Sh S PhD thesis

Sh S PhD thesis
Dissertation
submitted to the
Combined Faculties of the Natural Sciences and Mathematics
of the Ruperto-Carola-University of Heidelberg,
Germany
for the degree of
Doctor of Natural Sciences
Put forward by
Shahpoor Saeidian
born in Bijar (Iran)
Oral examination: June 18th 2008
Scattering resonances of ultracold atoms in
confined geometries
Referees: Prof. Dr. Peter Schmelcher
Prof. Dr. Jochen Schirmer
Zusammenfassung
Streuresonanzen ultrakalten Atomen in eingeschlossen Geometrien
Thema dieser Doktorarbeit ist sowohl die Untersuchung der Dynamik im quantenmechanischem Regime von ultrakalten Atomen in eingeschlossen Geometrien. Wir diskutieren das Verhalten von Grundzustandsatomen in einem dreidimensionalen magnetischen Quadrupolfeld. Solche Atome können für schwache
Felder näherungsweise wie neutrale Punktteilchen betrachtet werden. Ergänzend
zu den bekannten Resonanzen für positive Energien weisen wir die Existenz
kurzlebiger Resonanzen im negativen Energiebereich nach, wobei letztere ihren
Ursprung in einer fundamentalen Symmetrie des zugrundeliegenden HamiltonOperators haben. Desweiteren leiten wir eine Abbildung für die beiden Zweige
des Spektrums ab. Außerdem analysieren wir die atomaren Hyperfeinresonanzen in einem magnetischen Quadrupolfeld. Dies entspricht dem Fall, für
welchen sowohl die Hyperfein- als auch Zeemanwechselwirkung von vergleichbarer Größenordnung sind und beide berücksichtigt werden müssen. Schließlich
entwickeln wir für die Mehrkanalstreuung von zwei Atomen in einem zweidimensionalen harmonischen Einschluss eine allgemeine Gittermethode. Mit unserem Ansatz analysieren wir transversale An-/Abregungen im Zuge von Streuprozessen (unterscheidbare oder identische Atome), wobei alle wichtigen Partialwellen und deren Kopplung aufgrund von gebrochener Kugelsymmetrie in
Betracht gezogen werden. Besondere Aufmerksamkeit wird einer nicht-triviale
Erweiterung der CIR-Theorie gewidmet, welche ursprünglich nur für das Einmodenregime und den Grenzfall der Grundzustandsenergie entwickelt wurde.
Abstract
Scattering resonances of ultracold atoms in confined geometries
Subject of this thesis is the investigation of the quantum dynamics of ultracold
atoms in confined geometries. We discuss the behavior of ground state atoms
inside a 3D magnetic quadrupole field. Such atoms in enough weak magnetic
fields can be approximately treated as neutral point-like particles. Complementary to the well-known positive energy resonances, we point out the existence of
short-lived negative energy resonances. The latter originate from a fundamental symmetry of the underlying Hamiltonian. We drive a mapping of the two
branches of the spectrum. Moreover, we analyze atomic hyperfine resonances in
a magnetic quadrupole field. This corresponds to the case for which both the
hyperfine and Zeeman interaction, are comparable, and should be taken into account. Finally, we develop a general grid method for multichannel scattering of
two atoms in a two-dimensional harmonic confinement. With our approach we
analyze transverse excitations/deexcitations in the course of the collisional process (distinguishable or identical atoms) including all important partial waves
and their couplings due to the broken spherical symmetry. Special attention is
paid to suggest a non-trivial extension of the CIRs theory developed so far only
for the single-mode regime and zero-energy limit.
CONTENTS
1. Introduction and Outline of the Thesis . . . . . . . . . . . . . . . . . .
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
6
Part I
9
Scattering Theory and Numerical Methods
2. Scattering Theory: A Brief Reminder . . . .
2.1 Scattering by a Spherical Potential . . .
2.2 Cross Section . . . . . . . . . . . . . . .
2.3 Low Energy Scattering . . . . . . . . . .
2.4 Scattering Resonances . . . . . . . . . .
2.4.1 Shape (Breit-Wigner) resonances
2.4.2 Feshbach-Fano resonances . . . .
2.4.3 Decay of a resonant state . . . .
2.4.4 Virtual bound states . . . . . . .
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11
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3. Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . .
3.1 The Complex Scaling Method . . . . . . . . . . . . . . .
3.1.1 The c-product for time-independent Hamiltonian
3.2 The Linear Variational Principle . . . . . . . . . . . . .
3.2.1 The Hylleraas-Undheim theorem . . . . . . . . .
3.3 Solving the Algebraic Eigenvalue Problem . . . . . . . .
3.3.1 The Arnoldi method . . . . . . . . . . . . . . . .
3.3.2 The Shift-Invert method . . . . . . . . . . . . . .
3.3.3 Convergence of the eigenvalues . . . . . . . . . .
3.4 The Discrete Variable Representation Method . . . . . .
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31
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Part II
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Quantum Scattering Under 2D Confining Potential
4. Analytical Description of Atomic Scattering and Confinement-Induced
Resonances in Waveguides . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Hamiltonian and Two-body Scattering Problem in a Waveguide .
4.2 s-wave Scattering Regime: the Reference T -Matrix Approach . .
4.2.1 Eigenstates of the waveguide Hamiltonian . . . . . . . . .
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50
4.2.2
4.3
4.4
The Green’s function for the relative motion of two particles in a harmonic waveguide . . . . . . . . . . . . . . .
4.2.3 Multichannel scattering amplitudes and transition rates .
4.2.4 Single-channel scattering and effective one-dimensional interaction potential . . . . . . . . . . . . . . . . . . . . . .
Remarks on p-wave Scattering . . . . . . . . . . . . . . . . . . . .
Beyond the s-wave Approximation . . . . . . . . . . . . . . . . .
4.4.1 Effective quasi-1D scattering amplitudes . . . . . . . . . .
4.4.2 Comparison with the unconfined 3D case. The inclusion
of p-waves . . . . . . . . . . . . . . . . . . . . . . . . . . .
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65
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5. Numerical Description of Atomic Scattering and Confinement-Induced
Resonances in Waveguides . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Hamiltonian and Two-body Scattering Problem in a Waveguide .
5.2 Numerical approach . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Multichannel scattering of bosons . . . . . . . . . . . . . .
5.3.2 Multichannel scattering of fermions . . . . . . . . . . . . .
5.3.3 Multichannel scattering of distinguishable particles . . . .
5.4 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . .
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Part III
95
Atomic Resonances in a Quadrupole Magnetic Trap
6. Atomic Resonances in Magnetic Quadrupole Fields: An Overview . . 97
6.1 The Magnetic Quadrupole Field . . . . . . . . . . . . . . . . . . 99
6.1.1 Symmetry properties of the quadrupole magnetic field . . 101
7. Scattering Resonances of Spin-1 Particles in a Magnetic Quadrupole
Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.2 Symmetries and Degeneracies . . . . . . . . . . . . . . . . . . . . 104
7.3 Numerical Approach . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.4.1 Positive-energy resonances . . . . . . . . . . . . . . . . . . 106
7.4.2 Negative-energy resonances . . . . . . . . . . . . . . . . . 108
7.4.3 Comparing the two classes of resonances . . . . . . . . . . 109
7.4.4 Mapping among the two classes of resonances . . . . . . . 109
7.5 Summary and Concludes . . . . . . . . . . . . . . . . . . . . . . . 112
8. Atomic Hyperfine Resonances in a Magnetic Quadrupole
8.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . .
8.2 Symmetries and Degeneracies . . . . . . . . . . . .
8.3 Numerical Approach . . . . . . . . . . . . . . . . .
8.4 Results . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.1 Resonance positions in the Zeeman regime .
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Field
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8.5
8.4.2 Resonance positions in the intermediate regime . . . . . . 120
8.4.3 Resonance positions in the hyperfine Paschen-Back regime 123
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Appendix
127
A. SCATTERING THEORY IN FREE SPACE: SINGLE-MODE REGIME129
A.1 The Asymptotic Condition . . . . . . . . . . . . . . . . . . . . . 130
A.2 Orthogonality and Asymptotic Completeness . . . . . . . . . . . 131
A.3 The Scattering Operator . . . . . . . . . . . . . . . . . . . . . . . 132
A.4 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . 133
A.5 Scattering of two spinless particles . . . . . . . . . . . . . . . . . 135
A.5.1 Conservation of energy-momentum and the scattering amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
A.6 Invariance Principles and Conservation Laws . . . . . . . . . . . 138
A.6.1 Translational invariance and conservation of momentum . 138
A.6.2 Rotational invariance and the conservation of the angular
momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 138
A.6.3 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.6.4 Time reversal . . . . . . . . . . . . . . . . . . . . . . . . . 142
A.7 Scattering of the Two Particles With Spin . . . . . . . . . . . . . 144
A.7.1 The S operator for particles with spin . . . . . . . . . . . 145
A.7.2 The amplitudes and amplitude matrix . . . . . . . . . . . 146
A.7.3 The In and Out spinors . . . . . . . . . . . . . . . . . . . 148
A.8 Time-Independent Formulation of Quantum Scattering . . . . . . 149
A.8.1 Lippmann-Schwinger equation for G(z) . . . . . . . . . . 150
A.8.2 The T operator . . . . . . . . . . . . . . . . . . . . . . . . 151
A.8.3 Relation to the Møller operators . . . . . . . . . . . . . . 152
A.8.4 Relation to the scattering operator . . . . . . . . . . . . . 153
A.8.5 The stationary scattering states . . . . . . . . . . . . . . . 154
A.9 Identical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 159
B. SCATTERING THEORY IN FREE SPACE: MULTIMODE REGIME
B.1 Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Channel Hamiltonian and Asymptotic States . . . . . . . . . . .
B.2.1 Asymptotic condition . . . . . . . . . . . . . . . . . . . .
B.2.2 Orthogonality and asymptotic completeness . . . . . . . .
B.3 The Momentum-Space Basis Vectors . . . . . . . . . . . . . . . .
B.4 Conservation of Energy and the On-Shell T Matrix . . . . . . . .
B.5 Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.6 Rotational Invariance . . . . . . . . . . . . . . . . . . . . . . . .
B.7 Time-Reversal Invariance . . . . . . . . . . . . . . . . . . . . . .
B.8 Fundamentals of Time-Independent Multichannel Scattering . . .
B.8.1 The stationary scattering states . . . . . . . . . . . . . . .
B.8.2 The Lippmann-Schwinger equations . . . . . . . . . . . .
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B.8.3 The T operators . . . . . . . . . . . . . . . . . . . . . . .
B.8.4 Asymptotic form of the wave function; Collision without
rearrangement . . . . . . . . . . . . . . . . . . . . . . . .
B.8.5 Asymptotic form of the wave function; Rearrangement
collisions . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.9 Multichannel Scattering with Identical Particles . . . . . . . . . .
B.9.1 Transition probabilities and cross section . . . . . . . . .
185
C. ALKALI ATOMS IN A MAGNETIC GUIDE:MATRIX ELEMENTS
C.1 Radial Matrix Elements . . . . . . . . . . . . . . . . . . . . . . .
C.2 Angular Matrix Elements . . . . . . . . . . . . . . . . . . . . . .
C.3 Spin Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . .
196
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186
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191
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D. BOSE-FERMI MAPPING THEOREM . . . . . . . . . . . . . . . . . 200
E. BOUNDARY CONDITIONS . . . . . . . . . . . . . . . . . . . . . . . 202
F. FAST IMPLICIT MATRIX ALGORITHM . . . . . . . . . . . . . . . 204
G. ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . 206
4
1. INTRODUCTION AND OUTLINE OF THE THESIS
1.1 Introduction
Ultracold atomic gases offer a wealth of opportunities for studying quantum
phenomena at mesoscopic and macroscopic scales [64, 65, 66, 67, 68]. Laser
and evaporative cooling in conjunction with the application of magnetic and/or
electric fields allows for a precisely control over the external atomic degree of
freedom below temperatures of one micro-Kelvin. Eventually, this makes it possible experimental observation of some fascinating phenomena, Bose-EinsteinCondensation (BEC) of alkali-metal atoms being one of the most prominent examples [64, 65, 68, 67]. The resulting coherent matter waves exhibit a wealth of
intriguing properties and phenomena ranging from nonlinear excitation such as
solitons and vortices to an amazingly controllable and beautiful many-body dynamics. The later manifests itself e.g. in the recently discovered Mott-insulator
phase transition [91] and BEC-BCS crossover [92].
