weights . " < < endl ; s_e < < " ( Are you sure that your elements are triangular ?) " < < endl ; VecSimple ( Ptv ( real ) ) itg_x ( N ) ; VecSimple ( real ) itg_w ( N) ; int i = 1 , j = 1; real dx = 1.0/( real ) (n -1) ; real dx2 = dx * dx ; for ( int c =1; c <= N ; c ++) { // set up c o o r d i n a t e s real xi1 = dx *(i -1) ; real xi2 = dx *(j -1) ; itg_x ( c) . redim (2) ; itg_x ( c) (1) = xi1 ; itg_x ( c) (2) = xi2 ; // set up w e i g h t if ( (( i ==1) && ( j ==1) ) || (( i ==1) && ( j == n ) ) || (( i == n ) && ( j ==1) ) ) { if ( (( i ==1) && ( j == n ) ) || (( i == n ) && ( j ==1) ) ) itg_w ( c ) = 0.125* dx2 ; // s h a r p c o r n e r s ( 1 / 8 s q u a r e ) else itg_w ( c ) = 0.25* dx2 ; // ( 1 / 4 s q u a r e ) } else if ( ( i ==1) || ( j ==1) || ( i ==( n - j +1) ) ) itg_w ( c ) = 0.5* dx2 ; // e d g e s ( 1 / 2 s q u a r e ) else itg_w ( c ) = dx2 ; // i n t e r i o r ( 1 / 1 s q u a r e ) // a d v a n c e to next p o i n t if ( i == ( n - j +1) ) { j ++; i = 1; } else i ++; } // end of c =1.. N loop . // Save i n t e g r a t i o n rule into E l m I t g R u l e s FEM :: i t g _ r u l e s . itg_rules . ignoreRefill () ; itg_rules . setPointsAndWeights ( itg_x , itg_w ) ; } else { // use g a u s s i a n quad if n <2. s_e < < " Using quadrature from menu system . ( Phew . I don ’t like experimental setups .) " < < endl ; } // O u t p u t f o r m a t from menu . real_format = menu . get ( " real format " ) ; s_o - > setRealFormat ( real_format ) ; 190 B.2 – The HydroEigen class s_e - > setRealFormat ( real_format ) ; logfile - > setRealFormat ( real_format ) ; } // // F u n c t i o n for l o g g i n g both to s_e and l o g f i l e . // void HydroEigen :: addToLogFile ( const String & s ) { // I c o u l d w r i t e a joke here ; did you get it ? logfile < < s ; s_e < < s; } // // This f u n c t i o n e r a s e s rows / cols a B A N D E D m a t r i x a c c o r d i n g to e r a s e _ l o g [] // Some a s s u m p t i o n s are made on the m a t r i x s t o r a g e f o r m a t that I r e a l l y // have no r e a s o n to b e l i e v e in . However , I have n e v e r e x p e r i e n c e d a // c o u n t e r e x a m p l e. // void HydroEigen :: eraseRowAndCols ( MatBand ( NUMT ) & New , MatBand ( NUMT ) & Old ) { int rows = Old . getNoRows () ; int bw = Old . getBandwidth () ; bool symm_stor = false ; bool pivot_allow = false ; int cols2 = 2* bw - 1; int i , j , k ; int ndiag = cols2 > > 1; // n u m b e r of sub / s u p e r d i a g o n a l s . int middlecol = ndiag + 1; for ( k = rows ; k >=1; k - -) { // if row / col k is to be deleted , fix m a t r i x data in p l a c e ! if ( erase_log [k -1]) { // " move up " rows > k by one . for ( i = k ; i <= rows -1; i ++) { for ( j =1; j <= cols2 ; j ++) { Old (i ,j ) = Old ( i +1 , j) ; } } // fix u p p e r part of m a t r i x for ( i =k -1; i >= k - ndiag ; i - -) { // t o t a l s nd rows if ( i >=1) { for ( j= middlecol +( k -1) -i +1; j <= cols2 -1; j ++) Old (i , j ) = Old (i , j +1) ; Old (i , cols2 ) =0; } } // fix l o w e r part of m a t r i x for ( i = k ; i <= k + ndiag -1; i ++) { // t o t a l s nd rows if ( i <= rows ) { for ( j= middlecol - 1 + ( k - i ) ; j >=2; j - -) Old (i , j ) = Old (i , j -1) ; Old (i , 1) = 0; } } rows - -; // we have one less row now . } // end of if } // end of k - loop // u p d a t e o t h e r m a t r i x p a r a m e t e r s - - > New . New . redim ( rows , rows , bw , symm_stor , pivot_allow ) ; for ( i =1; i <= rows ; i ++) for ( j =1; j <= cols2 ; j ++) New (i , j ) = Old (i , j ) ; } // // E r a s e row and col n u m b e r k from s p a r s e m a t r i x _en p l a c e _ . // void HydroEigen :: eraseRowAndCol ( MatSparse ( NUMT ) & A , int k ) { SparseDS & pat = A . pattern () ; int n = pat . getNoRows () ; int m = pat . getNoColumns () ; // NOTE : c o u l d not use g e t N o N o n z e r o e s() b e c a u s e of // not c o r r e c t n u m b e r of e n t r i e s . I n s t e a d use d e f i n i t i o n of irow ( n +1) . // int nnz = pat . g e t N o N o n z e r o e s() ; / / l e n g t h of e n t r i e s a r r a y . int nnz = pat . irow ( n +1) -1; // // * * * first , e r a s e row k . * * * // // // // // must e r a s e e n t r i e s ( s ) from s = irow ( k ) to s = irow ( k +1) -1 , and also same i n d i c e s in jcol . then , we must c o r r e c t the e n t r i e s in irow , by s u b t r a c t i n g d e l t a from irow ( i ) , 191 Program Listings // i > k and r e m o v e irow ( k ) a l t o g e t h e r . // d e l e t e e n t r i e s ( s ) and jcol ( s ) int first = pat . irow ( k ) ; int last = pat . irow ( k +1) -1; int delta = last - first + 1; // n u m b e r of e l e m e n t s to r e m o v e from j for ( int s = first ; s <= nnz - delta ; s ++) { A ( s ) = A (s + delta ) ; pat . jcol (s ) = pat . jcol (s + delta ) ; } // fix irow ( i ) // take a copy . VecSimple ( int ) irow_copy (n -1) ; for ( int i =1; i < k ; i ++) irow_copy ( i ) = pat . irow ( i ) ; for ( int i =k ; i <= n -1; i ++) irow_copy ( i ) = pat . irow ( i +1) - delta ; // r e d i m pat . irow . and fill with m o d i f i e d copy . // cols are set to m -1 , a n t i c i p a t i n g r e m o v a l of c o l u m n ... pat . redimIrow (n -1 , m -1) ; for ( int i =1; i <= n -1; i ++) pat . irow (i ) = irow_copy ( i ) ; // last e n t r y of irow must be set p r o p e r y . pat . irow ( n +1 -1) = nnz - delta +1; // // * * * next , e r a s e c o l u m n k // // // // // // // // run b a c k w a r d s , s = nnz d o w n t o 1. for each s such that jcol ( s ) = k , jcol ( s ) and e n t r i e s ( s ) must be d e l e t e d . f u r t h e r m o r e , jcol ( s ’) must be l e s s e n e d by one for jcol ( s ’) > k . we have e r a s e d an element , so i r o w s must be f i x e d . we s u b t r a c t 1 from e v e r y irow ( i ) >= s . // u p d a t e v a r i a b l e s n = pat . getNoRows () ; nnz = pat . irow ( n +1) -1; // loop b a c k w a r d s ... for ( int s = nnz ; s >=1; s - -) { if ( pat . jcol ( s ) > k ) pat . jcol ( s ) - -; else if ( pat . jcol ( s ) == k ) { // r e m o v e e n t r y if at c o l u m n k . // ( s l i g h t s p e c i a l case if s = nnz . n o t h i n g // n e e d s to be done with e n t r i e s / jcol then .) for ( int t = s ; t <= nnz -1; t ++) { pat . jcol ( t ) = pat . jcol ( t +1) ; A ( t ) = A ( t +1) ; } // u p d a t e i r o w s .. for ( int i =1; i <= n +1; i ++) if ( pat . irow ( i ) >s ) pat . irow ( i ) - -; } } } // // W r a s e row / col no . k in a M a t D i a g ( NUMT ) o b j e c t . very s i m p l e i n d e e d ... // void HydroEigen :: eraseRowAndCol ( MatDiag ( NUMT ) & A , int k ) { Vec ( NUMT ) temp ( A . getNoRows () -1) ; for ( int i =1; i < k ; i ++) temp ( i ) = A ( i ) ; for ( int i =k ; i < A . getNoRows () ; i ++) temp ( i ) = A ( i +1) ; A . redim ( A . getNoRows () -1) ; for ( int i =1; i <= A . getNoRows () ; i ++) A ( i ) = temp ( i ) ; } // // S i m p l e stop - w a t c h f u n c t i o n s . // This s h o u l d be i m p r o v e d ; only s e c o n d s are c o u n t e d . // time_t HydroEigen :: getElapsedTime () { static time_t start = time ( NULL ) ; return time ( NULL ) - start ; } void HydroEigen :: reportElapsedTime () { addToLogFile ( aform ( " - - - elapsed time so far : % d s .\ n " , getElapsedTime () ) ) ; } // 192 B.2 – The HydroEigen class // e r a s e _ l o g is an a r r a y of b o o l s ; one per each u n k n o w n in l i n e a r s y s t e m . // e r a s e _ l o g [ k ] is set to true if u n k n o w n k c o r r e s p o n d s with a b o u n d a r y node . // void HydroEigen :: fillEraseLog () { int no_nodes = grid - > getNoNodes () ; int no_elms = grid - > getNoElms () ; erase_log = new bool [ no_nodes ]; for ( int k =0; k < no_nodes ; k ++) erase_log [ k ] = false ; // fill e r a s e _ l o g with true for b o u n d a r y n o d e s . for ( int e =1; e <= no_elms ; e ++) { int no_dof_per_elm = dof - > getNoDofInElm ( e ) ; for ( int e_dof =1; e_dof <= no_dof_per_elm ; e_dof ++) { int n = grid - > loc2glob (e , e_dof ); if ( grid - > boNode ( n) ) { int k = dof - > loc2glob (e , e_dof ) - 1; erase_log [ k ] = true ; } } } } // // this f u n c t i o n e n f o r c e s the h o m o g e n o u s BCs by r e m o v i n g all rows / cols in the // m a t r i c e s K and M that c o r r e s p o n d s to b o u n d a r y n o d e s . uses f i l l E r a s e L o g() . // void HydroEigen :: enforceHomogenousBC s ( Handle ( Matrix ( NUMT ) ) & K , Handle ( Matrix ( NUMT ) ) & M ) { addToLogFile ( " Erasing rows and columns ...\ n " ) ; // find the rows / cols that s h o u l d be r e m o v e d from the s y s t e m . // l o o p s t h r o u g h the n o d e s in the grid . fillEraseLog () ; addToLogFile ( aform ( " Total number of nonzeroes before == % d \ n " , K - > getNoNonzeroes () ) ) ; // // d i f f e r e n t t r e a t m e n t of M a t B a n d and M a t S p a r s e ... // if ( mat_type == " MatBand " ) { Handle ( MatBand ( NUMT )) K_new , M_new ; K_new . rebind ( new MatBand ( NUMT ) ( 1 ,1 ) ) ; M_new . rebind ( new MatBand ( NUMT ) ( 1 ,1 ) ) ; eraseRowAndCols ( K_new () , ( MatBand ( NUMT ) &) K () ) ; if (! lump ) eraseRowAndCols ( M_new () , ( MatBand ( NUMT ) &) M () ) ; addToLogFile ( " MatBand ( NUMT ) object K after BC incorporation :\ n" ) ; addToLogFile ( aform ( " bandwidth == % d \ n dimension == % d \ n nonzeroes K_new - > getBandwidth () , K_new - > getNoRows () , K_new - > getNoNonzeroes () ) ) ; ==% d \ n " , // copy back ... if (! lump ) M . rebind ( M_new () ) ; K . rebind ( K_new () ) ; } else if ( mat_type == " MatSparse " ) { for ( int k =K - > getNoRows () ; k >=1; k - -) if ( erase_log [k -1]) { s_e < < ". " ; eraseRowAndCol (( MatSparse ( NUMT ) &) (* K) , k ) ; if (! lump ) eraseRowAndCol (( MatSparse ( NUMT ) &) (* M ) , k ) ; } s_e < < endl ; addToLogFile ( " MatSparse ( NUMT ) object K after BC incorporation :\ n " ) ; int nz = (( MatSparse ( NUMT ) &) K () ) . pattern () . irow (K - > getNoRows () +1) -1; int rows = K - > getNoRows () ; addToLogFile ( aform ( " dimension == % d \ n nonzeroes == % d (% g percent ) \ n " , rows , nz , 100.0* nz /( real ) ( rows * rows ) ) ) ; } // BCs for l u m p e d mass m a t r i x ... if ( lump ) { addToLogFile ( " Mass matrix M is lumped . Enforcing BCs ...