Ultracold atomic gases, have also shown their potential for modelling solid
states systems. Periodic potentials which are usually found in crystalline matter
can be mimicked by optical lattices [91]. Gauge potentials which are reminiscent
of electromagnetic field coupling can be introduced into the many-body Hamiltonian by rotating the atomic gases [98]. Moreover, the capability of adjusting
the two-body coupling via Feshbach resonances [99] allows for modelling any
kind of many-particle Hamiltonian.
The possibility to control over the properties of many-body systems has
opened a new opportunity for studying and understanding of many-body phenomena in solid state physics, such as quantum-phase transition [91, 56], the
quantum Hall effect [100, 98] or the BEC-BCS crossover [92]. The latter takes
place in gases of fermionic atoms, which depending on their mutual interaction, either form Bose-Einstein condensate (BEC) of tightly bound molecules or
strongly correlated Cooper pairs (BCS-regime), a regime which has also been
found within superconductors although it is hidden from the view of the outside
world.
Ultracold atom gases also display interesting features of many-body systems when their dynamics are confined to one-dimension. By employing optical
dipole traps [15] and atom chips [16, 17, 18] it is possible to fabricate mesoscopic
structures in which the atoms are freezed to occupy a single or a few lowest
quantum states of a confining potential such that in one or more dimensions the
characteristic length possesses the order of the atomic de Broglie wavelength.
These configurations can be well described by effective one-dimensional sys-
tems. Well-known examples are quantum wires and atom waveguides or quasi
two-dimensional systems such as 2D electronic gases. The quantum dynamics of such systems is strongly influenced by the geometry of the confinement.
These systems are expected to play an important role for example in Atom
Interferometries, Quantum Computing applications, ultrasensitive detectors of
accelerators , and gravitational anomalies as well as to provide an opportunity
to study novel 1D many-body states.
The spin-statistics theorem, according to which identical particles with integer spin are bosons whereas those with half-integer spin are fermions, breaks
down in low-dimensional systems. The “Fermi-Bose duality” which is a very
general property of identical particles in 1D, maps strongly interacting bosons
to weakly-interacting fermions and vice versa. In recent years this esoteric subject has become highly relevant through experiments on ultracold atomic vapors
in atom waveguides.
Almost all phenomena discussed above are outcome of scattering events.
This explain the importance of studying quantum scattering theory in these
systems. Of particular interests are the scattering resonances. In Ref. [1], the
resonance has been defined as the most striking phenomenon in the whole range
of scattering experiments. When the lifetime of the particle-target system in
the region of interaction is larger than the collision time in a direct collision
process we call the phenomenon a resonance phenomenon. Scattering theory in
confined geometries are completely different from the free space. In a confined
geometry we may observe some resonances leading to strange phenomena which
never happen in free space. For example, two particles with strong coupling
in free space may pass through without any interaction in a confined geometry. This stimulated the development of quantum scattering theory in confined
geometries.
The work in this thesis studies scattering resonances in confined geometries
in low-energy limit.
1.2 Outline of the Thesis
In this work we study scattering resonances of atoms in confined geometries
including 3D magnetic quadrupole field as well as 2D harmonic confining potential.
Atom waveguides or 2D optical latices are well-known examples of quasi-1D
systems, which are very applicable in quantum optics, especially to produce 1D
Bose-Einstein condensate. These systems can be approximated for example by
assuming a 2D transverse potential (harmonic or unharmonic) as a confinement,
while along the third dimension the particles move freely.
Inhomogeneous fields (e.g., 3D magnetic quadrupole field) is another example which find its application in ultracold atomic physics for the purpose of
trapping and confinement.
The basic idea behind the control of ultracold atoms via inhomogeneous
magnetic fields is very simple. The neutral atoms couple to the magnetic field
6
via their magnetic moment. In the case of enough weak gradient field, when the
spatial variations of the field is small enough, it can be considered homogeneous
over the size of an atom, therefore the atom can be assumed as a point-like
particle. For ground state atoms, this assumption is in general justified.
In the case of strong hyperfine interactions (compared to the magnetic interactions) the atomic magnetic moment is due to the total angular momentum
(being composed of the electronic and nuclear angular momentum), which remains constant. Therefore the atom can be treated as a point-like particle with
total angular momentum F. However, in the case of small hyperfine interactions and/or strong gradient fields (e.g., for electronically excited atoms) this
simplification fails and an admixture of different hyperfine states F due to the
field interaction has to be expected. In this case F is not a constant of motion.
The thesis is divided into three parts. The first part is dedicated to give an
overview of scattering theory in free space, which is necessary to describe the
systems considered in next parts, along with the numerical methods which have
been used during this work.
In part II we study scattering of two particles under a 2D confining potential. Chapter 4 gives an overview of the works which have been done already to
describe such systems. In chapter 5 we develop a general grid method for multichannel scattering of two atoms confined by a transverse harmonic potential.
the method allows us to consider a rich spectral structure where several different
partial waves are participating in the scattering process. With our approach we
analyze transverse excitations and deexcitations in the course of the collisional
process (distinguishable or identical) including all important partial waves and
their coupling due to the broken spherical symmetry. Special attention is paid
to the analysis of the CIRs in the multimode regimes for non-zero collision energies, i.e., to suggest a non-trivial extension of the CIRs theory developed so
far only for the single-mode regime and zero-energy limit.
In part III we analyze quantum dynamics of atoms in a magnetic quadrupole
field. Chapter 6 gives an overview of the quadrupole magnetic field, and the
works which have been done already to analyze the dynamics of atoms in such
a field. In chapter 7 we present an analysis of the quantum dynamics of ground
state atoms in magnetic quadrupole fields. The internal atomic structure is
not resolved and consequently the atoms can be treated as if they were neutral fermions or bosons. Complementary to the well-known positive energy
resonances, it is shown that there exist short-lived negative energy resonances,
which originate from a fundamental symmetry of the underlying Hamiltonian.
We drive a mapping of the two classes. In chapter 8 we study the case for which
both interactions, the hyperfine and the field interaction, have to be taken into
account on equal level for the description of the neutral atoms in the field. We
focus on atoms possessing a single active valence electron with spin S = 1/2
and a nucleus with spin I = 3/2.
7
8
Part I
SCATTERING THEORY AND NUMERICAL
METHODS
10
2. SCATTERING THEORY: A BRIEF REMINDER
Probably much of our information about the interaction between particles being
derived from scattering experiments. In this chapter we study in brief the
scattering theory and derive some important relations which will be needed in
the next chapters. Here we follow [1-4].
Let us consider an experiment in which a particle is scattered from a target.
The particle can be, for example, an electron, an elementary particle, an atom
or a molecule and the target can be a nucleus, an atom, a molecule, a flat or
curved surface. In an elastic scattering experiment the energy of the particle is
conserved. In a non-elastic scattering experiment there is an energy exchange
between the particle and intrinsic degrees of freedom of the target, and the final
energy of the particle in its out asymptote limit can be smaller or larger than the
initial energy of the particle in its in asymptote limit. In a reactive scattering
experiment the particle and the target undergo a change during the rearrangement collision and become different species. The time-dependent description of
quantum scattering has a natural and instructive parallel in classical mechanics
and we begin with a first description of this classical theory. Figure 2.1 shows a
typical classical scattering of a particle by some fixed target. Even in the simple
case of elastic scattering three possibilities may be considered. The first possibility, presented in Fig. 2.1(a), is that of a particle which comes in from infinity,
gets caught in a spirally orbit and never emerges out of the attractive potential
well. As time passes the potential energy of the trapped particle drops down to
−∞ whereas the kinetic energy increases to +∞. This black hole phenomenon
is avoided when the interaction potential at the origin is less attractive than
V (r) = −r−2 .
In order to obtain a free particle in the in and out asymptotes, V (r) should
fall of quicker than r−3 at infinity. A direct scattering event is illustrated in
Fig. 2.1(b). One should add to these two, a 3th possibility, that is, due to
multiple-scattering events, the particle is temporarily trapped by the target
[Fig. 2.1(c)]. When the lifetime of the particle-target system in the region of
interaction is larger than the collision time in the direct collision process we call
the phenomenon a resonance phenomenon.
If the potential fall off faster than r−3 at infinity, the trajectory can be
roughly divided into three parts: (1) the approach of the particle along an
almost straight orbit until it reaches its region of interaction with the target,
(2) the possibly very complicated orbit and (3) the departure of the particle
along some other approximately straight orbit.
Although these three divisions are only roughly defined, it is clear that, in
Fig. 2.1: Different possible scattering orbits.
atomic scales, the region of interaction is certainly no longer than a few atomic
diameters and so is, in practice, completely unobservable. All that will be visible
is a pair of straight tracks corresponding to the free motion before and after the
collision. Therefore, in seeking a mathematical description of the scattering
process, we shall try (as far as possible) to suppress the precise details of the
orbit in the neighborhood of the target and concentrate on the relation between
the asymptotically free incoming and outgoing trajectories.
For every in or out asymptote, one may reasonably expect there will be a
corresponding actual orbit X(t). Although our principal interest is in the in
and out asymptote, we must recognize that the correspondence between them
is defined by the actual orbit, that is, the correspondence has the form
X(t)
−−−−−−−−−→ Xin (t)
t → −∞
X(t) −−−−−−−−−→ Xout (t)
t → +∞
(2.1)
Although the condition of asymptotic free motion is expressed mathematically by the limits t tends to infinity, this does not mean that one really has to
wait an infinite time to observe the asymptotic, free motion. Quite the contrary;
even for a very slow projectile (e.g., a thermal neutron) incident on a large target
(some big molecule) the total collision time will normally not exceed 10−10 s; this
means that if the collision is centered on t = 0 then for times before t ≈ −10−10 s
and after t ≈ 10−10 s the motion is experimentally indistinguishable from free
motion; that is in practice t becomes infinite at t ≈ ±10−10 s. The reason is
that, for any given orbit, even if there is no finite time beyond which X(t) is exactly equal to Xout (t), nevertheless our measuring devices have some minimum
resolutions, and there is a finite time beyond which the difference between X(t)
and Xout (t) is smaller than we can resolve.
In our discussion of quantum scattering we shall establish limits that are
to be interpreted in exactly the same way - that for times more than some
small amounts (usually 10−10 s or less) before or after the collision, the motion
is experimentally indistinguishable from free motion.
In the following the scattering of a single particle in free space by a fixed
target situated at the origin (or equivalently relative motion of two-body colli12
2
δ
l
ul(r)
1
0
−1
−2
0
100
200
300
r
Fig. 2.2: The effect of the short-range scattering potential on the radial function ul .
sion in the center of mass frame) will briefly be discussed. For more details, we
refer the reader to appendices A and B.
2.1 Scattering by a Spherical Potential
Scattering by a spherical potential probably is one of the easiest example of
3D scattering. In this case the scattering potential V (r) depends only on the
magnitude of the distance between the particle and the target. Since no internal
degrees of freedom of the particle and/or target can be excited by such an
interaction, they may be ignored. Only elastic scattering will be possible in this
model.
The time-independent Schrödinger equation, in spatial representation looks
like
h̄2 2
∇ ψ(x) + [E − V (r)]ψ(x) = 0
(2.2)
2µ
It is convenient to consider a restricted class of potentials that vanish [or at
least become negligible compared to the centrifugal term h̄2 l(l + 1)/2µr2 ] for
r > a. This does not include e.g. Coulomb potential, for which V (r) ∝ r−1 ,
requires some special treatment.
If the scattering potential were identically zero, the unique (apart from normalization) solution would be
X
ψ(x) = ψin (x) = eik·x =
(2l + 1)il jl (kr)Pl (cos θ)
(2.3)
l
q
where k = |k| = 2µE/h̄2 , jl is a spherical Bessel function, Pl is a Legendre
polynomial, and θ is the angle between k and x. Considering the solution to
Schrödinger equation (2.2), including the scattering potential, for scattering
of an incoming plane wave, the total wave function beyond the range of the
13
potential has the form
ψk (x) = eik·x + f (k, θ, ϕ)
eikr
r
(2.4)
where f (k, θ, ϕ) is the scattering amplitude which for a spherical symmetric
potential is given by (see appendix A)
Z
0
µ
f (k, θ, ϕ) = f (k, θ) = −
d3 x0 e−ikf ·x V (r0 )ψk (x0 ), kf = kb
x
(2.5)
2
2πh̄
Let us write the full solution in the form
X
ψ(x) =
(2l + 1)il Al Rl (kr)Pl (cos θ)
(2.6)
l
where the radial function Rl (r) satisfies the partial wave equation
·
¸
h̄2 1 d 2 d
h̄2 l(l + 1)
r
R
(r)
+
E
−
V
(r)
−
Rl (r) = 0
l
2µ r2 dr dr
2µ r2
(2.7)
For r > a, where V (r) = 0, the solution of (2.7) must be a linear combination
of spherical Bessel functions jl (kr) and nl (kr), we shall write as
Rl (r) = cos δl jl (kr) − sin δl nl (kr),
r≥a
(2.8)
Since the differential equation is real, the solution Rl (r) may be chosen real,
and so δl is real.
The asymptotic forms of the spherical Bessel functions in the limit kr → ∞
are
sin(kr − 12 πl)
jl (kr) →
(2.9)
kr
− cos(kr − 12 πl)
kr
Therefore the corresponding limit of Rl (r) is
nl (kr) →
Rl (r) →
sin(kr − 12 πl + δl )
kr
(2.10)
(2.11)
If the scattering potential were exactly zero, the form (2.11) would be valid all
the way to r = 0. But the function nl (kr) has a singularity at r = 0, which
is not allowed in a state function, so we would have δl = 0 if V (r) = 0 for
all r. Comparing the asymptotic limit of the zero-scattering solution jl (kr),
with (2.11), we see that the only effect of the short-range scattering potential
that appears at large r is the phase shift of the radial function by δl (see Fig.
2.2). Since the differential cross section is entirely determined by the asymptotic
form of the state function, it follows that the cross section must be expressible
in terms of these phase shifts.
14
If we substitute the series (2.3) and (2.6) into (2.4) , and replace the Bessel
functions with their asymptotic limits, we obtain
X
(2l + 1)il Pl (cos θ)Al
l
sin(kr − 21 πl + δl ) X
=
(2l + 1)il Pl (cos θ)
kr
l
sin(kr − 21 πl) f (k, θ)eikr
+
×
kr
r
ix
(2.12)
−ix
By using sin x = e −e
and equating the coefficients of e−ikr in the above
2i
equation one obtains
X
X
1
1
(2l + 1)il Pl (cos θ)Al ei( 2 πl−δl ) =
(2l + 1)il Pl (cos θ)ei 2 πl .
(2.13)
l
l
This equality must hold term by term, since the Legendre polynomials are linearly independent, so we have
Al = eiδl .
(2.14)
Equating the coefficients of eikr in (2.13) and using (2.14) then yields
X
f (k, θ) =
(2l + 1)fl (k)Pl (cos θ)
(2.15)
l
Here the partial amplitude fl is defined as
fl (k) =
eiδl (k) sin δl (k)
1
=
k
k cot δl (k) − ik
(2.16)
The phase shift δl for a scattering potential V (r) that may be nonzero for
r < a, but which vanishes for r > a, is obtained by solving (2.7) for the radial
function Rl (r) in the region r ≤ a and matching it to the form (2.8) at r = a.
There are two linearly independent solutions to (2.7), but only one of them remains finite at r = 0; so the solution for Rl (r) in the interval 0 ≤ r ≤ a is unique
l
except for normalization. At r = a both Rl and dR
dr must be continuous. For
our purposes it is sufficient to impose continuity on the logarithmic derivative,
d
1 dRl
dr log(Rl ) = Rl dr which is independent of the arbitrary normalization. This
yields the condition
γl =
k[cos δl jl0 (ka) − sin δl n0l (ka)]
cos δl jl (ka) − sin δl nl (ka)
(2.17)
where a prime indicates differentiation of a function with respect to its argument,
and γl denotes the logarithmic derivative evaluated at r = a from the interior.
The phase shift is then given by
tan δl =
kjl0 (ka) − γl jl (ka)
kn0l (ka) − γl nl (ka)
(2.18)
If the scattering potential is not identically zero for r > a, but is still of
short range, it is possible to define phase shifts as the limit of (2.17) as a → ∞,
15
remembering that γl depends on a. This limit will exist provided the potential
falls off more rapidly than r−1 .
It has been shown that for sufficiently large values of l the phase shift δl
decreases as the reciprocal of the factorial of l (see [3], Sec.19). This is very
rapid decrease, being faster than exponential, and it is ensures that series (2.15)
is convergent. However this sufficiently large value of l may be very large,
and the phase shift series is practical only if it converges in a small number
of terms. It can be shown that the phase shift δl will be small provided that
ka ¿ l. Therefore the real usefulness of the phase shift series is for the case
ka ¿ l and lies, in the fact that while only the complete set {δl } gives an
exact description of the scattering process, in practice the sum in (2.15) has
significant contributions from merely a few waves. Practically, this is clear since
higher partial waves are expelled from the core by the centrifugal barrier. They
are only able to penetrate and hence “feel” the potential if their momentum k
is large enough. Hence this method applies to low-energy scattering in the first
place.
2.2 Cross Section
The angular distribution of scattered particles in a particular process is described in terms of a differential cross section. Suppose that a flux of Jin
particles per unit area per unit time is incident on the target. The number of
particles per unit time scattered into a narrow cone of solid angle dΩ, outside
the incident beam centered about a direction specified by the angles θ and ϕ, at
a sufficiently large distance r from the target, is proportional to the incident flux
Jin and it might be written as Jin dσ(θ, ϕ). Dividing this number by the area
of the solid angle dΩ, we obtain the flux of scattered particles in the direction
(θ, ϕ),
dσ(θ, ϕ)
Jout = Jin 2
(2.19)
r dΩ
The factor dσ/dΩ is known as the differential cross section, and can be written
as
dσ
r2 Jout
(θ, ϕ) =
(2.20)
dΩ
Jin
from which it is apparent that it has the dimension of an area. Its value is
independent of the distance r from the target, because Jout is inversely proportional to r2 . This expression is convenient because the fluxes Jout and Jin are
measurable quantities and can be calculated theoretically. The probability flux
is given by
h̄
J = Im(ψ ∗ ∇ψ)
(2.21)
µ
The incident beam can be described by ψin = Aeik·x , for which the flux
Jin = |A|2
16
h̄k
µ
(2.22)
J
dΩ
s
D
J
θ
ϕ
.
i
O
z
Fig. 2.3: The scattering of a beam of particles by a target situated at the origin O.
is uniform. The outgoing spherical wave at large r has the asymptotic form
ψout = Af (k, θ, ϕ)eikr /r, for which the radial component of the flux is
(Jout )r =
h̄
∂ψout
h̄k |f (k, θ, ϕ)|2
Im(ψs∗
) = |A|2
µ
∂r
µ
r
(2.23)
Substituting (2.22) and (2.23) into (2.20) yields the differential cross section to
be
dσ
(θ, ϕ) = |f (k, θ, ϕ)|2
(2.24)
dΩ
Since we have neglected any internal degrees of freedom of the particles, we
have implicitly restricted our solution to the case of elastic scattering. The
result (2.24) will be modified when we treat inelastic scattering.
The total cross-section is defined as
Z
dσ
σ=
dΩ
(2.25)
dΩ
For spherically symmetric potential we have
Z π
|f (k, θ)|2 sin θdθ
σ = 2π
0
Z
= 2π
π
|
0
∞
X
(2l + 1)fl (k)Pl (cos θ)|2 sin θdθ
(2.26)
l=0
Now using the normalization of the Legendre polynomials
Z π
2
[Pl (cos θ)]2 sin θdθ =
2l
+1
0
17
(2.27)
we obtain for the total cross section
∞
X
σ = 4π
(2l + 1)|fl (k)|2
=
4π
k2
l=0
∞
X
(2l + 1) sin2 δl
(2.28)
l=0
so the total cross section is the sum of the cross-section for each l. It is easy to
prove the optical theorem for a spherically symmetric potential: Just take the
imaginary part of each side of (2.15) at (θ = 0), and using Pl (1) = 1 we obtain
1X
Imf (k, θ = 0) =
(2l + 1) sin2 δl (k)
(2.29)
k
or
4π
σ=
Imf (k, θ = 0)
(2.30)
k
2.3 Low Energy Scattering
The potential in the radial Schrödinger equation (2.7) is for s-wave (l = 0) only
the interaction potential V (r), whereas for other partial waves, this potential
is superimposed with the centrifugal barrier h̄2 l(l + 1)/2µr2 . At low energies
(the energy E has to be lower than the resulting barrier) partial waves for
higher l are not important. Classically this is clear, because a particle at such
low energies will not penetrate the barrier, but simply be reflected by it and
therefore the potential inside has no effect. Thus we expect qualitatively that
the scattering due to V (r) goes to zero for all partial waves l 6= 0, when the
energy is sufficiently low. We can expand (2.18) for low energy, i.e., small k,
using the behavior of jl (x) and nl (x) for x → 0
jl (x) ≈
xl
(2l − 1)!!
, nl (x) ≈
(2l + 1)!!
xl+1
(2.31)
where (2l + 1)!! = 1.3.5. · · · .(2l + 1) such that we obtain for k → 0
tan δl (k) ≈
(2l + 1)
l − aγl
(ka)2l+1
[(2l + 1)!!]2 l + 1 + aγl
(2.32)
If we also take the expansion of the tangent into account, we will obtain
δl (k) ∝ k 2l+1
(mod π)
when k → 0
(2.33)
This is known as threshold behavior and makes it clear that at low energy, with
a potential, which decreases rapidly enough, partial waves with l ≥ 1 do not
contribute to the scattering process. We have pure s-wave scattering. Now we
consider extremely low energies (k ≈ 0). Outside the range of the potential the
radial Schrödinger equation (2.7) satisfies
d2 u
=0
dr2
18
(2.34)
where we took for simplicity rRk=0,l=0 (r) = uk=0,l=0 (r) ≡ u(r). The solution
of this equation is
u(r) = const(r − a0 )
(2.35)
On the other hand, the usual asymptotic wavefunction for the l = 0 case, is
>
uk,l=0 = rRk,l=0
(r) ∝ sin(kr + δ0 )
(2.36)
Noting that
· µ
¶¸
µ
¶
δ0
δ0
lim sin(kr + δ0 ) = lim sin k r +
≈k r+
k→0
k→0
k
k
(2.37)
which looks like (2.35) - for low energies δ0 (k) ∝ k such that δ0 (k)/k =const and therefore we can understand (2.35) as an infinitely long wavelength limit of
the usual outside wavefunction (2.36). Furthermore we have
· µ
¶¸
u0k,l=0
δ0
= k cot k r +
(2.38)
uk,l=0
k
If we expand this quantity for very small k and then set r = 0, we obtain
lim k cot δ0 (k) = −
k→0
1
a0
(2.39)
where the quantity a0 is called the scattering length. A better result can be
obtained by keeping the next term in this expansion
k cot δ0 (k) ≈ −
1
1
+ R0 k 2
a0
2
(2.40)
Here R0 is called the effective range of the potential.
The physical meaning of the scattering length a0 is clear. The asymptotic wave function (2.36) has a series of zeros rn (k) = (−δ0 (k) + nπ)/k,
n = 0, ±1, ±2, · · ·. When k goes to zero, all rn (k) go to ±∞, except for one
which tends to a0 (which can be either positive or negative).
Figure 2.4 shows the s-wave scattering length a0 as a function of the potential
depth for the screened Coulomb potential
V (r) = −
V0 −r
e
r
(2.41)
When the potential has no bound state, as illustrated in Fig. 2.5, a repulsive
potential gives a > 0 [Fig. 2.5(a)], and an attractive potential gives a < 0
[Fig. 2.5(b)]. When there are bound states in an attractive potential, however,
the scattering length can be positive or negative [Fig. 2.5(d-f)]. If we increase
continuously the well depth, we find that the scattering length goes to infinity
in some discrete values of V0 . Each divergence corresponds to appearance of a
bound state (Levinson theorem). When V0 is slightly lower than the threshold
for the appearance of a new bound state, the scattering length is large and
19
50
40
30
20
a
0
10
0
−10
−20
−30
−40
−50
0
1
2
3
4
5
−V0
6
7
8
9
10
Fig. 2.4: S-wave scattering length as a function of the potential depth V0 for the potential (4.111).
negative [Fig. 2.5(c)]; while if V0 is slightly larger than this threshold, it is large
and positive [Fig. 2.5(d)]. The above results are quite general and can be shown
for other kind of potentials.
One of the exciting recent developments in ultracold atomic physics is that
the scattering length can be tuned, from −∞ to +∞, by changing an external
magnetic field, through the Feshbach resonance [7]. This realizes the usual
thought experiment of changing the coupling constant in the Hamiltonian.
It should be emphasized that a is a parameter that describes scattering in
3D in the limit of zero energy, where the shape of the potential is irrelevant.
At higher energies, the shape of the potential does make a difference, and one
must include other parameters, such as the effective range.