\ n " ) ; for ( int k =K - > getNoRows () ; k >=1; k - -) if ( erase_log [k -1]) eraseRowAndCol (( MatDiag ( NUMT ) &) (* M ) , k ) ; } } // // This f u n c t i o n is the main f u n c t i o n of the s o l v e r . // C a l c u l a t e s matrices , s o l v e s e i g e n v a l u e p r o b l e m and r e p o r t s . // void HydroEigen :: solveProblem () { int n , m ; // m a t r i x d i m e n s i o n s . // Init t i m e r . getElapsedTime () ; reportElapsedTime () ; // Make the m a t r i x K , i . e . , the left m a t r i x . addToLogFile ( " Creating the K matrix ...\ n " ) ; 193 Program Listings makeSystem (* dof , * lineq ); // f e t c h size . lineq - > A () . size (n , m ) ; // Make the mass m a t r i x M addToLogFile ( " Creating the M matrix ... " ) ; makeMassMatrix2 (M , lineq - >A () ) ; // F e t c h a copy of K . // For huge s y s t e m s this may be p r o b l e m a t i c. // I s h o u l d i n v e s t i g a t e the p o s s i b i l i t y of // s i m p l y m a k i n g a r e f e r e n c e i n s t e a d . lineq - > A () . makeItSimilar (K ) ; * K = lineq - >A () ; reportElapsedTime () ; // Tell user what ’s the deal with the grid . addToLogFile ( " Grid properties :\ n " ) ; addToLogFile ( aform ( " gridfile == % s \ n" , SimCase :: getMenuSystem () . get ( " gridfile " ) . c_str () ) ) ; addToLogFile ( aform ( " number of nodes == % d \ n number of elements == % d \ n " , grid - > getNoNodes () , grid - > getNoElms () )); // E n f o r c e BCs . addToLogFile ( " Enforcing homogenous Dirichlet BCs ...\ n" ) ; enforceHomogenousBC s (K , M) ; reportElapsedTime () ; // // Here we s t a r t the real p r o b l e m s o l v i n g part . // addToLogFile ( " Instantiating the EigenSolver object ...\ n " ) ; EigenSolver * solver ; if (! lump ) { // If M is full then the p r o b l e m is a g e n e r a l i z e d p r o b l e m . solver = new EigenSolver ( K () , M () , nev , SimCase :: getMenuSystem () ) ; } else { // If M is l u m p e d we e l i m i n a t e the r i g h t hand m a t r i x // by m u l t i p l y i n g with its inverse , w h i c h is r e a l l y // s i m p l e to to ... multInvMatDiag (M , K ) ; // // // // // // // // // // // // // // // // if ( m a t _ t y p e = = " M a t B a n d ") { for ( int p =1; p <= K - > g e t N o R o w s () ; p ++) for ( int q =1; q <= K - > g e t N o C o l u m n s() ; q ++) if ( ( ( M a t B a n d ( NUMT ) &) K () ) . i n s i d e B a n d ( p , q ) ) K - > elm (p , q ) /= M - > elm (p , p ) ; } else if ( m a t _ t y p e = = " M a t S p a r s e ") { M a t S p a r s e ( NUMT ) & X = ( M a t S p a r s e ( NUMT ) &) (* K ) ; S p a r s e D S & pat = X . p a t t e r n () ; for ( int p =1; p <= pat . g e t N o R o w s () ; p ++) { int f i r s t = pat . irow ( p ) ; int last = pat . irow ( p +1) -1; for ( int q = f i r s t ; q <= last ; q ++) X ( q ) /= M - > elm ( p , p ) ; } } solver = new EigenSolver ( K () , nev , SimCase :: getMenuSystem () ) ; } // Save m a t r i c e s if d e s i r e d . // No need to save M if l u m p e d ; inv ( M ) is a l r e a d y m u t i p l i e d into K . if ( savemat && ( mat_type == " MatSparse " ) ) { addToLogFile ( " Saving matrices in Matlab format ...\ n" ) ; saveMatrix ( ( MatSparse ( NUMT &) ) (* K ) , "K " , aform ( " % s_K . m " , casename . c_str () ) ) ; if (! lump ) saveMatrix ( ( MatSparse ( NUMT &) ) (* M ) , " M " , aform ( " % s_M . m " , casename . c_str () ) ) ; } // Skip d i a g o n a l i z a t i o n? if (! arpack_solve ) { addToLogFile ( " Skipping diagonalization with ARPACK !\ n " ) ; } else { // set up c o m p u t a t i o n a l mode . Note that E i g e n S o l v e r // in fact s u p p o r t s finding , e . g . , the _ l a r g e s t _ e i g e n v a l u e s // i n s t e a d of the l o w e s t . // The r e a s o n why we use a void * is that // due to not - so - good t e m p l a t i n g used // in A R P A C K ++ we c a n n o t i n c l u d e the c l a s s // d e f i n i t i o n s in H y d r o E i g e n . h ... the_solver = ( void *) solver ; solver - > setCompMode ( MODE_REGULAR ) ; solver - > setMatrixKind ( MATRIX_COMPLEX ); solver - > setSpectrumPart ( SPECTRUM_SMALLEST _R EA L ) ; // Go ! addToLogFile ( " Finding eigenvalues and eigenvectors ...\ n " ) ; solver - > solveProblem () ; addToLogFile ( " Done !\ n " ); reportElapsedTime () ; // R e p o r t on the r e s u l t s and f i n d i n g s . 194 B.2 – The HydroEigen class report () ; } } // // To c a l c u l a t e K m a t r i x ... // e v a l u a t e s the i n t e g r a n d s in the f i n i t e e l e m e n t f o r m u l a t i o n. // void HydroEigen :: integrands ( ElmMatVec & elmat , const FiniteElement & fe ) { /* The H a m i l t o n i a n for this p r o b l e m is : H = - n a b l a ^ 2 + g a m m a ^2 r ^ 2 / 4 + i g a m m a ( - y , x ) . n a b l a - 2/ r */ int i ,j ,k ; const int nsd = fe . getNoSpaceDim () ; const int nbf = fe . getNoBasisFunc () ; const real detJxW = fe . detJxW () ; const real gamma2 = gamma * gamma ; Ptv ( real ) coords = fe . getGlobalEvalPt () ; real X = coords (1) ; real Y = coords (2) ; static int gauge_int = ( gauge == " symmetric " ? 1 : 2) ; real nabla2 ; real r2 = X * X + Y * Y ; real r2c = sqrt ( r2 ) ; if ( r2c <= epsilon ) r2c = epsilon ; // C o m p l e x Im (0 ,1) ; Complex Im (0 ,1) ; for ( i = 1; i <= nbf ; i ++) for ( j = 1; j <= nbf ; j ++) { nabla2 = 0; for ( k = 1; k <= nsd ; k ++) nabla2 += fe . dN (i , k ) * fe . dN (j , k) ; // add c o n t r i b . from n a b l a ^2 ( s t i f f n e s s m a t r i x K ) elmat . A (i , j ) += nabla2 * detJxW ; if ( gauge_int == 1) { // * * * s y m m e t r i c g a u g e *** // add c o n t r i b . from h a r m o n i c o s c i l l a t o r term elmat . A (i , j ) += fe . N ( i) * fe . N ( j ) * r2 *0.25* gamma2 * detJxW ; // add c o n t r i b . from <A , nabla >. if ( angmom ) elmat .A (i , j ) += - Im * fe . N ( i ) * gamma *( -Y * fe . dN (j ,1) + X * fe . dN (j ,2) ) * detJxW ; } else { // * * * non - s y m m e t r i c g a u g e *** elmat . A (i , j ) += 2* Im * gamma * fe . N ( i ) * Y* fe . dN (j ,1) * detJxW ; elmat . A (i , j ) += gamma2 * fe . N ( i ) * fe . N (j ) * Y * Y * detJxW ; } // add c o n t r i b from c o u l o m b term if ( nucleus ) elmat . A (i , j ) += - fe . N (i ) * fe . N ( j ) *( 2.0/ sqrt ( r2 ) ) * detJxW ; } } // // S i m p l e r e p o r t f u n c t i o n . W r i t e s e i g e n v e c t o r s // as f i e l d s over grid . Also w r i t e s e i g e n v a l u e s. // void HydroEigen :: report () { // F e t c h the E i g e n S o l v e r o b j e c t p o i n t e r and cast it . EigenSolver & solver = *(( EigenSolver *) the_solver ); // In e v e c _ f i e l d we s t o r e a f i e l d c o r r e s p o n d i n g to the e i g e n v e c t o r // f o u n d . Handle ( FieldFE ) evec_field ; evec_field . rebind ( new FieldFE () ) ; // In this we s t u r e the n o d a l v a l u e s of the e i g e n v e c t o r f i e l d . Handle ( ArrayGen ( NUMT ) ) evec_values ; evec_values . rebind ( new ArrayGen ( NUMT ) ( grid - > getNoNodes () ) ) ; // Get the e i g e n v e c t o r s and e i g e n v a l u e s. Mat ( NUMT ) & eigenvectors = solver . getEigenvectors () ; Vec ( real ) eigenvals ( solver . getEigenvalues () . getNoEntries () ) ; // Copy e i g e n v a l u e s. ( We will m o d i f y this array , so we take a // copy i n s t e a d of a r e f e r e n c e .) for ( int k =1; k <= eigenvals . getNoEntries () ; k ++) eigenvals ( k ) = solver . getEigenvalues () ( k ) . Re () ; // i n d e x - - p e r m u t a t i o n of i n d i c e s that sort the e i g e n v a l u e s. // i n v _ i n d e x - - the i n v e r s e p e r m u t a t i o n. 195 Program Listings VecSimple ( int ) index ( eigenvals . size () ) , inv_index ( eigenvals . size () ); // Make the s o r t i n g i n d i c e s and the i n v e r s e . eigenvals . makeIndex ( index ) ; for ( int i =1; i <= index . size () ; i ++) inv_index ( index ( i ) ) = i ; // F a n c y huh ! Thanks , HPL ! // Sort a c c o r d i n g to i n d e x . eigenvals . sortAccording2index ( index ) ; addToLogFile ( " Expectation values of <r >:\ n " ) ; for ( int i = 1; i <= store_evecs ; i ++) { // // C r e a t e the f i e l d c o r r e s p o n d i n g to e i g e n v e c t o r. // int j2 = 1; // i n d e x into e i g e n v e c t o r c o m p o n e n t s . // loop t h r o u g h each grid p o i n t . for ( int j =1; j <= grid - > getNoNodes () ; j ++) { if (! erase_log [j -1]) { evec_values () ( j ) = eigenvectors ( inv_index ( i ) , j2 ) ; j2 ++; // go to next c o m p o n e n t ... } else { // do n o t h i n g with j2 , but s t o r e BC . evec_values () ( j ) = 0; } } // R e d i m f i e l d and put the f i e l d v a l u e s into it ... evec_field - > redim ( grid () , evec_values () , aform ( " % d " , i , eigenvals ( i ) ) . c_str () ); // . . . and off you go ! database - > dump ( evec_field () ) ; // W r i t e the e x p e c t a t i o n v a l u e < r >. addToLogFile ( aform ( real_format . c_str () , calcExpectation_r ( evec_field () , evec_field () ) . Re () ) if ( i < store_evecs ) addToLogFile ( " , " ) ; else addToLogFile ( " \ n " ) ; // S t o r e p r o b a b i l i t y d e n s i t y if n e e d e d . if ( store_prob ) { // c a l c u l a t e p r o b a b i l i t y d e n s i t y for ( int j =1; j <= grid - > g e t N o N o d e s () ; j ++) e v e c _ v a l u e s() ( j ) = pow ( e v e c _ v a l u e s() ( j ) . Re () ,2) + pow ( e v e c _ v a l u e s() ( j ) . Im () ,2) ; // // // // // Dump it . evec_field - > redim ( grid () , aform ( " prob_ % d " , i , eigenvals ( i ) ) . c_str () ) ; calcProbabilityDen si ty (* evec_field , * evec_field , false ) ; database - > dump ( evec_field () ) ; } } // Get e i g e n v a l u e s , p r i n t them to s t d o u t ... // a python - e v a l a b l e file . Os ev_file = Os ( aform ( " % s. eigenvalues " , casename . c_str () ) , NEWFILE ); addToLogFile ( " Eigenvalues ( real parts ) :\ n " ) ; for ( int k =1; k <= eigenvals . getNoEntries () ; k ++) { addToLogFile ( aform ( " %10.10 g " , eigenvals ( k ) ) ) ; ev_file < < aform ( " %10.10 g " , eigenvals ( k ) ) ; if ( k < eigenvals . getNoEntries () ) { addToLogFile ( " , " ) ; ev_file < < " , " ; } } addToLogFile ( " \ n " ) ; ev_file < < endl ; } // // This f u n c t i o n is not i m p l e m e n t e d. // void HydroEigen :: resultReport () {} // // c a l c u l a t e e x p e c t a t i o n v a l u e of r . // NUMT HydroEigen :: calcExpectation_r ( FieldFE & u , FieldFE & v ) { IntegrateOverGridFE integrator ; IntegrateOverGridFE integrator2 ; IntegrandOfExpec ta ti on _ r integrand ; InnerProdIntegran dC al c integrand2 ; integrand . setFields (u , v ); integrand2 . setFields (u , v) ; integrator . volumeIntegral ( integrand , * grid ) ; integrator2 . volumeIntegral ( integrand2 , * grid ) ; return integrand . getResult () / integrand2 . getResult () ; } // // c a l c u l a t e i n n e r p r o d u c t of u and v . // NUMT HydroEigen :: calcInnerProd ( FieldFE & u , FieldFE & v ) { 196 ); B.2 – The HydroEigen class IntegrateOverGridFE integrator ; InnerProdIntegrand Ca l c integrand ; integrand . setFields (u , v ) ; integrator . volumeIntegral ( integrand , * grid ) ; return integrand . getResult () ; } // // C r e a t e ( a p p r o x i m a t e) p r o b a b i l i t y d e n s i t y f i e l d . // void HydroEigen :: calcProbabilityDen si ty ( const FieldFE & u , FieldFE & prob , bool redim ) { real R , I ; int nno = grid - > getNoNodes () ; if ( redim ) prob . redim (* grid , " prob " ) ; for ( int i =1; i <= nno ; i ++) { R = u . values () ( i ) . Re () ; I = u . values () ( i ) . Im () ; prob . values () ( i ) = R * R + I * I ; } } // // save a s p a r s e m a t r i x in a Matlab - r e a d a b l e f o r m a t . // M a t r i x ( NUMT ) :: save is not e n t i r e l y r e l i a b l e ... // void HydroEigen :: saveMatrix ( MatSparse ( NUMT ) & A , const String & Aname , const String & fname ) { Os f ( fname , NEWFILE ) ; SparseDS & pat = A . pattern () ; int rows = pat . getNoRows () ; int cols = pat . getNoColumns () ; NUMT dummy ; bool reals = ( sizeof ( dummy ) == sizeof ( real ) ) ; f < < " % This is a matrix saved by HydroEigen :: saveMatrix () " < < endl ; f << "% gridfile == " < < gridfile < < " . " < < endl ; f < < endl ; f < < Aname < < " = sparse ( " < < rows < < " , " < < cols < < " ) ; " < < endl < < endl ; for ( int i =1; i <= rows ; i ++) { for ( int s = pat . irow (i ) ; s <= pat . irow ( i +1) -1; s ++) { f < < Aname < < " ( " < < i < < " , " < < pat . jcol ( s ) < < " ) = " ; if ( reals ) f < < A ( s) < < " ; " < < endl ; else f < < " complex " < < A ( s ) < < " ; " < < endl ; // c o m p l e x ( re , im ) ; } } f - > close () ; } // // m a k e M a s s M a t r i x 2. F i r s t a r g u m e n t is ref . to // m a t r i x that s h a l l hold mass matrix , s e c o n d // a r g u m e n t has the c o r r e c t p a t t e r n . // void HydroEigen :: makeMassMatrix2 ( Handle ( Matrix ( NUMT )) & Dest , const Handle ( Matrix ( NUMT ) ) & WithPattern ) { // Make mass m a t r i x a c c o r d i n g to menu c h o i c e s et . c . // L u m p e d m a t r i c e s are a l w a y s M a t D i a g ( NUMT ) o b j e c t s . // Not l u m p e d are same f o r m a t as r i g h t hand side m a t r i x . int n , m ; // m a t r i x size . WithPattern - > size (n , m) ; if (! lump ) { // Make a full mass m a t r i x . addToLogFile ( " ( not lumped ) \ n " ) ; if ( mat_type == " MatSparse " ) { // must COPY the s p a r s i t y p a t t e r n . m a k e I t S i m i l a r only // c o p i e s a _ r e f e r e n c e _ to the S p a r s e D S o b j e c t . Hence , // m o d i f y i n g one m a t r i x will d e s t r o y the s t r u c t u r e of the o t h e r . MatSparse ( NUMT ) & orig = ( MatSparse ( NUMT ) &) ( WithPattern () ) ; int nnz = orig . getNoNonzeroes () ; Dest . rebind ( new MatSparse ( NUMT )(n ,m , nnz ) ) ; SparseDS & pattern = (( MatSparse ( NUMT ) &) Dest () ). pattern () ; for ( int i =1; i <= m +1; i ++) pattern . irow ( i ) = orig . pattern () . irow ( i ) ; for ( int i =1; i <= nnz ; i ++) pattern . jcol ( i ) = orig . pattern () . jcol ( i ) ; } else { // The s t r u c t u r e of M a t B a n d is s i m p l e r to copy ... WithPattern - > makeItSimilar ( Dest ); } } else { 197 Program Listings // C r e a t e a l u m p e d mass matrix , i . e . , a M a t D i a d . addToLogFile ( " ( lumped ) \ n " ) ; M . rebind ( new MatDiag ( NUMT ) ( n ) ) ; } makeMassMatrix (* grid , * Dest , lump ) ; } // // M u l t i p l y A with i n v e r s e of D , w h i c h is d i a g o n a l . // void HydroEigen :: multInvMatDiag ( Handle ( Matrix ( NUMT ) ) D , Handle ( Matrix ( NUMT ) ) A ) { if ( mat_type == " MatBand " ) { for ( int p =1; p <= A - > getNoRows () ; p ++) for ( int q =1; q <= A - > getNoColumns () ; q ++) if ( (( MatBand ( NUMT ) &) A () ) . insideBand (p , q ) ) A - > elm (p , q ) /= D - > elm (p , p ) ; } else if ( mat_type == " MatSparse " ) { MatSparse ( NUMT ) & X = ( MatSparse ( NUMT ) &) (* A ) ; SparseDS & pat = X . pattern () ; for ( int p =1; p <= pat . getNoRows () ; p ++) { int first = pat . irow ( p ) ; int last = pat . irow ( p +1) -1; for ( int q = first ; q <= last ; q ++) X ( q ) /= D - > elm (p , p ) ; } } } // // End of c l a s s d e f i n i t i o n . // B.3 The TimeSolver class This is the class deﬁnition of TimeSolver that solves the time dependent Schrödinger equation. The main program is main.cpp and this instatiates the TimeSolver class. Notice that the full functionality from the base class HydroEigen is retained. B.3.1 TimeSolver.h # ifndef TimeSolver_h # define TimeSolver_h # include " HydroEigen . h " // // s i m p l e i n t e g r a n d for <x >. i n l i n e d . // class IntegrandOfExpect at i on _x : public InnerProdIntegrandC al c { public : IntegrandOfExpec ta ti on _ x () { } virtual void integrandsG ( const FiniteElement & fe ) { real detJxW = fe . detJxW () ; NUMT uval = u - > valueFEM ( fe ) ; NUMT vval = v - > valueFEM ( fe ) ; Ptv ( real ) x = fe . getGlobalEvalPt () ; result += conjugate ( uval ) * vval * x (1) * detJxW ; // i n t e g r a t e and s t o r e in r e s u l t . } }; // // s i m p l e i n t e g r a n d for <y >. i n l i n e d . // class IntegrandOfExpect at i on _y : public InnerProdIntegrandC al c { public : IntegrandOfExpec ta ti on _ y () { } virtual void integrandsG ( const FiniteElement & fe ) { real detJxW = fe . detJxW () ; NUMT uval = u - > valueFEM ( fe ) ; NUMT vval = v - > valueFEM ( fe ) ; Ptv ( real ) x = fe . getGlobalEvalPt () ; result += conjugate ( uval ) * vval * x (2) * detJxW ; // i n t e g r a t e and s t o r e in r e s u l t . } }; // just # define # define # define # define some h a n d y defs THETA_RULE 1 LEAP_FROG 2 IC_GAUSSIAN 1 IC_FIELD 2 // * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // 198 B.3 – The TimeSolver class // C l a s s d e c l a r a t i o n of T i m e S o l v e r - - - s o l v i n g the time // d e p e n d e n t öS c h r d i n g e r e q u a t i o n . This is d e r i v e d from // H y d r o E i g e n b e c a u s e the two p r o b l e m s s h a r e many i m p o r t a n t // p r o p e r t i e s , such as the H a m i l t o n i a n. // // * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * class TimeSolver : public HydroEigen { private : Handle ( Matrix ( NUMT ) ) A ; // u s u a l l y h o l d s m a t r i x from M a k e S y s t e m . M is d e f i n e d in H y d r o E i g e n . // Time p a r a m e t e r s . int time_method ; Handle ( TimePrm ) tip ; real dt ; real t_final ; // // // // real omega ; real delta ; real gamma0 ; // p a r a m e t e r s for // v a r y i n g m a g n e t i c f i e l d // g a m m a = g a m m a 0 * pow ( sin ( t /( pi * T ) ) , 2) * cos ( o m e g a * t + d e l t a ) real theta ; bool do_time_simulation ; // for t h e t a rule , \ in [ 0 , 1 ] . , 0 . 