We can obtain another relation for scattering length which is useful for identifying the scattering amplitude. To do this we divide (2.40) by k, differentiate
it with respect to k and take the limit k → 0
·
¸
µ
¶
∂δ0 (k)
1
R0
lim −(1 + cot2 δ0 (k))
= lim
+
(2.42)
k→0
k→0 a0 k 2
∂k
2
Now we use (2.40) again
−
·
¸
∂δ0 (k)
1/a0 + R0 k 2 /2
= lim 2
= a0
k→0 k + (−1/a0 + R0 k 2 /2)2
∂k
(2.43)
Thus we see, that another possibility of defining the scattering length consistently is
∂δ0 (k)
(2.44)
a0 = − lim
k→0
∂k
Therefore we can implicitly deduce that
f0 (k) = −
1
1 + ika0
20
(2.45)
(a)
(b)
V(r)
V0 = +0.5
a = +0.7
0
V0 = −0.5
a0 = −2.2
u(r)
u(r)
0
a0
0
a0
V(r)
r
r
(c)
(d)
u(r)
u(r)
0
0
1 bound state
V(r)
V = −0.8
0
a0 = −24
V = −0.9
0
a0 = +22
V(r)
r
r
(e)
(f)
u(r)
a0
0
a0
u(r)
0
V(r)
1 bound state
V = −1.5
0
a = +2.1
V = −6.4
0
a = −2.0
V(r)
0
2 bound states
0
r
r
Fig. 2.5: Scattering length a0 for the potential (4.111). The number of nodes of the
wavefunction u in the effective range of the potential coincides with the number of bound states of the potential.
21
The solution of the scattering problem at ultra-low energies is therefore equivalent to the determination of a single quantity: the scattering length a0 .
In the limit k → 0 the total cross section is
¯
¯2
¯
¯
1
¯ = 4πa20
σ = σ0 = 4π lim ¯¯
(2.46)
k→0 k cot δ0 (k) − ik ¯
For bosons we have to take the symmetry into account and we get 8πa20 . Due
2
to σl (k) ∝ sink2 δl ∝ k 4l for k → 0 and the fact that only odd l contribute to
the scattering process for fermions, the scattering cross section for polarized
fermions tends to zero at low temperatures, i.e., the polarized fermions do not
“see” each other at low temperatures. This makes the evaporative cooling of a
fermi gas difficult.
To obtain an implicit definition for l-wave scattering length al let study
the analytical properties of fl (k), for small k. Using the lowest-order Born
approximation, we replace ψk (r0 ) in (2.5) by
X
0
eik·x =
il (2l + 1)jl (kr0 )Pl (cos θ0 )
(2.47)
l
to obtain
Z
0
0
µ
f (k, θ) = −
d3 x0 e−ikf ·x V (r0 )eik ·x
2
2πh̄ Z
X
0
µ
=−
d3 x0 e−ikf ·x V (r0 )
il (2l + 1)jl (kr0 )Pl (cos θ0 )
2
2πh̄
l
(2.48)
Labelling the angle between kf and x0 by γ, we have
X
0
e−ikf ·x =
(−i)l (2l + 1)jl (kr0 )Pl (cos γ)
(2.49)
l
Now using the theorem
b · Yb ) =
Pl (X
l
4π X ∗ b
Ylm (X)Ylm (Yb )
2l + 1
(2.50)
m=−l
b = (θ0 , ϕ0 ) and Yb = (θ, ϕ) as the direction of x0 and kf respectively,
and taking X
we obtain
l
4π X ∗ 0 0
Pl (γ) =
Ylm (θ , ϕ )Ylm (θ, ϕ)
(2.51)
2l + 1
m=−l
By inserting (2.49) and (2.51) into (2.48), and integrating over θ0 and ϕ0 , the
nonzero m terms give zero, in fact the only nonzero term is that with the same
l as the term in the ψk (x0 ) expansion, giving
Z ∞
∞
2µ X
(2l + 1)Pl (cos θ)
r2 drV (r)[jl (kr)]2
(2.52)
f (k, θ) = − 2
h̄ l=0
0
22
or
fl (k) ' −
2µ
h̄2
=−
Z
2µ
h̄2
drV (r)[rjl (kr)]2
·
Z
drV (r)r2
¸
¢ 2
(kr)l ¡
1 + O[(kr)2 ]
(2l + 1)!!
(k → 0)
(2.53)
Assuming uniform convergence, one may interchange the limits and conclude the general momentum dependence fl (k) = O(k 2l ) as k → 0. Via
fl (k) ∼ 1/k cot δl (k) (2.16), one can reformulate this in terms of the phase
shifts. Eventually, that leads to
k 2l+1 cot δl (k) = −
1
1
+ Rl k 2 + O(k 4 ),
(al )2l+1
2
al 6= 0, Rl ∈ R
(2.54)
or
tan δl (k)
= −(al )2l+1 + O(k 2 )
k 2l+1
(2.55)
which is an implicit definition of the l-wave scattering length al and the effective
range Rl . It is only for s-wave scattering (l = 0) that the amplitude does
not vanish for low energies (or, alternatively, the slope of the phase shift). In
that limit, a0 is the only relevant parameter, no matter what the shape of the
potential. It makes sense, in the wider scheme of things, because as k → 0, the
wavelength λ becomes much larger than the potential’s range and thus cannot
probe its short-scale structure.
2.4 Scattering Resonances
The scattering cross sections as a function of energy, may exhibit some sharp
peaks superimposed on a smooth background. These peaks correspond to resonance states and occur when one of the phase shifts passes rapidly through π/2.
In Ref.[1], the resonances has been defined as the most stricking phenomenon
in the whole range of scattering experiment. When the lifetime of the particletarget system in the region of interaction is larger than the collision time in a
direct collision process we call the phenomenon a resonance phenomenon. A
resonance state is defined as a long-lived state of a system which has sufficient
energy to break-up into two or more subsystems. In elastic and inelastic scattering experiments the subsystems are associated with the scattering particle
and the target. Resonances are observed in atomic, nuclear and particle physics.
There are many different theoretical approaches to the resonance phenomenon,
all of them having in common that the sharp variation of the cross section at a
resonant energy Er , is in some way related to the existence of a nearly bound
state of the projectile-target system with energy Er . When the projectile is sent
in with energy Er it can be temporarily captured into this meta-stable state;
and it is this possibility that is considered the cause of violent variations in cross
section.
23
V(r)
E1
0
E0
r
Fig. 2.6: Schematic representation of a spherically symmetric potential which supports
shape-resonances.
2.4.1
Shape (Breit-Wigner) resonances
An illustrative plot of spherically symmetric potential which supports resonances, V (r), is given in Fig. 2.6. Such potential describing for example decay
of radioactive nuclei or of unstable particles [8]. Naturally, we shall refer to the
nucleus as the target in this case. r is, for example, the distance between an
α particle and the nucleus. E0 and E1 represent bound and resonance energies
respectively.
A very similar potential can describe both the molecular interaction in a
diatom for which the total angular momentum, j, is much larger than zero and
gives rise to a centrifugal potential barrier [9], and the scattering of an electron
from a neutral diatom [10]. In such cases, the lifetime of the metastable resonance states can vary from femtoseconds (e.g. ∼ 10−15 s for the autoionization
of H2− in its ground state) to millions of years (e.g. 4.5 × 109 y for the decay of
the 238 u isotope to Thorium). In all of these cases the temporarily trapping of
V(r)
θ = θ1; φ = φ1
E
0
θ = θ0; φ = φ0
r
Fig. 2.7: Schematic representation of two couples adiabatic potentials.
24
the particle inside the potential well is a quantum phenomenon which is known
as the tunnelling phenomenon. The quantum equation of motion are reduced to
the classical one as h̄ → 0. While h̄ is taken to zero, the penetration probability
of the quantum particle through the potential barrier (into the well or out of
it) is reduced and in the limit of h̄ = 0 no resonance states will be observed.
These kind of resonances are known as shape-type resonances. The temporarily
trapping of the particle inside the potential well can occur also when the energy
of the particle is larger than the height of the potential barrier or even in the
absence of a potential barrier (e.g. scattering of a Gaussian wave packet from a
finite square well [5]).
Let’s consider scattering by a spherical potential and neglect any internal
degrees of freedom of the particles. In order for δl to achieve the value π/2,
which maximizes contribution to the cross section from the partial wave l, it
is necessary for the denominator of (2.18) to vanish. Suppose this happens
at the energy E = Er . Then in a neighborhood of E = Er , we may write,
approximately, γl ≈ c − b(E − Er ), where c = kn0l (ka)/nl (ka). It is clear that
the approximation can be valid only if nl (ka) 6= 0. In the neighborhood of
E = Er , (2.18) becomes
kjl0 (ka) − γl jl (ka)
nl (ka)b(E − Er )
kj 0 (ka) − cjl (ka)
≈ l
nl (ka)b(E − Er )
k[jl0 (ka)nl (ka) − n0l (ka)jl (ka)]
=
n2l (ka)b(E − Er )
tan δl ≈
(2.56)
this expression can be simplified by means of the Wronskian relation, jl0 (z)nl (z)−
n0l (z)jl (z) = −z −2 , which follows directly from the differential equation satisfied
by the Bessel functions. Thus we obtain
tan δl '
1
2Γ
Er − E
(2.57)
where 21 Γ = (ka2 b[nl (ka)]2 )−1 and h̄2 k 2 /2µ = Er . Without further approximation this yields
Γ
sin δl exp(iδl ) =
.
(2.58)
2(Er − E) − iΓ
The contribution of this resonant partial wave l to the total cross section is
σl =
4π(2l + 1)
Γ2
k2
4(Er − E)2 + Γ2
(2.59)
This is the famous Breit-Wigner formula. Although we obtained this result for
a spherical potential, it is also correct in more general cases.
If Γ is small, this term will produce a sharp narrow peak in the total cross
section.
25
2.4.2 Feshbach-Fano resonances
Let us consider a non-spherical symmetric interaction potential, between the
particle and the target. In this case it may happen that a bound state of the
particle-target system in a fixed orientation (e.g. a square integrable eigenfunction of the one-dimensional time-independent Schrödinger equation
H(r, θ0 , ϕ0 )|ψi = E|ψi
(2.60)
for fixed polar coordinates θ0 and ϕ0 of the relative motion) is embedded in the
continuum of the system in another orientation when θ = θ1 and ϕ = ϕ1 . The
two adiabatic potentials are schematically presented in Fig. 2.7.
Due to the coupling of the r-coordinate with the angle θ and ϕ the bound
state which is pushed up into the continuum becomes a resonance state. These
kind of resonances are known as Feshbach-Fano type resonances and can also be
obtained in classical calculations. Unlike the shape-type resonances, the lifetime
of the Feshbach metastable states get nonzero value as h̄ is taken to the limit
of h̄ = 0.
For a Feshbach-Fano resonance, in contrast with a shape resonance, the
corresponding profile in the cross-section has the so-called Fano shape, i.e. it
can be fitted with a function proportional to
(qΓ + 2(E − Er ))2
4(E − Er )2 + Γ2
(2.61)
The Er and Γ parameters are the standard Breit-Wigner parameters (position
and width of the resonance, respectively). The so-called Fano parameter, q is
interpreted as the ratio between the resonant and background scattering probability. In the case the background scattering probability is vanishing, the q
parameter becomes infinite and the Fano formula is boiling down to the usual
Breit-Wigner formula.
Within a method is named as the Feshbach-Fano partitioning theory one can
partitions the wave function (and therefore the associated quantities like cross
section and phase shift) into the resonant and the background components
|ψi = P |ψi + Q|ψi
(2.62)
with asymptotic conditions
1
lim hx|P |ψi =
r→∞
(2π)3/2
µ
¶
eikr
ik·x
e
+ [fP (k, θ, ϕ) + fQ (k, θ, ϕ)]
r
(2.63)
and
lim hx|Q|ψi = 0
r→∞
(2.64)
Here fP and fQ are the background and the resonance scattering amplitude
respectively. P and Q are projectors on the background and the resonant subspace respectively. The subspaces onto which P and Q project are sets of states
obeying the continuum and the bound state boundary conditions respectively.
26
In this method the Schrödinger equation governing the whole scattering process (defined by the Hamiltonian H) is solved in two steps: First by solving the
scattering problem ruled by the background Hamiltonian P HP
P HP |φi = E|φi
(2.65)
with asymptotic condition
1
lim hx|φi =
r→∞
(2π)3/2
µ
¶
eikr
ik·x
e
+ fP (k, θ, ϕ)
r
(2.66)
It is often supposed that the solution of this problem is trivial or at least fulfilling
some standard hypotheses which allow to skip its full resolution. Second by
solving the resonant scattering problem corresponding to the effective complex
(energy dependent) Hamiltonian
H ef f (E) = QHQ + QHP
1
P HQ
E + i0 − P HP
(2.67)
where E + i0 = limη→0+ E + iη. One can show that
1
QHP |φi
E − H ef f (E)
(2.68)
1
1
P HQ
QHP |φi
E + i0 − P HP
E − H ef f (E)
(2.69)
Q|ψi =
and
P |ψi = |φi +
By taking the limit r → ∞ of hr|P |ψi and comparing with (2.63) one can easily
calculate the resonance scattering amplitude fQ .
The resonance positions are the solutions (in the lower complex-E plane) of
the so-called implicit equation
det[H ef f (E) − E] = 0
(2.70)
with E = ² − iΓ/2.
2.4.3 Decay of a resonant state
The physical nature of a resonance scattering state can be understood by examining its behavior in time. Instead of a stationary (monoenergetic) state, we
now consider a time-dependent state involving a spectrum of energies that is
much broader
Z
(+)
ψ(x, t) = A(k)ψk (x)e−iEt/h̄ d3 k
(2.71)
where E = h̄2 k 2 /2µ. The function A(k) should be nonzero only for values of k
that are collinear with the initial beam. This state function can be divided into
27
an incident wave and a scattered wave [see (2.4)], the scattered wave will be of
the form
Z
eikr −iEt/h̄ 3
ψout (x, t) ∼ A(k)f (k, θ, ϕ)
e
d k
(2.72)
r
in the limit of large r.
Suppose now that all phase shifts δl are small except for l = l0 which is a
resonant. Then the scattering amplitude will be dominated by l0 , and using
(2.15) and the resonance approximation (2.58) we obtain
Z
ψout (x, t) = (2l + 1)Pl (cos θ)
A(k)
eikr
Γ
e−iEt/h̄ d3 k
kr [2(Er − E) − iΓ]
(2.73)
here θ is the angle of x relative to the incident beam. This integral can most
conveniently be analyzed by going to polar coordinates and using E = h̄2 k 2 /2µ
as a variable of integration, so we put
d3 k = k 2 dΩk dk =
This yields
ψout (x, t) ∼
where
Z
µ
F (r, t)
(2l + 1)Pl (cos θ)
r
h̄2
∞
F (r, t) =
µ
dΩk kdE
h̄2
α(E)Γ
0
and
exp[i(kr − Et/h̄)]
dE
2(Er − E) − iΓ
(2.74)
(2.75)
(2.76)
Z
α(E) =
A(k)dΩk
(2.77)
The precise time dependence of F (r, t) is determined by the details of the initial
states through the function α(E) and can be quite complicated. We assume
α(E) to be a smooth function of energy, nearly constant over an energy range Γ,
and so it is reasonable to replace α(E) by α(Er ) in the integral. In the resonance
approximation, the integral is dominated by contribution in the energy range
Er ± Γ. Therefore we replace k in the exponential by its Taylor series k ≈
kr + µ(E − Er )/h̄2 kr where Er = h̄2 kr2 /2µ. Introducing a dimensionless variable
of integration, z = (E − Er )/Γ, and a retarded time τ = t − r/vr , where
vr = h̄kr /µ, we can rewrite (2.76) as
Z ∞
exp(−iτ Γz/h̄)
F (r, t) = −α(Er )Γexp(ikr r − iEr t/h̄)
dz
(2.78)
2z + i
−Er /Γ
If Γ ¿ Er , the lower limit can be replaced by −∞. The integral then can be
evaluated for positive τ by closing the counter of integration with an infinite
semicircle in the lower half of the complex z−plane. From the residue of the
pole at z = −i/2 we obtain the time dependence exp(−τ Γ/2h̄). For negative τ ,
the counter must be closed in the upper half-plane, where there are no poles,
28
and so the integral vanishes. Thus the time dependence of the scattered wave
(2.73) at large distance will be
ψout (x, t) ∝ e−Γt/2h̄
ψout (x, t) = 0
for t > r/vr
for t < r/vr
(2.79)
It is zero before t = r/vr because that is the time needed for propagation from
the scattering region to the detector. For times greater than this, the detection
probability decay exponentially.
2.4.4 Virtual bound states
The physical picture of a scattering resonance that we derive from the above
analysis is of a particle being temporarily captured by the scattering potential
in a virtual bound state whose mean lifetime is h̄/Γ. It is possible to exhibit
a closer connection between bound states and resonances. Suppose that the
potential supports a bound state at the negative energy E = −Eb . As the
strength of the potential is reduced the binding energy Eb will decrease and
eventually vanish. As the potential strength is further reduced, a resonance
or virtual bound state, appears at positive energy. Further reduction in the
potential strength results in Γ increasing, so that the virtual bound state has
so short a lifetime that it is no longer significant.
We shall illustrate this connection only for E = 0, which is the boundary
between bound states and scattering states. It is apparent from (2.18) that in the
limit k → 0 we have tan δl → 0 for almost all values of the logarithmic derivative
γl . The exception occurs if the denominator vanishes, in which case the phase
shift has the zero energy limit π/2, and we have a zero-energy resonance. In
this case, we must have γl = kn0l (ka)/nl (ka) → −(l + 1)/a in the limit k → 0.
29
30
3. NUMERICAL METHODS
3.1 The Complex Scaling Method
In the previous chapter a resonance state, has been defined as a non-stationary
(or quasi-bound) state with a lifetime long enough to be well characterized, and
long enough to make its explicit recognition of experimental and theoretical
importance. The simplest, and most naive, mathematical description of such
states is that they resemble bound stationary states in that they are localized
in space (at t = 0) and their time evolution is given by
ψ res (x, t) = e−iEr t/h̄ φres (x)
(3.1)
which is the usual stationary state time dependence, except that now the energy,
Er , of the resonant state is complex:
Er = ² − i
Γ
2
(3.2)
where ² and Γ are real, and Γ ≥ 0 . The presence of the −i Γ2 forces the
probability density
|ψ res (x, t)|2 = |φres (x)|2 e−Γt/h̄
(3.3)
to decay exponentially to zero as time passes at a constant r. Therefore the
particles disappear from any given point in the coordinate space. From this
point of view, one needs to solve the time-dependent schrödinger equation to
obtain the lifetime of a resonance state. Nevertheless it is also possible to
estimate the resonance lifetime from time-independent calculation. In [6] it
has been shown that the resonance phenomenon as obtained in a scattering
experiment are associated with the poles of the scattering matrix S in fourthquarter of the complex k-plane (i.e., Re(k) > 0, Im(k) < 0), where k is the
wavevector given by
(h̄k)2
E − Et =
(3.4)
2µ
with Et being the threshold energy, which for resonance state is a complex
number. These resonance poles are complex eigenvalues of the Hamiltonian H
res
H(x)φres
n (x) = En φn (x),
i
En = εn − Γn ,
2
(3.5)
where εn is the resonance position above the threshold, Γ/2 is the inverse lifetime. Equation (3.5) is the basic equation in the resonance theory for timeindependent Hamiltonian. The resonances are associated with complex eigenvalues of the Hamiltonian which describes the physical systems. One may expect that, the Hamiltonian of a physical system should be Hermitian, with real
eigenvalues. However a physical Hamiltonian is hermitian only when it acts on
bounded functions (not necessarily square integrable) or, when box normalization is used, on a functional space of all possible square integrable functions (i.e.,
Hilbert space) [6]. The φres which are associated with complex eigenvalues are
not in the Hermitian domain of the Hamiltonian (i.e., not in the Hilbert space)
as at asymptotic region they diverges exponentially
φres (x → ∞) ∼ eik·x + S(k)eikr → ∞,
(r = |x|)
(3.6)
Due to (3.3) and (3.6) the number of particles is conserved only when both the
reaction coordinate r and the time t, approach the limit of infinity.
Most of the computational algorithms in quantum mechanics, such as variational methods, have been developed for Hermitian operators, and they can not
be used to solve (3.5), even for the 1D case. During the past decays there has
been rapid development and application of a theory variously known as complex
scaling, complex coordinates, coordinate-rotation, and dilatation analyticity, to
problems of resonances in atomic and molecular physics and in chemistry. One
of the major purposes of the introduction of complex scaling theory is to extend
the variational principle and of other well-known theorems in quantum mechanics to non-Hermitian operators by carrying out similarity transformation
S which makes the resonance wavefunctions φres , square integrable functions.
That is
i
(SHS−1 )(Sφres ) = E(Sφres ), E = ε − Γ
(3.7)
2
such that Sφres → 0 as r → ∞ and thus satisfy our feeling that resonances are
localized, at least at complex values of the coordinates. One example for the
complex scaling operator is
S = eiηr∂/∂r
(3.8)
such that
Sf (r) = f (reiη )
(3.9)
for any analytical function f (r). As it has been illustrated in figure 3.1, bound
states and scattering thresholds (corresponding to the possibility of fragmentation of different subsystems in differing states of excitation) are independent
under the dilatation transformation. However the segments of continua beginning at each scattering threshold rotate by an angle 2η into the lower half plane
(η ≥ 0), each about its individual threshold. This follows from the fact that
due to (3.4), φres becomes square integrable only for the complex wavevector
k = |k|e−iη and since for finite range potential a continuum is related to the
kinetic energy E ∝ k 2 /2 and not to the potential, it rotates like
E∝
k2
k2
→ e−2iη .
2
2
32
(3.10)
As the continua rotate, new, complex, discrete eigenvalues of H(η) may appear in the lower half complex energy plane, such that once revealed , maintain
their positions. These eigenvalues correspond to poles of the S−matrix which
we associate with resonances. They are hidden on a higher sheet if η = 0, and
will be exposed if the cuts pass through the angle
·
¸
Γ
θ = 2η = arctan
(3.11)
2(ε − Et )
where as before ε is the resonance position above the threshold and Γ/2 is the
inverse lifetime.
Im(E)
Re(E)
........ . .
Resonances
Rotated Continuum
Fig. 3.1: Schematic representation of the eigenvalues of the rotated Hamiltonian.
Since resonance states of the complex scaled Hamiltonian are square integrable the linear variational principle can be applied. An eigenstate |Ei of
the stationary Schrödinger equation can now be expanded in terms of a set of
basis-functions
X
|Ei =
ci |χi i
(3.12)
i
leading to the generalized spinor eigenvalue problem
H(η)c = ESc
(3.13)
which yields the complex eigenvalues Ei , and the complex vectors ci . S is the
corresponding matrix representation of the overlap matrix
(S)ij = hχi |χj i
(3.14)
The basis set must be chosen in such a way that the exact wave function can
be approximated to a sufficient degree of accuracy by as small as possible number of functions. Therefore the form of the basis set must be adapted to the
33
geometry and the symmetries of the system. We note once that as H(η) is not
Hermitian, its spectral resolution involves both its left and right eigenfunctions.
However, elementary manipulation gives the results that the left eigenfunction
di corresponding to eigenvalue Ei is just cTi where T is the transpose. (Note
that the transpose is not the usual Hermitian conjugate, which is the transpose
complex conjugate.) Thus
X
(3.15)
H(η) =
Ei ci cTi
i
This is a bi-orthogonal expansion, and cTi cj = δij (rather than the usual for
Hermitian problems.)
3.1.1
The c-product for time-independent Hamiltonian
The inner product is defined in quantum mechanics as the scaler product, that
is
Z
hf |gi =
f ∗ gdτ
(3.16)
all space
where f, g are functions in the Hilbert space. When f and g are exponentially
divergent functions (as the resonance eigenfunctions of H are) they are not in
the Hilbert space, and they are NOT in the Hermitian domain of H. By scaling the coordinates in the time-independent Schrödinger equation by a complex
factor, the exponentially divergent resonance states become square integrable.
However, in spite of the fact that the complex-scaled resonance states are in
the generalized Hilbert space, they are not associated with an Hermitian operator [H(reiη ) is non-Hermitian] and the above scaler Hermitian product should
be replaced by a generalized inner product (the so-called c-product). In the
c-product, the complex conjugate of the terms in the function which are complex not as a result of the complex scaling are taken. In the c-product first
the complex conjugate of the function is taken and only then the coordinates or
momentum are analytically continued into the complex plan. If for example, the
eigenfunctions of the unscaled Hamiltonian are real and the COMPLEX eigenfunctions of the complex-scaled Hamiltonian are expanded in terms of REAL
functions (as it is the case in our calculation of resonances of an atom in external
magnetic field), then the inner product is given by
Z
(f |g) ≡ hf ∗ |gi =
f gdτ
(3.17)
all space
Knowing the resonance eigenfunctions of the complex-scaled Hamiltonian
one can calculate corresponding expectation value of an arbitrary observable A
within the generalized inner c-product
hAi =
(Sφres |A|Sφres )
.
(Sφres |Sφres )
34
(3.18)
The expectation value which is obtained in this way is in general complex.
The real part represents the average value, whereas the imaginary part can be
interpreted as the uncertainty of our observable in a measurement when the
system is prepared in the corresponding resonance state [6].