5 = = Crank - N i c h o l s o n // i n d i c a t e s w h e t h e r og not a time dep . s i m u l a t i o n s h o u l d be made . // R e p o r t v a r i a b l e s . real dt_rep ; int n_rep ; // time i n t e r v a l b e t w e e n r e p o r t s . // n u m b e r of r e p o r t s d u r i n g s i m u l a t i o n . i n d i c a t e s what m e t h o d to use , T H E T A _ R U L E or L E A P _ F R O G n u m e r i c a l c l o c k ... time step f i n a l time // I n i t i a l c o n d i t i o n p a r a m e t e r s . int ic_type ; // I C _ G A U S S I A N or I C _ F I E L D String field_database ; // name of the d a t a b a s e that h o l d s f i e l d s to load . int field_no ; // n u m b e r of f i e l d to use as IC . String gaussian_parameters ; // p a r a m e t e r s for g a u s s F u n c IC . Handle ( FieldFE ) * fields ; // a r r a y of f i e l d h a n d l e s . h o l d s f i e l d s l o a d e d from d a t a b a s e . int no_of_fields ; // n u m b e r of f i e l d s to load , i . e . , l e n g t h of f i e l d s a r r a y . // f i e l d s et . c .; r e c a l l that the grid is d e f i n e d in H y d r o E i g e n . // H a n d l e ( F i e l d F E ) u ; // u is d e f i n e d in H y d r o E i g e n ! Handle ( FieldFE ) u_prev ; // p r e v i o u s s o l u t i o n . Handle ( Vec ( NUMT ) ) scratch ; Handle ( Vec ( NUMT ) ) scratch2 ; // s c r a t c h v e c t o r s . // for the i n t e g r a n d s () f u n c t i o n . // d e f i n i n g t h e s e v a r i a b l e s here s a v e s some time // when a s s e m b l i n g the l i n e a r s y s t e m at each time step . MassMatIntg * mass_integrand ; // H a n d l e ( . . . ) not impl . ? ? ? ElmMatVec elmat2 ; ElmMatVec elmat3 ; public : // c o n s t r u c t o r s and d e s t r u c t o r s. TimeSolver () { mass_integrand = NULL ; }; ~ TimeSolver () { if ( mass_integrand ) delete mass_integrand ; }; // // " c o m p u l s o r y " m e t h o d s . // virtual void define ( MenuSystem & menu , int level = MAIN ) ; virtual void scan () ; virtual void solveProblem () ; // v i r t u a l void r e p o r t () ; virtual void resultReport () { }; virtual void fillEssBC () ; // fill e s s e n t i a l BCs . virtual void setIC () ; // set i n i t i a l c o n d i t i o n . virtual void integrands ( ElmMatVec & elmat , const FiniteElement & fe ) ; // o v e r l o a d e d i n t e g r a n d s . // // P r o v i d e m a g n e t i c f i e l d as f u n c t i o n of time . // virtual real gammaFunc ( real t ) ; // C a l c u l a t e e x p e c t a t i o n v a l u e when g i v e n a m a t r i x . NUMT calcExpectationFr om M at ri x ( FieldFE & u , FieldFE & v , Matrix ( NUMT ) & A ) ; // // R e p o r t m e t h o d s . Used d u r i n g s i m u l a t i o n . // virtual void initialReport () ; virtual void finalReport () ; virtual void reportAtThisTimeSte p ( int t_index , int r_index ) ; Ptv ( real ) calcExpectation_pos ( FieldFE & u , FieldFE & v ) ; protected : // // Load f i e l d s ( and grid !) from d a t a b a s e // void loadFields ( String & db_name ) ; }; # endif B.3.2 TimeSolver.cpp # include " TimeSolver . h " # include < IntegrateOverGridFE .h > // * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // 199 Program Listings // C l a s s d e f i n i t i o n of T i m e S o l v e r . // // * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * // // Main p r o b l e m s o l v i n g m e t h o d . // void TimeSolver :: solveProblem () { if (! do_time_simulation ) { s_e < < " Skipping time dependent simulation . Diagonalizing instead ! " < < endl ; HydroEigen :: solveProblem () ; return ; } // H a n d l e ( M a t r i x ( NUMT ) ) A , M ; A . rebind ( NULL ) ; M . rebind ( NULL ) ; // h o l d s the l i n e a r s y s t e m and mass m a t r i x . // bool m a t r i x _ h a s _ c h a n g e d = true ; bool first_step_lf = ( time_method == LEAP_FROG ) ; // real l a s t _ r e p o r t _ t i m e = 0 ; // used for r e p o r t book - k e e p i n g . int last_report_index = 0; // used for r e p o r t book - k e e p i n g . int report_index = 0; int t_index = 0; // h o l d s n u m b e r of s t e p s t a k e n . tip - > initTimeLoop () ; // i n i t i a l i z e t i m e l o o p . // make M and A . gammaFunc ( tip - > time () ) ; // u p d a t e m a g n e t i c f i e l d . s_e < < " Make M and A ... " < < endl ; do_time_simulation = false ; makeSystem (* dof , * lineq ) ; // make H ^0. do_time_simulation = true ; A . rebind ( lineq - > A () ) ; // f e t c h p o i n t e r to the e l e m e n t m a t r i x . s_e < < " Make mass matrix ... " < < endl ; A - > makeItSimilar ( M ) ; // copy s t r u c t u r e of A into M . makeMassMatrix (* grid , M () , lump ) ; setIC () ; // set i n i t i a l c o n d i t i o n . linsol = u - > values () ; // s t a r t v e c t o r for f i r s t time step . initialReport () ; // r e p o r t v a r i o u s s t u f f reportAtThisTimeSte p (0 , 0) ; // r e p o r t i n i t i a l c o n d i t i o n . while (! tip - > finished () ) { // loop u n t i l t = t _ f i n a l tip - > increaseTime () ; // t - > t + dt addToLogFile ( " % " ) ; addToLogFile ( aform ( " Solving for t == % g \ n " , tip - > time () ) ) ; // b u i l d H a m i l t o n i a n and mass m a t r i x at this time step . s_e < < " Essential BCs ... " < < endl ; fillEssBC () ; // ess . BCs . // f i r s t step i leap - frog s h o u l d be a theta - rule step . // c r e a t e mass m a t r i x also . if ( first_step_lf ) { time_method = THETA_RULE ; } // // - - - s o l v e for leap frog m e t h o d . - - // if ( time_method == LEAP_FROG ) { s_e < < " ( Using LEAP_FROG ) " < < endl ; s_e < < " ( " ; s_e < < " making system ... " ; // make c o e f f i c i e n t m a t r i x M and r i g h t hand side -2* i * dt * H * u makeSystem (* dof , * lineq ) ; // // // if s o l v i n g a l u m p e d s y s t e m in the leap - frog m e t h o d is very easy ! : -) t h e r e f o r e we s e p a r a t e the two c a s e s . (! lump ) { s_e < < " solving linear system ... " ; lineq - > solve ( true ) ; } else { s_e < < " solving linear system ( by inspection ) ... " ; Vec ( NUMT ) & rhs = ( Vec ( NUMT ) &) lineq - > b () ; // a l i t t l e bit ugly but ... for ( int p =1; p <= linsol . size () ; p ++) linsol ( p ) = pow ( lineq - > A () . elm (p , p ) , -1) * rhs ( p ) ; } // add u _ p r e v to s o l u t i o n of l i n e a r s y s t e m . // l i n s o l now then the new u ’s f i e l d v a l u e s . linsol . add ( linsol , u_prev - > values () ) ; * u_prev = * u ; // u p d a t e u _ p r e v for next time step . dof - > vec2field ( linsol , * u ) ; // s t o r e new s o l u t i o n in f i e l d o b j e c t . s_e < < " ) " < < endl ; } // // - - - s o l v e for t h e t a rule m e t h o d . - - // if ( time_method == THETA_RULE ) { s_e < < " ( Using THETA_RULE ) " < < endl ; s_e < < " ( " ; // make the s y s t e m : rhs = [ M - i * dt *(1 - t h e t a ) * H (t - dt ) ]* u , 200 B.3 – The TimeSolver class // A = M + i * t h e t a * dt * H ( t ) s_e < < " making system ... " ; makeSystem (* dof , * lineq ) ; s_e < < " solving linear system ... " ; lineq - > solve ( true ) ; // l i n s o l h o l d s s o l u t i o n * u_prev = * u ; // u p d a t e u _ p r e v for next time step dof - > vec2field ( linsol , * u ) ; // c o n v e r t to f i e l d o b j e c t . s_e < < " ) " < < endl ; } // if we did a theta - rule step in the f i r s t leap - frog step ... if ( first_step_lf ) time_method = LEAP_FROG ; first_step_lf = false ; // s o l v e d a new time step we did ! t_index ++; report_index = ( int ) floor ( tip - > time () / dt_rep ) ; // r e p o r t if ( tip - > finished () || ( report_index > last_report_index ) ) { // u p d a t e l a s t _ r e p o r t _ t i m e last_report_index = report_index ; reportAtThisTimeStep ( t_index , report_index ) ; } } // end of time loop finalReport () ; // r e p o r t v a r i o u s s t u f f . } // // R e p o r t in time - loop // void TimeSolver :: reportAtThisTimeSte p ( int t_index , int r_index ) { Handle ( FieldFE ) prob ; prob . rebind ( new FieldFE ) ; String prefix = aform (" % s_data . " , casename . c_str () ) ; s_e < < " Reporting ... " ; // S t o r e c u r r e n t s o l u t i o n of d e s i r e d . if ( store_evecs >0) { database - > dump (* u , &( tip () ) ) ; s_e < < " ( u saved ) " ; } // S t o r e p r o b a b i l i t y d e n s i t y if d e s i r e d . if ( store_prob ) { // Calc . prob . d e n s i t y from c u r r e n t s o l u t i o n . calcProbabilityDen si ty (* u , * prob , true ) ; database - > dump (* prob , &( tip () ) ) ; s_e < < " ( u * u saved ) "; } Ptv ( real ) pos = calcExpectation_pos (* u ,* u ) ; addToLogFile ( " \ n % " ) ; addToLogFile ( aform ( " reporting at t (% d ) == %10.10 g ;\ n " , t_index , tip - > time () ) ); addToLogFile ( prefix ) ; addToLogFile ( aform ( " t (% d ) = %10.10 g ;\ n " , r_index +1 , tip - > time () ) ) ; addToLogFile ( prefix ) ; addToLogFile ( aform ( " u_norm (% d ) = %10.10 g ;\ n " , r_index +1 , calcInnerProd (* u ,* u ) . Re () ) ) ; // a d d T o L o g F i l e( a f o r m (" u _ e n e r g y (% d ) = % 1 0 . 1 0 g ;\ n " , r _ i n d e x +1 , c a l c E x p e c t a t i o n F r o m M a t r i x(* u , * u , * A ) . Re () ) ) ; addToLogFile ( prefix ) ; addToLogFile ( aform ( " gamma (% d ) = %10.10 g ;\ n " , r_index +1 , gammaFunc ( tip - > time () ) ) ) ; addToLogFile ( prefix ) ; addToLogFile ( aform ( " u_pos (% d ,:) = [%10.10 g , %10.10 g ];\ n " , r_index +1 , pos (1) , pos (2) ) ) ; if (( t_index > 0) && !( lump && ( time_method == LEAP_FROG ) ) ) { // r e t r i e v e p e r f o r m a n c e info . LinEqStatBlk & perf = lineq - > getPerformance () ; SolverStatistics & stats = perf . solver_info ; addToLogFile ( prefix ); addToLogFile ( aform ( " solve_time (% d ) = %10.10 g ;\ n" , r_index +1 , stats . cputime ) ) ; addToLogFile ( prefix ); addToLogFile ( aform ( " solve_niter (% d) = % d ;\ n " , r_index +1 , stats . niterations )) ; } addToLogFile ( prefix ) ; addToLogFile ( aform ( " system_time (% d ) = %10.10 g ;\ n " , r_index +1 , cpu_time_makeSystem ) ) ; addToLogFile ( prefix ) ; do_time_simulation = false ; s_e < < " Making Hamiltonian ... " < < endl ; gammaFunc ( tip - > time () ); makeSystem (* dof , * lineq ) ; // make H ^\ ell . do_time_simulation = true ; addToLogFile ( aform ( " u_energy (% d ) = %10.10 g ;\ n " , r_index +1 , calcExpectationFr om Ma t ri x (* u , * u , lineq - > A () ) . Re () ) ) ; } // // E x t e n d f u n c t i o n a l i t y of s o l v e r menu . 