3.2 The Linear Variational Principle
The quantum mechanical problems, can not be solved exactly, except for some
special cases. Therefore we need to use some methods of approximation. One
important tool is the linear variational principle which allows for solving the
stationary Schrödinger equation
H|ψj i = Ej |ψj i
(3.19)
In this method, a set of normalized basis functions {|φi i} is employed to construct the trial wavefunction |ψi as
X
|ψi =
ci |φi i
(3.20)
i
with the expansion coefficient ci serving as variational parameters. The basis
set could in general be non orthonormal. Introducing the column vector c with
component ci , we find for a normalized |ψi
X
X
c∗i cj hφi |φj i =
c∗i cj Sij = c† Sc.
(3.21)
1 = hψ|ψi =
i,j
i,j
S is the so-called overlap matrix, which in the case of an orthonormal set of
basis functions, is just the unity matrix. The expectation value of H is given by
X
X
c∗i cj hφi |H|φj i =
c∗i cj Hij = c† Hc
(3.22)
hHi = hψ|H|ψi =
i,j
i,j
According to the variational principle finding the optimum variational parameters is equivalent to finding the minimum of the expectation value hHi under
the constraint (3.21), which is equivalent to minimize the function L, defined
by
X
X
L=
c∗i cj Hij − χ[
c∗i cj Sij − 1]
(3.23)
i,j
i,j
here χ is the corresponding Lagrange parameter. Evaluating the derivative of
L with respect to c∗i and putting it equal to zero we obtain a set of equations
whose solution yields the optimal expansion coefficient ci :
X
∂L
=
[Hij − χSij ]cj = 0.
∗
∂ci
i
(3.24)
They can be written more compactly as a single matrix equation:
Hc = χSc.
35
(3.25)
Therefore the task of finding the optimum parameters reduces to a generalized
algebraic eigenvalue problem. In order to illuminate the actual meaning of the
generalized eigenvalues χ, let’s consider the matrix element
hψi |H|ψj i = ci † Hcj = χj c†i Scj = χj hψi |ψj i = χj δij
(3.26)
This shows that χj is the energy expectation value of the state |ψj i, or, on
the other words, it can be considered as an approximation for the eigenenergy
Ej of the Hamiltonian H. In the case of a complete basis functions, |φj i, χj
would represent the exact eigenenergies. Usually for the systems which will be
considered in this work, we need infinite set of basis functions, and since in
practice any computation can involve only a finite set of basis functions, the
eigenvalue χj represent only upper bounds of the exact eigenvalues such that:
E1 ≤ χ1 ≤ E2 ≤ χ2 ≤ E3 ≤ χ3 ≤ ... ≤ EN ≤ χN
N is the number of the basis functions {|φi i}. This is known as the HylleraasUndheim theorem whose prove is presented in the following section.
3.2.1 The Hylleraas-Undheim theorem
0
Let assume H and H to be the representations of the Hermitian operator H
in the subspace spanned by the basis M0 = {|φ1 i, ..., |φN −1 i} and M = M0 ∪
{|φN i}, respectively. Then the real eigenvalues λ01 ≤ λ02 ≤ .. ≤ λ0N −1 of H0 and
λ1 ≤ λ2 ≤ .. ≤ λN of H satisfy the relation
λ1 ≤ λ01 ≤ λ2 ≤ λ02 ≤ ... ≤ λN −1 ≤ λ0N −1 ≤ λN
For proving we follow Ref. [11]. Without loss of generality, the basis M0 is
chosen such that H0 is diagonal with its eigenvalues being the diagonal elements
in ascending order, i.e. λ0i ≤ λ0i+1 . Then H takes the form



H=

H0
HN 1
HN 2

H1N
H2N
..
.
···




(3.27)
HN N
The eigenvalues λi of the matrix H are the zeros of the determinant
¯ 0
¯
¯ λ1 − λ
¯
0
···
H1N
¯
¯
0
¯
¯
0
λ
−
λ
·
·
·
H
2N
2
¯
¯
f (λ) = det(H − λ1) = ¯
¯
..
..
.
.
.
.
¯
¯
.
.
.
.
¯
¯
¯ HN 1
HN 2 · · · HN N − λ ¯
= (HN N − λ)
N
−1
Y
n=1
36
(λ0n − λ) −
N
−1
X
m=1
|HmN |2
N
−1
Y
(3.28)
(λ0n − λ)
n6=m
Evaluating f (λ) at the points λ = λ0k yields
f (λ) = −|HkN |2
N
−1
Y
(λ0n − λ0k )
n6=k
≤ 0 if k odd
≥ 0 if k even
(3.29)
One then finds the following behavior for f (λ)
f (λ) →
(±)N ∞ if
λ → ∓∞
which is obvious when writing f (λ) in the form
#
"
N
−1
N
−1
Y
X
|HmN |2
f (λ) =
(λn − λ) (HN N − λ) −
(λm − λ)
n=1
m=1
(3.30)
(3.31)
from (B.80) it follows that there is an odd number of zeros between two distinct
eigenvalues of H0 . On the other hand f (λ) has exactly N real zeros λ1 , λ2 , ..., λN
and therefore the λ0i must be ordered according to
λ1 ≤ λ01 ≤ λ2 ≤ λ02 ≤ ... ≤ λN −1 ≤ λ0N −1 ≤ λN
Hence, the eigenvalues λ0 constitute upper bounds for the λ.
3.3 Solving the Algebraic Eigenvalue Problem
By employing the variational principle, solving the stationary Schrödinger equation (3.19) is reduced to an generalized eigenvalue problem (3.25),which can be
solved by means of the Arnoldi method, a numerical procedure designed for
solving large scale eigenvalue problem. In conjunction with Arnoldi method, we
use the so-called shift-invert procedure which allows for the calculation of eigenvalues lying in arbitrary regions of the spectrum without the need to find the
entire bottom up. For the numerical implementation of these two methods, the
routines from the software packages ARPACK [12], and superLU [13] have been
used. The latter has been designed to solve large inhomogeneous linear system
of equations, performing an LU-decomposition of the corresponding matrices.
3.3.1 The Arnoldi method
The idea of the Arnoldi method is to reduce a large-scale eigenvalue problem
AX = λSX
(3.32)
to one with a significantly lower dimension, which is then soluble with comparatively little effort. The basis for this procedure is the so-called Arnoldi
decomposition. By representing a matrix A ∈ C n×n in the so-called Krylovspace Kk (A, v) := Span{v, Av, ..., Ak−1 v}, where v ∈ C n \{0} is an arbitrary
vector, we obtain a so-called k-step Arnoldi decomposition of the form
AVk = Vk Hk + fk eTk
37
(3.33)
here the matrix Vk ∈ C n×k has orthonormal columns, obeys VkT fk = 0 and the
matrix Hk ∈ C k×k is an upper Hessenberg matrix with non-negative subdiagonal elements. In the case of a Hermitian matrix A, Hk is reduced to a real,
symmetric and tri-diagonal matrix, and (3.33) is referred to as k-step Lanczos
Factorization. Given vector s satisfying the eigenvalue equation
Hk s = θs
(3.34)
kAu − θuk = k(AVk − Vk Hk )sk = kfk k|eTk s|
(3.35)
the vector u = Vk s satisfies
The pair (u, θ) is denoted as a Ritz pair representing an approximate eigenpair
of the matrix A. The accuracy degree of this approximation can be measured by
the so-called Ritz estimate kfk k|eTk s|. If this value drops below a given threshold
(commonly machine precision), the Ritz pair (u, θ) is considered a solution of
the eigenvalue problem (3.34). If necessary the value of k should be increased
to establish convergence.
In summary, the eigenvalue problem should only be solved for k-dimensional
Hessenberg matrix Hk with k being much smaller than the dimension of the
matrix A (Hk is even symmetric and tridiagonal if A is Hermitian), for which
efficiently working algorithms are available. The method converges firstly for
those eigenvalues of the matrix A which are the largest or smallest in magnitude.
Thus, it can be most efficiently used to calculate the extermal eigenvalues of A.
3.3.2
The Shift-Invert method
For physical problems with high dimensionality it is not possible to calculate
the entire spectrum of the Hamiltonian. Employing the shift-invert method,
in the case we need to calculate the eigenvalues just in a certain region of the
spectrum, we can transfer the initial Matrix, such that those eigenvalues which
we are interested in be the first to converge. For this purpose we perform the
transformation
Hc = ²Sc ↔ (H − σS)c = (² − σ)Sc
↔ (H − σS)−1 Sc = λc
(3.36)
where λ = 1/(² −σ). The eigenvector c remains unchanged. The old eigenvalues
²i are related to the new ones λi by
²i = σ + 1/λi .
(3.37)
According to this relation, the eigenvalues lying close to σ correspond to eigenvalues of the operator (H − σs)−1 S with largest magnitude and thus converge
first.
38
3.3.3
Convergence of the eigenvalues
Given an operator H, the eigenvalues ²i obtained from the linear variational
method, constitute upper bounds of the exact eigenenergy Ei of H, thus Ei can
be approximated by ²i . As the number of basis functions increases, the accuracy of this approximation increases. To obtain a measure of the convergence
behavior, of the eigenvalues, we introduce the convergence parameter Ki as
¯
¯
¯ ²G1 − ²G2 ¯
¯ i
i ¯
Ki = ¯ G1
(3.38)
¯
1 ¯
¯ ²i − ²G
i−1
which is a measure of the difference of the same eigenvalue ²i for two different
basis set size G1 and G2 (G2 > G1 ) relative to the difference to the next lower
2
eigenvalue ²i−1 . During the computation an eigenvalue ²G
is considered as
i
converged once the corresponding convergence parameter Ki is found to be
smaller than a certain value (e.g., 0.01).
3.4 The Discrete Variable Representation Method
The essence of all numerical schemes in solving differential equations and integration is in the discretization of the continuous variables, that is the spatial
coordinates and/or time. Let us assume the simple case of a single-variable
function f (x). The first order derivative of this function around a point x is
defined from the limit
f 0 (x) = lim
∆x→0
f (x + ∆x) − f (x)
∆x
(3.39)
if it exists. Now if we divide the space into discrete points xk with an evenly
spaced interval h = xk+1 − xk and label the function at the lattice points as
fk = f (xk ), we obtain the simplest expression for the first-order derivative
fk0 =
fk+1 − fk
+ O(h)
h
(3.40)
The above formula is referred to as the two-point formula for the first-order
derivative; it can be easily derived by taking the Taylor expansion of fk+1 around
xk . The accuracy can be improved if we expand fk+1 and fk−1 around xk and
take the difference
fk+1 − fk−1 = 2hfk0 +
h3 (3)
f + ···
6 k
(3.41)
After a simple rearrangement, we have
fk0 =
fk+1 − fk−1
+ O(h2 )
2h
(3.42)
which is commonly known as the three-point formula for the first-order derivative. The accuracy of the expression will increase to even higher orders in h if
39
more points are used. For example a seven-point formula (what we have used in
our calculations) can be derived by including the expansions of fk+3 , fk−3 , fk+2
and fk−2 around xk . After some combinations one obtains [14]
fk0 =
1
(−fk−3 + 9fk−2 − 45fk−1 + 45fk+1 − 9fk+2 + fk+3 ) + O(h6 ) (3.43)
60h
The approximate expressions for the second-order derivative can be obtained
with different combinations of fi . The three-point formula for the second-order
derivative is given by the combination
fk+1 − 2fk + fk−1 = h2 fk00 + O(h4 )
(3.44)
with the Taylor expansions of fk±1 around xk . The above equation gives the
second-order derivative as
fk00 =
fk+1 − 2fk + fk−1
+ O(h2 )
h2
(3.45)
With the similar method one obtains in the seven-point approximation to the
second order derivative
1
(2fk−3 −27fk−2 +270fk−1 −490fk +270fk+1 −27fk+2 +2fk+3 )+O(h6 )
180h2
(3.46)
Let us turn to numerical integration. In general, we want to obtain the
numerical value of an integral defined in the region [a, b] as
Z b
S=
f (x)dx
(3.47)
fk00 =
a
We can divide the region [a, b] into n slices with an evenly spaced interval h. If
we take the lattice points as xk with k = 0, 1, . . . , n, we can write the integral
as a summation of integrals over all the slices
Z
b
f (x)dx =
a
n−1
X Z xk+1
k=0
f (x)dx
(3.48)
xk
Of course, if we can develop a numerical scheme that evaluates the integral over
several slices accurately, we will have solved the problem. Let us first consider
each slice separately. The simplest quadrature is obtained if we approximate
f (x) in the region [xk , xk+1 ] linearly, that is, f (x) ' fk + (x − xk )(fk+1 − fk )/h.
After integration every slice with this linear function, we have
S=
n−1
hX
(fk + fk+1 + O(h2 )
2
(3.49)
k=0
where O(h2 ) comes from the error in the linear interpolation of the function.
The above quadrature is commonly referred to as the trapezoid rule, which has
an overall accuracy up to O(h2 ).
40
We can obtain a quadrature with a higher accuracy by working on two slices
together. If we apply the Lagrange interpolation to the function f (x) in the
region of [xk−1 , xk+1 ] we have
f (x) =
(x − xk )(x − xk+1 )
(x − xk−1 )(x − xk+1 )
fk−1 +
fk
(xk−1 − xk )(xk−1 − xk+1 )
(xk − xk−1 )(xk − xk+1 )
(x − xk−1 )(x − xk )
fk+1 + O(h3 ) (3.50)
+
(xk+1 − xk−1 )(xk+1 − xk )
If we carry out the integral for every pair of slices together with the integrand
given from the above equation, we have [14]
n/2−1
h X
S=
(f2l + 4f2l+1 + f2l+2 ) + O(h4 )
3
(3.51)
l=0
which is known as the Simpson rule. The third-order term vanishes because of
cancellation. In order to pair up all the slices, we have to have an even number
of slices.
41
42
Part II
QUANTUM SCATTERING UNDER 2D CONFINING
POTENTIAL
44
4. ANALYTICAL DESCRIPTION OF ATOMIC SCATTERING
AND CONFINEMENT-INDUCED RESONANCES IN
WAVEGUIDES
Recently the field of ultracold few-body confined systems has progressed remarkably. By employing optical dipole traps [15] and atom chips [16, 17, 18]
it is possible to fabricate mesoscopic structures in which the atoms are frozen
to occupy a single or a few lowest quantum states of a confining potential such
that in one or more dimensions the characteristic length possesses the order of
the atomic de Broglie wavelength. These configurations can be well described
by effective low-dimensional systems. The quantum dynamics of such systems
is strongly influenced by the geometry of the confinement. This situation is encountered ,e.g., in certain quantum wires for electronic transport, or in atomic
waveguides or optical lattices for ultracold atoms. Atom waveguides are fundamental component of atom optics, and are expected to play an important
role for example in atom interferometry, quantum computing applications, ultrasensitive detectors of accelerators , and gravitational anomalies as well as to
provide an opportunity to study novel 1D many-body states. To ensure proper
coherence (long-range order) as atomic beams propagate through the waveguides, effort should be made to avoid decoherence -inducing mechanism such as
collisional losses and collisional phase shifts. This clearly requires a detailed understanding of the effects of quasi-one-dimensional confinement on atom-atom
collisions (see Refs. [19-45] and references therein). In particular, we will see
that in the few-mode regime, necessary for coherent propagation, the free-space
estimate of the collisional effects are no longer valid and a waveguide-specific
theory is needed. This stimulated the development of quantum scattering theory in low dimensions. The spin-statistics theorem, according to which identical
particles with integer spin are bosons whereas those with half-integer spin are
fermions, breaks down in low-dimensional systems. The “Fermi-Bose duality”
is a very general property of identical particles in 1D, maps strongly interacting bosons to weakly-interacting fermions and vice versa. In recent years this
esoteric subject has become highly relevant through experiments on ultracold
atomic vapors in atom waveguides.
Of particular interest is the single-mode regime, where only the ground state
with respect to the confining direction can be actually populated. This occurs
for a sufficiently dilute gas in a regime of low temperatures and densities under
a sufficiently strong transverse confinement, when both the chemical potential
and the thermal energy kB T are less than the transverse oscillator level spac-
ing h̄ω. In this case the two-body scattering properties are strongly modified
and short-range correlation are greatly enhanced. The dynamics of such system can then be mapped to an effective one-dimensional (1D) Hamiltonian with
zero-range interactions. With this assumption, the dynamics of ultracold atoms
in low-dimensional structures, e.g., tight waveguides has been studied using
the simplification that the atoms occupy only the ground state of the transverse confining potential. Nevertheless, the virtual transverse excitations in the
course of the collision process can play a crucial role in the scattering leading
to the so-called confinement-induced resonances (CIRs), predicted by Olshanii
[27, 28]. It was shown [27, 28] that CIRs appear if the binding energy of the
pseudopotential, approximating the two-atom molecular state in the presence
of the confinement, coincides with the energy spacing between the levels of the
transverse harmonic potential. In the vicinity of the CIR the coupling constant
g1D can be tuned from −∞ to +∞ by varying the strength of the confining
potential over a small range. This can result in a total atom-atom reflection,
thereby creating a gas of impenetrable bosons. CIRs have also been studied
for the three-body [29, 30] and the four-body [31] scattering under confining
potential, as well as for a pure p-wave scattering of fermions [32]. Experimental
evidence for the CIRs for bosons [33, 34] and fermions [35] has recently been
reported.
A general analytical treatment of low-energy scattering under action of a
general cylindrical confinement involving all partial waves and their coupling,
was first provided in Ref. [36] for a spherically symmetric short-range potential.
The effect of the CM motion on the s-wave collision of two distinguishable
atoms (i.e. a two-species mixture) in a harmonic confinement as well as for
two identical atoms in a non-parabolic confining potential has been investigated
in Ref. [37] in the zero-energy limit neglecting the s and p wave mixing. A
detailed study including the effect of the CM nonseparability and taking into
account the s and p wave mixing for harmonic confinement was performed in the
single-mode regime in Ref. [40]. Recently a so-called dual-CIR was discovered
[38, 39, 40], which is characterized by a complete transmission (suppression
of quantum scattering) in the waveguide due to destructive interference of s
and p waves although the corresponding collisions in free space involve strong
interactions.
The problem of atomic pair collisions under the action of a harmonic trap in
the multimode regime when the energy of the atoms exceeds the level spacing of
the transverse trapping potential is much more intricate than the single-mode
regime due to several open transverse channels. It demands the development of
a multichannel scattering theory accounting for the possible transitions between
the levels of the confining potential. Using as a starting point the formalism
for scattering in restricted geometries suggested in Refs. [41, 42], the multichannel scattering problem for bosons in a harmonic confining potential has
been analyzed analytically by Olshanii et al. [43] in the s-wave pseudopotential
approximation and the zero-energy limit. Without detailization of the interatomic interaction but using only two input parameters - the s-wave two-body
scattering length in free space and the trap frequency - they have derived an ap46
proximate formula for the scattering amplitude describing two boson collisions
confined by transverse harmonic trap in the multi-mode regime.
In [45], a general grid method has been developed for multichannel scattering
of two atoms confined by a transverse harmonic trap with a single frequency for
both atoms. The method applies to arbitrary atomic interactions, permitting
a rich spectral structure where several different partial waves are participating
in the scattering process, or even the case of an anisotropy of the interaction.
Using this method the transverse excitations and deexcitations in the course of
the collisional process (distinguishable or identical atoms) has been analyzed
including all important partial waves and their couplings due to the broken
spherical symmetry. Special attention has been paid to the analysis of the CIRs
in the multimode regimes for non-zero collision energies.
In this chapter we study scattering theory of identical as well as distinguishable atoms confined by a transverse harmonic trap and discuss the scattering
parameters and the transition probabilities characterizing the two-body collisions in the trap. Special attention is paid to the analysis of the CIRs in zero
collision- energies.
4.1 Hamiltonian and Two-body Scattering Problem in a
Waveguide
Let us consider collisions of two atoms under the action of the transverse harmonic confinement. We address both cases, distinguishable and indistinguishable atoms under the action of the same confining potential, i.e. the trap is
characterized by a single frequency ω for every atom. The corresponding Hamiltonian is given by
H=−
h̄2 2
h̄2 2 1
1
∇1 −
∇2 + m1 ω 2 ρ21 + m2 ω 2 ρ22 + V (x1 − x2 )
2m1
2m2
2
2
(4.1)
where mi is the mass of the ith atom, V (x1 − x2 ) is the two-body potential
describing the interaction between two colliding atoms in free space, and xi =
(ρi , zi ) = (ri , θi , ϕi ) is the coordinate of the ith atom.
The Hamiltonian is separable with respect to the relative/CM coordinate
and momenta x, X and p, P variables. This can be done by the canonical
transformation





X
m1 m2
0
0
x1
 x 


1 
0
0 


 M −M
  x2 
(4.2)
 P = M  0


0
M
M
p1 
p
0
0
m2 −m1
p2
where M = m1 + m2 is the total mass. The transformed Hamiltonian takes the
form
H = HCM + Hrel
(4.3)
where
HCM = −
h̄2 2
1
2
∇X + M ω 2 R⊥
2M
2
47
(4.4)
g1D /[ a1⊥ ]
10
8
6
4
2
0
−2
−4
−6
−8
−10
−3
CIR
−2
−1
0
as /a⊥
1
2
3
Fig. 4.1: The effective quasi-1D coupling constant g1D as a function of the s-wave scattering length as for single-channel scattering of two bosons under a harmonic
confinement.
and
Hrel = −
h̄2 2 1 2 2
∇ + µω ρ + V (x)
2µ x 2
(4.5)
Here µ = m1 m2 /(m1 + m2 ) is the reduced mass, and ρ and R⊥ are the relative
and center of mass radial coordinates respectively. The center of mass Hamiltonian is that of a simple harmonic oscillator whose solution is known, hence
the problem, reduces to scattering of a single effective particle with the reduced
mass µ and collision energy E > h̄ω, off a scatterer V (x) at the origin, under
transverse harmonic confinement with frequency ω, U (ρ) = 12 µω 2 ρ2
·
¸
h̄2 2
− ∇x + U (ρ) + V (r) ψ(x) = Eψ(x)
2µ
(4.6)
Here the energy of the relative two-body motion E = E⊥ + Ek is a sum of
the transverse E⊥ and longitudinal collision Ek energies. Due to our definition
of the confining potential, the transverse excitation energies E⊥ can take the
possible values E⊥ = E − Ek = h̄ω(2n + |m| + 1) > 0 of the discrete spectrum
of the 2D oscillator 21 µω 2 ρ2 .
Determining the eigenstates of this Hamiltonian, in particular within the
s-wave scattering approximation for V (x), is the central goal of this section.
4.2 s-wave Scattering Regime: the Reference T -Matrix Approach
The T -matrix formulation allows for a self-consistent description of the lowenergy part of the spectrum that uses the free-space low-energy scattering properties of the interaction potential as the only input. In addition the low-energy
(s-wave) limit is isolated to a single well-defined approximation without requiring the ad-hoc introduction of regularization via a pseudo-potential. This
48
self-consistent low-energy treatment has been outlined first in [43]. In their
work, they also solved for the T -matrix using the standard Huang-Fermi pseudopotential, showing that the pseudo-potential reproduces the exact result in this
situation.
In this section we follow [43] and [44]. Let the unperturbed Hamiltonian H
be a Hamiltonian for a single nonrelativistic particle in presence of a trapping
potential U
· 2 2
¸
−h̄ ∇x
hx|H|ψi =
+ U (x) hx|ψi
(4.7)
2µ
Assume also that the particle is ‘perturbed’ by a scatterer given by
hx|V |ψi = V (x)hx|ψi
(4.8)
localized around x = 0. In what follows we will derive a low-energy approximation for the T -matrix of the scatterer V in presence of H. By making use of
the Lupu-Sax formula (A.119), one can drive the correct form of the T -matrix
in the low-energy s-wave regime. We begin our derivation by first specifying a
“reference” background Hamiltonian H 0 as
· 2
¸
−h̄ 4x
hx|H 0 |ψi =
+ E hx|ψi
(4.9)
2µ
This Hamiltonian is that of a free particle, but with an explicit energy dependence included so that the eigenstates have zero wavelength at all energies. We
note that this reference Hamiltonian agrees with the free-space Hamiltonian in
the zero-energy limit. While this Hamiltonian may seem strange, it is a valid reference Hamiltonian which turns out to be useful because the resulting T -matrix
is energy independent for any scattering potential. The Green’s function for
this Hamiltonian is given by
hx|GH 0 (E)|x0 i = −
µ
1
2 |x − x0 |
2πh̄
(4.10)
as can be derived by direct substitution into [E − H 0 ]GH 0 (E) = I. In turn
the T -matrix of the interaction potential V in presence of H 0 is independent of
energy and can therefore be expressed as
hx|TH 0 ,V (E)|x0 i = gD(x, x0 )
(4.11)
where the kernel D is defined as normalized to unity,
Z
dxdx0 D(x, x0 ) = 1
(4.12)
The normalized coefficient g is then related, through the zero-energy scattering
amplitude, to the three-dimensional scattering length as according to
g=
2πh̄2 as
µ
49
(4.13)
Imagine that the kernel D(x, x0 ) is well localized within some radius R. In perturbative expansion at low energies this kernel only participates in convolutions
with slow (as compared to R) functions, in which case it can be approximated
by a δ-function,
D(x, x0 ) ≈ δ(x)δ(x0 )
(4.14)
This straightforward approximation is the key to the s-wave scattering approximation. This effectively replaces the exact reference T -matrix by its longwavelength limit, so that the reference T -matrix assumes the form
hx|TH 0 ,V (E)|x0 i ≈ gδ(x)δ(x0 ) [k, k 0 ¿ 1/R]
(4.15)
which is equivalent to
TH 0 ,V (E) = g|0ih0|
(4.16)
where |0i is the position eigenstate corresponding to the location of the scatterer.