201 Program Listings // void TimeSolver :: define ( MenuSystem & menu , int level ) { menu . addItem ( level , " simulation in time " , " set to false to solve eigenvalue problem instead " , " true " ) ; int level1 = level +1; MenuSystem :: makeSubMenuHeader ( menu , " Params for time dependent solve " , " time " , level1 , ’T ’) ; menu . addItem ( level1 , " time method " , " time integration method " , " theta - rule " ) ; // or " leap - frog " menu . addItem ( level1 , " T " , " final time " , " 1.0 " ) ; menu . addItem ( level1 , " dt " , " time step " , " 0.1 " ) ; menu . addItem ( level1 , " omega " , " omega " , " 0 " ) ; menu . addItem ( level1 , " delta " , " delta " , " 0 " ) ; menu . addItem ( level1 , " gamma0 " , " gamma0 " , " 0 " ) ; menu . addItem ( level1 , " n_rep " , " number of reports " , " 10 " ) ; menu . addItem ( level1 , " theta " , " theta " , " 0.5 " ) ; menu . addItem ( level1 , " ic type " , " initial condition type " , " gaussian " ) ; // or " f i e l d " menu . addItem ( level1 , " field database " , " field database " , " none " ) ; // d a t a b a s e that s h o u l d hold e i g e n v e c t o r s. menu . addItem ( level1 , " ic field no " , " number of field to use as ic " , " 1 " ) ; // 1 s h o u l d be g r o u n d s t a t e ... menu . addItem ( level1 , " gaussian " , " gaussian parameters " , " -x -10 - y 0 - sx 2 - sy 2 - kx 0 - ky 1 " ) ; // used to initialize gaussFunc . // Set up p r e v i o u s f u n c t i o n a l i t y as well . HydroEigen :: define ( menu , level ) ; } // // E x t e n d f u n c t i o n a l i t y of scan () . // void TimeSolver :: scan () { MenuSystem & menu = SimCase :: getMenuSystem () ; int ok = false ; do_time_simulation = menu . get ( " simulation in time " ) . getBool () ; // // If user sets ’ s i m u l a t i o n in time ’ to false , the H y d r o E i g e n s o l v e r is used i n s t e a d . // if (! do_time_simulation ) { // we do not want to do a time d e p e n d e n t s i m u l a t i o n but r a t h e r a time i n d e p e n d e n t s i m u l a t i o n . s_e < < " Calling HydroEigen :: scan () . " < < endl ; // call the old scan () . HydroEigen :: scan () ; s_e < < " Back from HydroEigen :: scan () ." < < endl ; s_e < < " Skipping initialization of time dependent solver . " < < endl ; return ; } // Get the time m e t h o d from menu . // A b o r t if not " theta - rule " or " leap - frog " String s = menu . get ( " time method " ) ; ok = false ; if ( s == " theta - rule " ) { time_method = THETA_RULE ; ok = true ; } else if ( s == " leap - frog " ) { time_method = LEAP_FROG ; ok = true ; } if (! ok ) fatalerrorFP ( " TimeSolver :: scan () " , " Illegal time method ! " ) ; // Get time p a r a m e t e r s , set up tip . t_final = menu . get ( " T " ) . getReal () ; dt = menu . get ( " dt " ) . getReal () ; theta = menu . get ( " theta " ). getReal () ; gamma0 = menu . get ( " gamma0 " ) . getReal () ; omega = menu . get ( " omega " ). getReal () ; delta = menu . get ( " delta " ). getReal () ; tip . rebind ( new TimePrm () ) ; tip - > scan ( aform ( " dt =% g , t in [% g ,% g ] " , dt , 0.0 , t_final ) ) ; // Set up r e p o r t p a r a m e t e r s . n_rep = ( int ) menu . get ( " n_rep " ) . getReal () ; if ( n_rep < 1) n_rep = 1; // at l e a s t 1 r e p o r t ( at the end of sim ) dt_rep = t_final /( real ) n_rep ; if ( dt_rep < dt ) { dt_rep = dt ; // not too many r e p o r t s ... n_rep = ( int ) ( t_final / dt ) ; } // Set up IC p a r a m e t e r s . field_database = menu . get ( " field database " ) ; field_no = ( int ) menu . get (" ic field no " ). getReal () ; s = menu . get ( " ic type " ) ; gaussian_parameters = menu . get ( " gaussian " ) ; ok = false ; if ( s == " gaussian " ) { ic_type = IC_GAUSSIAN ; ok = true ; } if ( s == " field " ) { ic_type = IC_FIELD ; ok = true ; } if (! ok ) fatalerrorFP ( " TimeSolver :: scan () " , " Illegal IC type ! " ) ; // Call the old scan () . Set up grid et . c . 202 B.3 – The TimeSolver class s_e < < " Calling HydroEigen :: scan () . " < < endl ; HydroEigen :: scan () ; s_e < < " Back from HydroEigen :: scan () . " < < endl ; // Load f i e l d s from d a t a b a s e . // This o v e r w r i t e s the grid read by H y d r o E i g e n :: scan () if f i e l d s are f o u n d / read . if ( ic_type == IC_FIELD ) loadFields ( field_database ) ; // set up u and u _ p r e v . u_prev . rebind ( new FieldFE ) ; u_prev - > redim (* grid , " u_prev " ) ; u . rebind ( new FieldFE ) ; u - > redim (* grid , " u " ) ; // set up s c r a t c h v e c t o r s scratch . rebind ( new Vec ( NUMT ) ) ; scratch - > redim ( grid - > getNoNodes () ) ; scratch - > fill (0.0) ; scratch2 . rebind ( new Vec ( NUMT ) ) ; scratch2 - > redim ( grid - > getNoNodes () ) ; scratch2 - > fill (0.0) ; // for i n t e g r a n d s () . mass_integrand = new MassMatIntg ( lump ) ; // d e l e t e d in d e s c t r u c t o r... elmat2 . attach (* dof ) ; elmat3 . attach (* dof ) ; } // // G a m m a as f u n c t i o n of t . // real TimeSolver :: gammaFunc ( real t ) { // g a m m a = 1 ; if ( gamma0 ==0) gamma =0; else gamma = gamma0 * pow ( sin ( M_PI * t / t_final ) , 2) * cos ( ( omega *( t -0.5* t_final ) + delta ) *2* M_PI ); return gamma ; } // // Set h o m o g e n o u s BCs . // void TimeSolver :: fillEssBC () { dof - > initEssBC () ; const int nno = grid - > getNoNodes () ; for ( int i =1; i <= nno ; i ++) if ( grid - > boNode ( i ) ) dof - > fillEssBC (i , 0.0) ; } // // S i m p l e g a u s s i a n f u n c t o r . // class GaussFunc : public FieldFunc { public : Ptv ( real ) r0 ; // c e n t r e Ptv ( real ) sigma ; // w i d t h Ptv ( real ) x ; // work v e c t o r Ptv ( real ) k0 ; // m o m e n t u m GaussFunc () { // d e f a u l t c o n s t r u c t o r: exp ( - r ^ 2 / 2 ) r0 . redim (2) ; sigma . redim (2) ; x . redim (2) ; k0 . redim (2) ; r0 . fill (0.0) ; sigma . fill (0.5) ; x . fill (0.0) ; k0 . fill (0.0) ; } virtual void init ( real x0 , real y0 , real s1 , real s2 , real k1 , real k2 ) { // exp ( -( x - x0 ) ^2/ s i g m a 1 ^2 - ( y - y0 ) ^2/ s i g m a 2 ^2) r0 . redim (2) ; sigma . redim (2) ; x . redim (2) ; k0 . redim (2) ; k0 (1) = k1 ; k0 (2) = k2 ; r0 (1) = x0 ; r0 (2) = y0 ; sigma (1) = s1 ; sigma (2) = s2 ; sigma (1) = 0.5/( sigma (1) * sigma (1) ); sigma (2) = 0.5/( sigma (2) * sigma (2) ); x . fill (0.0) ; }; virtual NUMT valuePt ( const Ptv ( real ) & r , real t = DUMMY ) { // e v a l u a t e g a u s s i a n NUMT temp ; x (1) = r (1) - r0 (1) ; x (2) = r (2) - r0 (2) ; temp = exp ( Complex (0 , 1) * x . inner ( k0 ) ) ; x (1) = x (1) * x (1) ; x (2) = x (2) * x (2) ; return exp ( - x . inner ( sigma ) ) * temp ; }; virtual void scan ( String & s ) ; }; void GaussFunc :: scan ( String & s ) { real x = 0 , y = 0 , sx = 1 , sy = 1 , kx = 0 , ky = 0; int ntok = s . getNoTokens ( " " ) ; for ( int i =1; i <= ntok ; i ++) { String token , token2 ; s . getToken ( token , i , " " ) ; 203 Program Listings if if if if if if ( token ( token ( token ( token ( token ( token == == == == == == " -x " ) { s . getToken ( token2 , i +1 , " " ) ; x = " -y " ) { s . getToken ( token2 , i +1 , " " ) ; y = " - sx " ) { s. getToken ( token2 , i +1 , " " ) ; sx " - sy " ) { s. getToken ( token2 , i +1 , " " ) ; sy " - kx " ) { s. getToken ( token2 , i +1 , " " ) ; kx " - ky " ) { s. getToken ( token2 , i +1 , " " ) ; ky token2 . getReal () ; } token2 . getReal () ; } = token2 . getReal () ; = token2 . getReal () ; = token2 . getReal () ; = token2 . getReal () ; } } } } } init (x , y , sx , sy , kx , ky ) ; } // // Set i n i t i a l c o n d i t i o n s . // void TimeSolver :: setIC () { // If i c _ t y p e == I C _ G A U S S I A N , set up a g a u s s i a n i n i t i a l // c o n d i t i o n . If i c _ t y p e == IC_FIELD , we c h o o s e a l o a d e d // e i g e n v e c t o r as IC . if ( ic_type == IC_GAUSSIAN ) { GaussFunc gaussian ; gaussian . scan ( gaussian_parameters ) ; u_prev - > fill ( gaussian , 0.0) ; } else { if (( field_no > no_of_fields ) || ( field_no <1) ) field_no = 1; // copy f i e l d from d a t a b a s e . * u_prev = *( fields [ field_no - 1]) ; } // e n s u r e unit norm . u_prev - > mult ( 1.0/ sqrt ( calcInnerProd (* u_prev , * u_prev ) . Re () ) ) ; // set u = u _ p r e v . * u = * u_prev ; } // // c a l c u l a t e ( u , Av ) . uses s c r a t c h a s s u m e d to be of same l e n g t h as u and v . // NUMT TimeSolver :: calcExpectationF ro mM a tr ix ( FieldFE & u , FieldFE & v , Matrix ( NUMT ) & A ) { Vec ( NUMT ) & u_values = u . valuesVec () ; Vec ( NUMT ) & v_values = v . valuesVec () ; A . prod ( v_values , * scratch ) ; return u_values . inner (* scratch ) ; } // // f u n c t i o n that l o a d s the f i e l d s s t o r e d in d a t a b a s e d b _ n a m e . // they are t y p i c a l l y c r e a t e d with H y d d r o E i g e n s o l v e r s over the // same grid and H a m i l t o n i a n. // void TimeSolver :: loadFields ( String & db_name ) { real t = 0.0; int nsd = 0; String field_name ; String field_type ; int component =0 , max_component =0; int test = 0; if ( db_name == " none " ) { no_of_fields = 0; return ; } // open the s i m r e s file , e . g . , S I M U L A T I O N . SimResFile s ( db_name ) ; s_e < < " Dataset name == " < < s . getDatasetName () < < endl ; int n = s . getNoFields () ; // get n u m b e r of f i e l d s s t o r e d . bool load_it [ n ]; // i n d i c a t e s what f i e l d s s h o u l d be l o a d e d . s_e < < " I check " < < n < < " fields from " < < db_name < < " . " < < endl ; no_of_fields = 0; for ( int i =0; i < n ; i ++) { s . locateField ( i +1 , field_name , t , nsd , field_type , component , max_component ) ; // e i g e n v e c t o r s are s t o r e d with t h e i r n u m b e r as f i e l d name , the n u m b e r >0. test = atoi ( field_name . c_str () ) ; if ( test >0) { load_it [ i ] = true ; no_of_fields ++; // i n c r e a s e t o t a l n u m b e r of f i e l d s . } else { load_it [ i ] = false ; } } s_e < < " I found " < < no_of_fields < < " appropriate fields to load "; fields = new Handle ( FieldFE ) [ no_of_fields ]; int j = 0; for ( int i =0; i < n ; i ++) { if ( load_it [ i ]) { // a l l o c a t e h a n d l e s . fields [j ]. rebind ( new FieldFE ) ; // a l l o c a t e a new f i e l d . SimResFile :: readField ( fields [ j ]() , s , aform ( " % d " , j +1) , t ) ; 204 // read it . B.3 – The TimeSolver class // e n s u r e unit norm . fields [ j ] - > mult ( 1.0/ sqrt ( calcInnerProd (* fields [ j ] , * fields [j ]) . Re () ) ) ; j ++; s_e < < " . " ; } } s_e < < " ok " < < endl ; s_e < < " Fetching the grid . " < < endl ; grid . rebind ( fields [0] - > grid () ) ; // b e c a u s e we have a new grid , we must u p d a t e some s t u f f . // init the u n k n o w n u and the dof . a c t u a l l y not used in the s i m u l a t o r . // n e e d e d in the D e g F r e e F E o b j e c t . // u . r e b i n d ( new F i e l d F E (* grid ," u ") ) ; dof . rebind ( new DegFreeFE (* grid , 1) ) ; // 1 for 1 u n k n o w n per node // init the l i n e a r s y s t e m . ( h o l d s m a t r i x in e i g e n v a l u e f o r m u l a t i o n) // FEM s y s t e m s are not s o l v e d in H y d r o E i g e n , but the s t r u c t u r e is n e e d e d . lineq . rebind ( new LinEqAdmFE () ) ; lineq - > scan ( SimCase :: getMenuSystem () ) ; linsol . redim ( grid - > getNoNodes () ) ; linsol . fill (0.0) ; lineq - > attach ( linsol ) ; } // // Do an i n i t i a l r e p o r t . // void TimeSolver :: initialReport () { String prefix = aform (" % s_data . " , casename . c_str () ) ; addToLogFile ( " %\ n " ) ; addToLogFile ( aform ( " % s simulation log from case % s\ n " , " % " , casename . c_str () )) ; addToLogFile ( " \ n % erase variables we use .\ n " ) ; addToLogFile ( aform ( " clear % s_data ;\ n" , casename . c_str () ) ) ; addToLogFile ( aform ( " % scasename = ’% s ’;\ n " , prefix . c_str () , casename . c_str () ) ); addToLogFile ( " %\ n \ n " ) ; MenuSystem & menu = SimCase :: getMenuSystem () ; addToLogFile ( " % simulation parameters : \ n " ) ; addToLogFile ( prefix ) ; addToLogFile ( aform ( " gridfile = ’% s ’;\ n " , menu . get (" gridfile " ) . c_str () ) ) ; if ( time_method == THETA_RULE ) { addToLogFile ( aform ( "\ n % s using the theta - rule .\ n" , " % " ) ) ; addToLogFile ( prefix ); addToLogFile ( aform ( " theta = %10.10 g ;\ n " , theta ) ); } else { addToLogFile ( " % using the leap - frog method .\ n " ) ; } addToLogFile ( " % is mass matrix lumped ?\ n " ) ; addToLogFile ( prefix ) ; addToLogFile ( " lump = ") ; if ( lump ) addToLogFile ( " 1;\ n " ); else addToLogFile ( " 0;\ n " ); addToLogFile ( " \ n " ) ; addToLogFile ( prefix ) ; addToLogFile ( aform ( " dt = %10.10 g ;\ n " , dt ) ) ; addToLogFile ( prefix ) ; addToLogFile ( aform ( " t_final = %10.10 g ;\ n " , t_final ) ) ; addToLogFile ( prefix ) ; addToLogFile ( aform ( " gamma0 = %10.10 g ;\ n " , gamma0 ) ); addToLogFile ( prefix ) ; addToLogFile ( aform ( " omega = %10.10 g ;\ n " , omega ) ) ; addToLogFile ( prefix ) ; addToLogFile ( aform ( " delta = %10.10 g ;\ n " , delta ) ) ; if ( ic_type == IC_GAUSSIAN ) { addToLogFile ( " \ n % using a Gaussian for IC .\ n " ) ; addToLogFile ( prefix ); addToLogFile ( " gaussian_params = ’ ") ; addToLogFile ( menu . get ( " gaussian " ) ); addToLogFile ( " ’;\ n " ); } else { addToLogFile ( " % using a field from database for IC .\ nfield_database = ’ " ) ; addToLogFile ( prefix ); addToLogFile ( menu . get ( " field database " ) ) ; addToLogFile ( " ’;\ n " ); addToLogFile ( prefix ); addToLogFile ( " field_number = " ) ; addToLogFile ( menu . get ( " field no " ) ); addToLogFile ( " ;\ n " ) ; } } // // Do a f i n a l r e p o r t . // void TimeSolver :: finalReport () { 205 Program Listings String prefix = aform ( " % s_data . " , casename . c_str () ) ; addToLogFile ( " \ n % finally ...\ n " ) ; addToLogFile ( prefix ) ; addToLogFile ( aform ( " n_rep = length (% st ) ;\ n " , prefix . c_str () ) ) ; addToLogFile ( prefix ) ; addToLogFile ( aform ( " solve_total = sum (% ssolve_time ) ;\ n " , prefix . c_str () ) ) ; addToLogFile ( prefix ) ; addToLogFile ( aform ( " system_total = sum (% ssystem_time ) ;\ n " , prefix . c_str () ) ) ; } // // new i n t e g r a n d s for time d e p e n d e n t p r o b l e m s . // r e u s e s H y d r o E i g e n :: i n t e g r a n d s () as well . // void TimeSolver :: integrands ( ElmMatVec & elmat , const FiniteElement & fe ) { // if we s o l v e a e i g e n v a l u e problem , use H y d r o E i g e n :: i n t e g r a n d s () if (! do_time_simulation ) { HydroEigen :: integrands ( elmat , fe ) ; return ; } instead . int nbf = fe . getNoBasisFunc () ; // i n i t i a l i z e s c r a t c h o b j e c t s . MassMatIntg mass_integrand ( lump ) ; elmat2 . attach (* dof ) ; elmat3 . attach (* dof ) ; elmat2 . refill ( elmat . elm_no ) ; elmat3 . refill ( elmat . elm_no ) ; if ( time_method == LEAP_FROG ) { // goal : A = M , b = -2* i * dt * H ( t ) * u ( t ) // c r e a t e the leap - frog e l e m e n t m a t r i x and v e c t o r . // e l e m e n t m a t r i x = mass m a t r i x . mass_integrand . integrands ( elmat , fe ) ; // add c o n t r i b u t i o n from mass m a t r i x . // e l e m e n t v e c t o r = -2* i * dt * H ( t ) * u // c r e a t e a new E l m M a t V e c to hold the H a m i l t o n i a n c o n t r i b u t i o n s. gammaFunc ( tip - > time () - dt ) ; // u p d a t e g a m m a . HydroEigen :: integrands ( elmat2 , fe ) ; // fill e l m a t 2 . b with u ’ s v a l u e s . for ( int i =1; i <= nbf ; i ++) elmat2 .b ( i ) = u - > values () ( elmat . loc2glob_u ( i ) ) ; elmat2 . A . prod ( elmat2 .b , elmat3 . b ) ; // e l m a t 2 . b = e l m a t 2 . A * e l m a t 2 . b elmat3 . b . mult ( Complex (0 , -2* dt ) ) ; // m u l t i p l y Hu by -2 i * dt elmat . b . add ( elmat .b , elmat3 . b ) ; // u p d a t e e l m a t . b } else if ( time_method == THETA_RULE ) { // c r e a t e the theta - rule e l e m e n t m a t r i x and v e c t o r // b = [ M - i * dt *(1 - t h e t a ) * H ( t - dt ) ]* u gammaFunc ( tip - > time () - dt ) ; mass_integrand . integrands ( elmat2 , fe ) ; // add c o n t r i b u t i o n from mass m a t r i x . // add c o n t r i b to M * u in rhs . for ( int i =1; i <= nbf ; i ++) elmat2 .b ( i ) = u - > values () ( elmat . loc2glob_u ( i ) ) ; elmat2 . A . prod ( elmat2 .b , elmat3 . b ) ; elmat . b . add ( elmat .b , elmat3 . b ) ; // e l m a t . b += M * u elmat2 . A . fill (0.0) ; elmat2 . b . fill (0.0) ; elmat3 . b . fill (0.0) ; HydroEigen :: integrands ( elmat2 , fe ) ; // H a m i l t o n i a n. elmat2 . A . mult ( Complex (0 , - dt *(1.0 - theta ) ) ) ; // . . . t i m e s - i * dt *(1 - t h e t a ) for ( int i =1; i <= nbf ; i ++) elmat2 .b ( i ) = u - > values () ( elmat . loc2glob_u ( i ) ) ; elmat2 . A . prod ( elmat2 .b , elmat3 . b ) ; elmat . b . add ( elmat .b , elmat3 . b ) ; // e l m a t . b += - i * dt *(1 - t h e t a ) * H * u // A = M + i * dt * t h e t a * H ( t ) gammaFunc ( tip - > time () ) ; // u p d a t e g a m m a . elmat2 . A . fill (0.0) ; // e r a s e v a l u e s . HydroEigen :: integrands ( elmat2 , fe ) ; // fill it elmat2 . A . mult ( Complex (0 , dt * theta ) ) ; elmat . A . add ( elmat .A , elmat2 . A ) ; // e l m a t . A = - i * dt * t h e t a * H ( t ) ; mass_integrand . integrands ( elmat , fe ) ; // add M . } 206 B.4 – The EigenSolver class } // // c a l c u l a t e e x p e c t a t i o n v a l u e of p o s i t i o n . // Ptv ( real ) TimeSolver :: calcExpectation_pos ( FieldFE & u , FieldFE & v ) { IntegrateOverGridFE integrator ; IntegrateOverGridFE integrator2 ; IntegrandOfExpect at i on _x integrand_x ; IntegrandOfExpect at i on _y integrand_y ; InnerProdIntegrand Ca l c integrand2 ; integrand_x . setFields (u , v ) ; integrand_y . setFields (u , v ) ; integrand2 . setFields (u , v ) ; Ptv ( real ) result ; result . redim (2) ; integrator . volumeIntegral ( integrand_x , * grid ) ; integrator . volumeIntegral ( integrand_y , * grid ) ; integrator2 . volumeIntegral ( integrand2 , * grid ) ; result (1) = ( integrand_x . getResult () / integrand2 . getResult () ) . Re () ; result (2) = ( integrand_y . getResult () / integrand2 . getResult () ) . Re () ; return result ; } B.3.3 main.cpp # include < TimeSolver .h > int main ( int argc , const char * argv []) { initDiffpack ( argc , argv ) ; global_menu . init ( " Time solver test " , " TimeSolver ") ; TimeSolver sim ; global_menu . multipleLoop ( sim ) ; return 0; } B.4 The EigenSolver class This class extends the class deﬁnition found in Ref. [5] to include standard and generalized complex eigenvalue problems. B.4.1 EigenSolver.h # ifndef _EIGENSOLVER_H_ # define _EIGENSOLVER_H_ # include # include # include # include < FEM .h > < Matrix_Complex .h > < LinEqSolver .h > < LinEqAdm .h > // d i s p l a y w a r n i n g m e s s a g e s # define WARNING ( s ) { s_e < < " > > > EigenSolver warning : " < < s < < endl ; } // c o n s t a n t s used in the c l a s s d e f i n i t i o n # define PROBLEM_UNDEFINED 0 # define PROBLEM_STANDARD 1 # define PROBLEM_GENERALIZED 2 # define MATRIX_SYMMETRIC 1 # define MATRIX_NONSYMMETRIC 2 # define MATRIX_COMPLEX 3 # define MODE_REGULAR 1 # define MODE_SHIFT_INVERT 2 # define MODE_BUCKLING 3 # define MODE_CAYLEY 4 # define MODE_COMPLEX_SHIFT 5 # define S P E C T R U M _ S M A L L E S T _ M A G N I TU D E 0 # define S P E C T R U M _ S M A L L E S T _ A L G E B RA I C 1 # define SPECTRUM_SMALLEST_R E AL 2 # define SPECTRUM_SMALLEST_I M AG 3 # define SPECTRUM_LARGEST_ M AG NI T UD E 4 # define SPECTRUM_LARGEST_ A LG EB R AI C 5 # define SPECTRUM_LARGEST_RE AL 6 # define SPECTRUM_LARGEST_IM AG 7 // // c l a s s d e f i n i t i o n of E i g e n S o l v e r. // class EigenSolver { private : int n , nev ; // Handle ( Matrix ( NUMT ) ) A , B ; // int problem_kind ; // int matrix_kind ; // int comp_mode ; // int spectrum_part ; // Mat ( NUMT ) eigenvectors ; // d i m e n s i o n of problem , n u m b e r of e i g e n v a l u e s to seek problem defining matrices s t a n d a r d or g e n e r a l i z e d symm , non - symm or c o m p l e x regular , shift - and - i n v e r t et . c . i n d i c a t e s what part of s p e c t r u m to find s t o r e the e i g e n v e c t o r s here 207 Program Listings Vec ( NUMT ) eigenvalues ; Handle ( Vec ( NUMT ) ) V , W ; real sigma , sigma_im ; // s t o r e the e i g e n v a l u e s here // aux v e c t o r s for matrix - v e c t o r o p e r a t i o n s // real ( and i m a g i n a r y ) part of s h i f t // l i n e a r e q u a t i o n s o b j e c t s . Handle ( Matrix ( NUMT ) ) C , D; Handle ( LinEqSolver ) linear_solver ; Handle ( LinEqSolver_prm ) linear_solver_prm ; Handle ( LinEqAdm ) lineq ; bool c_has_changed ; // i n d i c a t e s w h e t h e r we do the f i r s t l i n e a r s o l v e or not public : // c o n s t r u c t o r s... EigenSolver ( Handle ( MenuSystem ) menu_handle = NULL ) ; EigenSolver ( Matrix ( NUMT ) & the_A , int the_nev = 1 , Handle ( MenuSystem ) menu_handle = NULL ) ; EigenSolver ( Matrix ( NUMT ) & the_A , Matrix ( NUMT ) & the_B , int the_nev = 1 , Handle ( MenuSystem ) menu_handle = NULL ) ; // m e t h o d s for s e t t i n g the m a t r i c e s bool setA ( Matrix ( NUMT ) & the_A ) ; bool setB ( Matrix ( NUMT ) & the_A ) ; // m e t h o d s for s e t t i n g / g e t t i n g n u m b e r of e i g e n v a l u e s to seek bool setNev ( int the_nev ) ; int getNev () ; // set the m a t r i x c h a r a c t e r i s t i c s bool setMatrixKind ( int kind ) ; // set the real and i m a g i n a r y part of the s p e c t r u m s h i f t void setSigma ( real the_sigma ) ; void setSigmaIm ( real the_sigma_im ) ; // set the c o m p u t a t i o n a l mode bool setCompMode ( int mode ) ; bool setSpectrumPart ( int part ) ; // s o l v e the problem , s i l l y r e p o r t and s i l l y p r i n t i n g of m a t r i c e s . void solveProblem () ; void report () ; void sillyPrint ( Matrix ( NUMT ) & matrisen ); // r e t r i e v e r e f e r e n c e s to the i n t e r n a l e i g e n t h i n g s s t o r a g e Mat ( NUMT ) & getEigenvectors () ; Vec ( NUMT ) & getEigenvalues () ; protected : void resetAllMembers () ; // c l e a r s e v e r y t h i n g , incl . p o i n t e r s . // i n i t i a l i z e the c h a i n of o b j e c t s c o n s t i t . l i n e a r s o l v e r . void initLinearSolver ( Handle ( MenuSystem ) menu_handle = NULL ) ; void removeA () ; // t h r e e m e t h o d s that kill m a t r i c e s ... void removeB () ; void removeMatrix ( Handle ( Matrix ( NUMT ) ) h ) ; // matrix - v e c t o r o p e r a t i o n s m e t h o d s . p a s s e d to A R P A C K ++ o b j e c t s . void multAx ( NUMT * , NUMT *) ; void multBx ( NUMT * , NUMT *) ; void multInvCDx ( NUMT * , NUMT *) ; void multInvCx ( NUMT * , NUMT *) ; }; // E i g e n S o l v e r # endif B.4.2 EigenSolver.cpp # include " EigenSolver . h " // i n c l u d e the A R S y m G e n E i g c l a s s t e m p l a t e # include " argsym . h " # include " argcomp . h " // // // // // // // // // // // // // I m p l e m e n t a t i o n of E i g e n S o l v e r c l a s s . Author: Simen Kvaal. C u r r e n t l y only s y m m e t r i c p r o b l e m s are s u p p o r t e d . The rest of the p r o b l e m s may e a s i l y be a d d e d in the S o l v e P r o b l e m m e t h o d . Last u p d a t e : Aug . 1 2 , 2 0 0 3 // // r e s e t all m e m b e r s . h e l p s k e e p i n g t h i n g s c l e a n . // void EigenSolver :: resetAllMembers () { removeA () ; removeB () ; n = 0; nev = 0; problem_kind = PROBLEM_UNDEFINED ; matrix_kind = MATRIX_SYMMETRIC ; comp_mode = MODE_REGULAR ; spectrum_part = S P E C T R U M _ S M A L L E S T _ M A G N I T U DE; eigenvalues . redim (0) ; eigenvectors . redim (0 ,0) ; removeMatrix ( C ) ; 208 B.4 – The EigenSolver class removeMatrix ( D ) ; sigma = 0; sigma_im = 0; } // // i n i t i a l i z e c h a i n of o b j e c s c o n s t i t u t i n g // the l i n e a r s o l v e r . // void EigenSolver :: initLinearSolver ( Handle ( MenuSystem ) menu_handle ) { // c r e a t e the L i n E q S o l v e r o b j e c t linear_solver_prm . rebind ( LinEqSolver_prm :: construct () ) ; if ( menu_handle != NULL ) linear_solver_prm - > scan (* menu_handle ) ; // i n i t F r o m C o m m a n d L i n e A r g(" - s " , l i n e a r _ s o l v e r _ p r m - > b a s i c _ m e t h o d , " G a u s s E l i m ") ; a c t u a l l y may use the s e t t i n g s in the menu s y s t e m i n s t e a d ! linear_solver . rebind ( linear_solver_prm - > create () ) ; /// c o m m e n t e d out ; i t h i n k we // c r e a t e L i n E q A d m o b j e c t for s o l v i n g l i n e a r s y s t e m s . lineq . rebind ( new LinEqAdm ( EXTERNAL_STORAGE ) ) ; lineq - > attach ( linear_solver () ) ; c_has_changed = true ; } // // d e f a u l t c o n s t r u c t o r // EigenSolver :: EigenSolver ( Handle ( MenuSystem ) menu_handle ) { // make it c l e a n ... resetAllMembers () ; initLinearSolver ( menu_handle ) ; V . rebind ( new Vec ( NUMT ) (0) ) ; W . rebind ( new Vec ( NUMT ) (0) ) ; } // // c o n s t r u c t o r for s t a n d a r d p r o b l e m s . // EigenSolver :: EigenSolver ( Matrix ( NUMT ) & the_A , int the_nev , Handle ( MenuSystem ) menu_handle ) { resetAllMembers () ; initLinearSolver ( menu_handle ) ; V . rebind ( new Vec ( NUMT ) (0) ) ; W . rebind ( new Vec ( NUMT ) (0) ) ; // a t t e m p t to set A . if (! setA ( the_A ) ) { WARNING ( " Could not set A . Bailing out of constructor . " ) ; return ; } nev = the_nev ; } // // c o n s t r u c t o r for g e n e r a l i z e d p r o b l e m s . // EigenSolver :: EigenSolver ( Matrix ( NUMT ) & the_A , Matrix ( NUMT ) & the_B , int the_nev , Handle ( MenuSystem ) menu_handle ) { resetAllMembers () ; initLinearSolver ( menu_handle ) ; V . rebind ( new Vec ( NUMT ) (0) ) ; W . rebind ( new Vec ( NUMT ) (0) ) ; // a t t e m p t to set A . if (! setA ( the_A ) ) { WARNING ( " Could not set A . Bailing out of constructor . " ) ; return ; } // a t t e m p t to set B . if (! setB ( the_B ) ) { WARNING ( " Could not set B . Solver is now a standard solver . " ) ; nev = the_nev ; return ; } else { nev = the_nev ; } } // // r e m o v e m a t r i c e s ... // void EigenSolver :: removeA () { removeMatrix ( A ) ; } void EigenSolver :: removeB () { removeMatrix ( B ) ; } void EigenSolver :: removeMatrix ( Handle ( Matrix ( NUMT ) ) h ) { h . detach () ; h . rebind ( NULL ) ; } // // set m a t r i c e s ... 209 Program Listings // also sets p r o b l e m _ k i n d and n , m a k i n g a c o m p l e t e // p r o b l e m s p e c i f i c a t i o n. // bool EigenSolver :: setA ( Matrix ( NUMT ) & the_A ) { int N , M ; // get the m a t r i x size . the_A . size (N , M ) ; // the m a t r i x must be s q u a r e ... if ( N == M ) { A . rebind ( the_A ) ; n = N; } else { return false ; } // set p r o b l e m kind . if ( B . getPtr () ) problem_kind = PROBLEM_GENERALIZED ; else problem_kind = PROBLEM_STANDARD ; return true ; } // // set B . A must be a l r e a d y set . // bool EigenSolver :: setB ( Matrix ( NUMT ) & the_B ) { int N , M ; // get the m a t r i x size . the_B . size (N , M ) ; // the m a t r i x must be s q u a r e and of same size as A ... if (( N == M ) && ( N == n ) ) { B . rebind ( the_B ) ; problem_kind = PROBLEM_GENERALIZED ; return true ; } else { return false ; } } // set nev and r e t u r n true if s u c c e s s . // nev must be in r a n g e 1 . . n -1 bool EigenSolver :: setNev ( int the_nev ) { if (( the_nev >= 1) && ( the_nev <= n -1) ) { nev = the_nev ; return true ; } else { WARNING ( " nev is out of range . " ) ; return false ; } } // get nev ... int EigenSolver :: getNev () { return nev ; } // set s p e c t r u m part ... bool EigenSolver :: setSpectrumPart ( int part ) { if (( part == S P E C T R U M _ S M A L L E S T _ M A G N I T U DE) || ( part == S P E C T R U M _ S M A L L E S T _ A L G E B R A IC) || ( part == SPECTRUM_SMALLEST _R EA L ) || ( part == SPECTRUM_SMALLEST _I MA G ) || ( part == SPECTRUM_LARGEST _ MA GN I TU DE ) || ( part == SPECTRUM_LARGEST _ AL GE B RA IC ) || ( part == SPECTRUM_LARGEST_R EA L ) || ( part == SPECTRUM_LARGEST_I MA G ) ) { spectrum_part = part ; return true ; } WARNING ( " Invalid spectrum part . " ) ; return false ; } // set m a t r i x kind ... bool EigenSolver :: setMatrixKind ( int kind ) { if (( kind == MATRIX_SYMMETRIC ) || ( kind == MATRIX_NONSYMMETRIC ) || ( kind == MATRIX_COMPLEX ) ) { matrix_kind = kind ; } else { WARNING ( " Illegal matrix kind . " ) ; return false ; } return true ; } // // s o l v e the p r o b l e m ! // note that only s y m m e t r i c p r o b l e m s // are i m p l e m e n t e d at this point , // but a d d i n g more m a t r i x k i n d s is // r e a l l y easy . // void EigenSolver :: solveProblem () { 210 B.4 – The EigenSolver class bool supported_mode = false ; char * descriptive_array [] = { " SM " , " SA " , " SR " , " SI " , " LM " , " LA " , " LR " , " LI " } ; char * descriptive = descriptive_array [ spectrum_part ]; // c r e a t e a p o i n t e r to the base c l a s s in the // A R P A C K ++ h i e r a r c h y . ARrcStdEig < real , real > * solver = NULL ; ARrcStdEig < real , arcomplex < real > > * solverComplex = NULL ; typedef void ( EigenSolver ::* real_multfunc ) ( real * , real *) ; typedef void ( EigenSolver ::* Complex_multfunc ) ( arcomplex < real > * , arcomplex < real > *) ; // c r e a t e i n s t a n c e of p r o p e r // c l a s s b a s e d on m a t r i x kind and p r o b l e m kind // and r e g u l a r mode . // a t t e m p t at c o m p l e x m a t r i x i m p l e m e n t a t i o n. s e e m s to work ! if ( matrix_kind == MATRIX_COMPLEX ) { if ( problem_kind == PROBLEM_STANDARD ) { if ( comp_mode == MODE_REGULAR ) { ARCompStdEig < real , EigenSolver > * temp = new ARCompStdEig < real , EigenSolver >( n , nev , this , ( Complex_multfunc ) & EigenSolver :: multAx , descriptive ) ; solverComplex = temp ; supported_mode = true ; WARNING ( " ARCompStdEig in regular mode created . " ) ; } } if ( problem_kind == PROBLEM_GENERALIZED ) { if ( comp_mode == MODE_REGULAR ) { // use the r e g u l a r mode c o n s t r u c t o r. ARCompGenEig < real , EigenSolver , EigenSolver > * temp = new ARCompGenEig < real , EigenSolver , EigenSolver >(n , nev , this , ( Complex_multfunc ) & EigenSolver :: multInvCDx , this , ( Complex_multfunc ) & EigenSolver :: multBx , descriptive ) ; solverComplex = temp ; // set up h e l p e r m a t r i c e s . D . rebind ( A () ) ; B - > makeItSimilar ( C ) ; C () = B () ; // note d i f f e r e n t l i n e a r s y s t e m in the r e g u l a r case . supported_mode = true ; WARNING ( " ARCompGenEig in regular mode created . " ) ; supported_mode = true ; } } } // s y m m e t r i c p r o b l e m s . - - > r e q u i r e s real m a t r i c e s ! if ( matrix_kind == MATRIX_SYMMETRIC ) { // s t a n d a r d p r o b l e m s . if ( problem_kind == PROBLEM_STANDARD ) { if ( comp_mode == MODE_REGULAR ) { // use the r e g u l a r mode c o n s t r u c t o r // must cast p o i n t e r due to d i f f e r e n t NUMT ... ARSymStdEig < real , EigenSolver > * temp = new ARSymStdEig < real , EigenSolver >(n , nev , this , ( real_multfunc ) & EigenSolver :: multAx , descriptive ) ; solver = temp ; supported_mode = true ; WARNING ( " ARSymStdEig in regular mode created . " ) ; } if ( comp_mode == MODE_SHIFT_INVERT ) { // use the shift - and - i n v e r t mode c o n s t r u c t o r ARSymStdEig < real , EigenSolver > * temp = new ARSymStdEig < real , EigenSolver >(n , nev , this , ( real_multfunc ) & EigenSolver :: multInvCx , sigma , descriptive ) ; solver = temp ; A - > makeItSimilar ( C ) ; Handle ( Matrix ( NUMT ) ) Eye ; A - > makeItSimilar ( Eye ) ; Eye () = 0.0; for ( int i =1; i <= n ; i ++) Eye - > elm (i , i ) = 1.0; add ( C () , A () , - sigma , Eye () ) ; // C = A - s i g m a * I supported_mode = true ; WARNING ( " ARSymStdEig in shift - and - invert mode created . " ) ; } } // g e n e r a l i z e d p r o b l e m s . if ( problem_kind == PROBLEM_GENERALIZED ) { if ( comp_mode == MODE_REGULAR ) { // use the r e g u l a r mode c o n s t r u c t o r. ARSymGenEig < real , EigenSolver , EigenSolver > * temp = new ARSymGenEig < real , EigenSolver , EigenSolver >( n , nev , this , ( real_multfunc ) & EigenSolver :: multInvCDx , this , ( real_multfunc )& EigenSolver :: multBx , descriptive ) ; solver = temp ; // set up h e l p e r m a t r i c e s . D . rebind ( A () ) ; B - > makeItSimilar ( C ) ; C () = B () ; // note d i f f e r e n t l i n e a r s y s t e m in the r e g u l a r case . supported_mode = true ; WARNING ( " ARSymGenEig in regular mode created . " ) ; } if ( comp_mode == MODE_SHIFT_INVERT ) { // use shift - and - invert - mode c o n s t r u c t o r. ARSymGenEig < real , EigenSolver , EigenSolver > * temp = new ARSymGenEig < real , EigenSolver , EigenSolver >( ’S ’ , n , nev , this , ( real_multfunc ) & EigenSolver :: multInvCx , 211 Program Listings this , ( real_multfunc ) & EigenSolver :: multBx , sigma , descriptive ); solver = temp ; // set up l i n e a r s y s t e m . B - > makeItSimilar ( C ) ; add ( C () , A () , - sigma , B () ) ; supported_mode = true ; WARNING ( " ARSymGenEig in shift - and - invert mode created . " ) ; } if ( comp_mode == MODE_BUCKLING ) { ARSymGenEig < real , EigenSolver , EigenSolver > * temp = new ARSymGenEig < real , EigenSolver , EigenSolver >( ’B ’ , n , nev , this , ( real_multfunc ) & EigenSolver :: multInvCx , this , ( real_multfunc ) & EigenSolver :: multAx , sigma , descriptive ); solver = temp ; // set up l i n e a r s y s t e m . B - > makeItSimilar ( C ) ; add ( C () , A () , - sigma , B () ) ; supported_mode = true ; WARNING ( " ARSymGenEig in buckling mode created . ") ; } if ( comp_mode == MODE_CAYLEY ) { // use b u c k l i n g mode c o n s t r u c t o r. ARSymGenEig < real , EigenSolver , EigenSolver > * temp = new ARSymGenEig < real , EigenSolver , EigenSolver >( n , nev , this , ( real_multfunc ) & EigenSolver :: multInvCx , this , ( real_multfunc ) & EigenSolver :: multAx , this , ( real_multfunc ) & EigenSolver :: multBx , sigma , descriptive ); solver = temp ; // set up l i n e a r s y s t e m . B - > makeItSimilar ( C ) ; add ( C () , A () , - sigma , B () ) ; supported_mode = true ; WARNING ( " ARSymGenEig in Cayley mode created . " ) ; } } } if (! supported_mode ) { WARNING ( " Your computational mode is not supported ! Sorry . " ) ; return ; } // find the e i g e n v a l u e s and e i g e n v e c t o r s. // copy them to i n t e r n a l a r r a y s i n s i d e E i g e n S o l v e r. void * eval_ptr ; void * evec_ptr ; int converged ; if ( solver ) { solver - > FindEigenvalues () ; solver - > FindEigenvectors () ; eval_ptr = ( void *) solver - > RawEigenvalues () ; evec_ptr = ( void *) solver - > RawEigenvectors () ; converged = solver - > ConvergedEigenvalues () ; } if ( solverComplex ) { solverComplex - > FindEigenvalues () ; solverComplex - > FindEigenvectors () ; eval_ptr = ( void *) solverComplex - > RawEigenvalues () ; evec_ptr = ( void *) solverComplex - > RawEigenvectors () ; converged = solverComplex - > ConvergedEigenvalue s () ; } eigenvalues . redim ( converged ) ; eigenvectors . redim ( converged , n ) ; for ( int i =0; i < converged ; i ++) eigenvalues ( i +1) = (( NUMT *) eval_ptr )[ i ]; for ( int i =0; i < converged ; i ++) for ( int j =0; j < n ; j ++) eigenvectors ( i +1 , j +1) = (( NUMT *) evec_ptr ) [ n * i + j ]; // c l e a n up m e m o r y . if ( solver ) delete solver ; if ( solverComplex ) delete solverComplex ; } // // matrix - v e c t o r m u l t i p l i c a t i o n: w < - Av // void EigenSolver :: multAx ( NUMT * v , NUMT * w ) { s_e < < " . " ; // matrix - v e c t o r o p e r a t i o n s m e t h o d s . p a s s e d to A R P A C K ++ o b j e c t s . void multAx ( real * , real *) ; void multBx ( real * , real *) ; void multInvCDx ( real * , real *) ; void multInvCx ( real * , real *) ; // let V use v as u n d e r l y i n g pointer , and W use w . V - > redim (v , n ) ; W - > redim (w , n ) ; // m u l t i p l y : y = Ax A - > prod ( V () , W () ) ; 212 B.4 – The EigenSolver class } // // matrix - v e c t o r m u l t i p l i c a t i o n: w < - Bv // void EigenSolver :: multBx ( NUMT * v , NUMT * w ) { s_e < < ". " ; // let V use v as u n d e r l y i n g pointer , and W use w . V - > redim (v , n ) ; W - > redim (w , n ) ; // m u l t i p l y : y = Ax B - > prod (V () , W () ) ; // s i l l y P r i n t ( V () ) ; s_o < < " - - - Bx - - - > "; // s i l l y P r i n t ( W () ) ; s_o < < endl ; } // // matrix - v e c t o r m u l t i p l i c t i o n: w < - inv ( C ) Dv , v < - Av // void EigenSolver :: multInvCDx ( NUMT * v , NUMT * w ) { s_e < < "* " ; // let V use v and W use w as u n d e r l y i n g p o i n t e r . V - > redim (v , n ) ; W - > redim (w , n ) ; // m u l t i p l y : y = Ax D - > prod (V () , W () ) ; // let v < - - w V () = W () ; // s o l v e w = B ^ -1 y // C x = b lineq - > attach ( C () , W () , V () ) ; lineq - > solve ( c_has_changed ) ; if ( c_has_changed ) c_has_changed = false ; } // // matrix - v e c t o r m u l t i p l i c t i o n: w < - inv ( C ) v // void EigenSolver :: multInvCx ( NUMT * v , NUMT * w ) { s_e < < "* " ; // let V use v and W use w as u n d e r l y i n g p o i n t e r . V - > redim (v , n ) ; W - > redim (w , n ) ; // s o l v e w = B ^ -1 y // C x = b lineq - > attach ( C () , W () , V () ) ; // C W = V lineq - > solve ( c_has_changed ) ; if ( c_has_changed ) c_has_changed = false ; } // // make a s i m p l e report , d i s p l a y i n g m a t r i c e s and so on . // void EigenSolver :: report () { s_o < < " n == " < < n < < endl ; s_o < < " nev == " < < nev < < endl < < endl ; s_o < < " sigma == " < < sigma < < endl ; s_o < < " A == " < < endl ; if ( A . getPtr () ) sillyPrint ( A () ) ; else s_o < < " [ undefined ] " ; s_o < < " B == " < < endl ; if ( B . getPtr () ) sillyPrint ( B () ) ; else s_o < < " [ undefined ] " ; s_o < < endl ; } // // m e t h o d that p r i n t s m a t r i x in a s i l l y way // void EigenSolver :: sillyPrint ( Matrix ( NUMT ) & matrisen ) { int m , n; matrisen . size (n , m ) ; s_o < < "[ " ; 213 Program Listings for ( int i =1; i <= n ; i ++) for ( int j =1; j <= m ; j ++) { s_o < < matrisen . elm (i , j ) ; if (( i == n ) && ( j == m ) ) s_o < < " ] " < < endl ; else s_o < < " , " ; } } // // set the s h i f t // void EigenSolver :: setSigma ( real the_sigma ) { sigma = the_sigma ; } void EigenSolver :: setSigmaIm ( real the_sigma_im ) { sigma = the_sigma_im ; } // // sets the c o m p u t a t i o n a l mode . // bool EigenSolver :: setCompMode ( int mode ) { bool success = false ; // r e g u l a r mode and shift - and - invert - m o d e s are a l w a y s ok . if (( mode == MODE_REGULAR ) || ( mode == MODE_SHIFT_INVERT ) ) success = true ; // two more m o d e s for g e n e r a l i z e d , s y m m e t r i c p r o b l e m s . if (( problem_kind == PROBLEM_GENERALIZED ) && ( matrix_kind == MATRIX_SYMMETRIC ) ) { if (( mode == MODE_CAYLEY ) || ( mode == MODE_BUCKLING ) ) success = true ; } // one more mode for g e n e r a l i z e d , non - s y m m e t r i c p r o b l e m s . if (( problem_kind == PROBLEM_GENERALIZED ) && ( matrix_kind == MATRIX_NONSYMMETRIC ) ) { if ( mode == MODE_COMPLEX_SHIFT ) success = true ; } if ( success ) comp_mode = mode ; return success ; } // // get r e f e r e n c e s to e i g e n t h i n g s. // Mat ( NUMT ) & EigenSolver :: getEigenvectors () { return eigenvectors ; } Vec ( NUMT ) & EigenSolver :: getEigenvalues () { return eigenvalues ; } 214 Bibliography [1] Home page for FemLab. 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