In expression (4.15) k and k 0 refer to the wavevectors of any matrices which
multiply the T -matrix from the left and right, respectively.
If we now substitute the above expression for the reference T -matrix into the
Lupu-Sax formula (A.119) for the T -matrix under the background Hamiltonian
H we arrive at
TH,V (E) =
∞
X
n
[g|0ih0|GH 0 (E)] g|0ih0|
n=0
−1
= [1 − gh0|GH (E)|0i + gh0|GH 0 (E)|0i]
g|0ih0|
Making use of (4.10), we introduce the function χ(²), defined as
¸
·
µ
χ(E) = lim hx|GH (E)|0i +
x→0
2πh̄2 |x|
(4.17)
(4.18)
from which we obtain the following simple expression for the T -matrix of the
scatterer V in presence of the trap
hx|TH,V (E)|ψi ≈
gδ(x)
hx|ψi [E ¿ h̄2 /µR2 ]
1 − gχ(E)
(4.19)
From comparing the equations the free-space and bound Green’s functions
obey, one can show that the singularity in bound Green’s function is the same as
that in the free-space Green’s function. Hence, χ(E) is the value of the regular
part of the bound Green’s function at the origin.
4.2.1 Eigenstates of the waveguide Hamiltonian
The Hamiltonian for the relative motion of two atoms in a harmonic waveguide
contains two parts, the longitudinal free Hamiltonian Hz and the transverse
confinement Hamiltonian H⊥ ,
H = Hz + H⊥
50
(4.20)
Transmission Coef. T
0.05
0.04
0.03
0.02
CIR
0.01
0
0.5
1
as /a⊥
1.5
Fig. 4.2: The Transmission Coefficient T as a function of the s-wave scattering length
as for single-channel scattering of two bosons under a harmonic confinement.
where
Hz = −
h̄2 ∂ 2
2µ ∂z 2
(4.21)
and
h̄2 ∂ 2
1 ∂
1 ∂2
1
( 2+
+ 2
) + µω 2 ρ2
(4.22)
2µ ∂ρ
ρ ∂ρ ρ ∂ϕ2
2
The eigenstates of the transverse Hamiltonian, denoted by |nmi, are well known
and satisfy both
H⊥ |nmi = h̄ω(2n + |m| + 1)|nmi
(4.23)
H⊥ = −
as well as
Lz |nmi = h̄m|nmi
(4.24)
where Lz is the operator for angular momentum along the z-axis. We note that
in this representation the radial and the azimuthal quantum numbers n and m
independently take the values
n = 0, 1, 2, ...∞
(4.25)
m = 0, ±1, ±2, ... ± ∞
(4.26)
and
The eigenfunctions are given in spatial-representation by
· 2
¸−1/2
2
2
πa⊥ (n + |m|)!
ρ
2
2
imϕ
hρϕ|nmi =
( )|m| e−ρ /(2a⊥ ) L|m|
(4.27)
n (ρ /a⊥ )e
n!
a⊥
p
Here Lm
h̄/µω, is the
n (x) are the generalized Laguerre polynomials and a⊥ =
transverse oscillator length. Lastly we note that the value of |nmi at the origin
is given by
δm,0
(4.28)
h0|nmi = √
πa⊥
which is independent of n.
51
4.2.2 The Green’s function for the relative motion of two particles in a
harmonic waveguide
For the case of transverse harmonic confinement function we can evaluate χ(E)
by expanding the matrix element hx|G(E)|0i onto the set of states
|nmzi = |nmi ⊗ |zi
(4.29)
where |nmi is the eigenstate of the transverse potential given by (4.27) and |zi
is the eigenstate of the operator z, satisfying hz|z 0 i = δ(z − z 0 ). For simplicity
we take GH (E) ↔ G(E),
X
hρϕ|nmihnmz|G(E)|n0 m0 0ihn0 m0 0|0i
(4.30)
hx|G(E)|0i =
n,m,n0 ,m0
In the other hand the bound Green’s function is the solution to the equation
[E − H⊥ − Hz ]G(E) = I
(4.31)
where I is the identity matrix. Expanding this equation onto the set of states
(4.29) gives
·
¸
h̄2 ∂ 2
E − h̄ω(2n + |m| + 1) +
hnmz|G(E)|n0 m0 z 0 i = hnmz|n0 m0 z 0 i
2µ ∂z 2
= δn,n0 δm,m0 δ(z − z 0 ) (4.32)
We proceed by making the ansatz for the Green’s function
0
hnmz|G(E)|n0 m0 z 0 i = δn,n0 δm,m0 αnm eiγnm |z−z |
(4.33)
Differentiating (4.33) twice with respect to z and substituting the result into
(4.32) gives
·
¸
2
h̄2 γnm
E − h̄ω(2n + |m| + 1) −
hnmz|G(E)|n0 m0 z 0 i
2µ
+i
h̄2 γnm αnm
hnmz|n0 m0 z 0 i = hnmz|n0 m0 z 0 i
µ
(4.34)
This equation is satisfied provided that
2
γnm
=
2µ
[E − h̄ω(2n + |m| + 1)]
h̄2
and
αnm = −i
µ
h̄ γnm
2
(4.35)
(4.36)
Equation (4.35) is quadratic, and therefore has in general two solutions. We
should choose the solutions which propagate outwards from z = z 0 .
52
By introducing the dimensionless energy
ε=
E
1
−
2h̄ω 2
(4.37)
we find that retarded Green’s function can be expressed as
p
|m|
i a2
ε−n− 2 |z−z 0 |
⊥
µa⊥ e
0 0 0
q
hnmz|G(ε)|n m z i = −i 2
δn,n0 δm,m0
2h̄
ε − n − |m|
(4.38)
2
With this expression for the bound Green’s function and using the value of
|n, mi at the origin (4.28) which is independent of n, we obtain for the matrix
element hx|G(ε)|0i
∞
X
√
ghx|G(ε)|0i = − πas
hρϕ|n0i
e
− a2
√
↓
⊥
√
↓
n=0
n−ε|z|
n−ε
(4.39)
√
where the modified complex square root ↓ is defined for an arbitrary complex
number c as
q
p
↓
|c|eiφ = |c|eiφ/2 − 2π < φ ≤ 0
note that the usual square root is defined as
q
p
↓
|c|eiφ = |c|eiφ/2 0 ≤ φ < 2π
Now by inserting (4.39) and (4.13) into (4.18) we arrive at the expression
√
#
"
∞
− 2 ↓ n−ε|z|
X
√
as
e a⊥
gχ(ε) = lim − πas
+
(4.40)
hρϕ|n0i √
↓
x→0
|x|
n−ε
n=0
While both terms in (4.40) diverges in the limit x → 0, their difference remains
finite and leads to an expression for χ(ε) in terms of a generalized Zeta function.
To proof this we assume, that the directional single-variable limits x =
sn, s → 0 exist for all directions n and they are all equal to each other. This
assumption allows us to deal with the limit along the z-axis only
√
#
"
∞
− 2 ↓ n−ε|z|
as
a s X e a⊥
√
−
(4.41)
gχ(ε) = − lim
↓
|z|
|z|→0 a⊥
n−ε
n=0
where we have used the identity (4.28). We proceed by first replacing the as /|z|
term in (4.41) with an integral expression via the identity
as
as
=
|z|
a⊥
Z
− a2
∞
dn
ε
53
e
⊥
√
√
n−ε|z|
n−ε
(4.42)
This allows us to write
√
√
"N
#
2 ↓
− 2
n−ε|z|
X e− a⊥ n−ε|z| Z N
as
e a⊥
√
gχ(ε) = −
lim lim
−
dn √
↓
a⊥ |z|→0 N →∞ n=0
n−ε
n−ε
ε
(4.43)
It is now tempting to interchange the limit signs and thus get rid of the coordinate dependence. In order to be able to do that one have to prove the uniformity
with respect to |z| of the N → ∞ convergence of the expression in the square
brackets Ξ(N, |z|), i.e., to prove that for every ² there exists N ∗ , the same for
all |z|, such that Ξ(N, |z|) − limN →∞ Ξ(N, |z|) < ² for all N > N ∗ . Such a proof
does exist, although we do not exhibit it here.
We arrive at the following expression for χ(ε)
"N
#
X
√
as
1
√
−2 N −ε
gχ(ε) = −
lim
(4.44)
a⊥ N →∞ n=0 ↓ n − ε
One can now make use of the following theorem involving the Hurwitz Zeta
function, an analytic generalized Zeta function described in the mathematical
literature [46]
"N
#
X
1
1
1
ζ(s, α) = lim
−
(4.45)
N →∞
(n + α)s
1 − s (N + α)s−1
n=0
Re(s) > 0,
−2π < arg(n + α) ≤ 0
In particular,
"
ζ(1/2, α) = lim
N →∞
N
X
n=0
√
1
√
−2 N +α
↓
n+α
#
(4.46)
While this expression, valid for any N , may be taken as a definition of the Hurwitz Zeta function, it does not constitute an efficient method for computation.
With this definition and taking s = 1/2 we arrive at
gχ(ε) = −
as
ζ(1/2, −ε)
a⊥
(4.47)
By substituting this expression into (4.19) we arrive at the final expression
for the long-wavelength T -matrix in the waveguide
T (ε) =
g|0ih0|
1+
as
a⊥ ζ(1/2, −ε)
(4.48)
4.2.3 Multichannel scattering amplitudes and transition rates
From equations (A.118) and (4.48) we find that the scattered wavefunction takes
the form
hnmz|G(ε)|0i
h0|ψ0 (ε)i
(4.49)
hnmz|ψout (ε)i = g
[1 + aa⊥s ζ(1/2, −ε)]
54
Now the matrix element hnmz|G(ε)|0i can be determined by making use of
(4.28) and (4.38), yielding
X Z
hnmz|G(ε)|0i =
dz 0 hnmz|G(ε)|n0 m0 z 0 ihn0 m0 z 0 |0i
n0 ,m0
i
2
√
ε−n|z|
µ
e a⊥
= −i √ 2 δm,0 √
ε−n
2 πh̄
(4.50)
Inserting this expression into (4.49) then gives
√
√
i 2
ε−n|z|
πas
e a⊥
hnmz|ψout (ε)i = −i
δm,0 √
h0|ψ0 (ε)i
as
[1 + a⊥ ζ(1/2, −ε)]
ε−n
(4.51)
Let us now assume that the incident wave has the longitudinal wave vector k
and the transverse quantum numbers n and m, according to
hx|ψ0 (ε)i = hρϕ|nmieikz
where the relation
µ
ε=
a⊥ k
2
¶2
+n+
(4.52)
|m|
2
(4.53)
gives the dependence of the scaled energy ε on the incident wave vector k. This
incident wave is nonzero at the origin only for m = 0, hence only incident waves
with zero angular momentum will scatter. The value of the m = 0 incident
wave at the origin conveniently takes the n-independent value,
δm,0
h0|ψ0 (ε)i = √
πa⊥
(4.54)
Assuming henceforth m = 0, we can now express (4.51) as
i
hn0 m0 z 0 |ψout (ε)i = −i
[ aa⊥s
2
√
ε−n|z|
δm0 ,0
e a⊥
√
+ ζ(1/2, −ε)]
ε−n
(4.55)
From this expression it follows that the full wavefunction of the relative motion
in s-wave regime takes the form
hx|ψ(ε)i =
∞
X
h
i
e
ikn0 |z|
hρϕ|n0 0i δn0 ,n eikz + fnn
0e
(4.56)
n0 =0
Here we have introduced the even-wave transversely inelastic scattering amplie
tudes f e (kn0 ← kn )n0 ←n = fnn
0 , given by
e
fnn
0 = −
2i
1
h
i
a⊥ kn0 a⊥ + ζ(1/2, − ¡ a⊥ kn ¢2 − n)
as
2
55
(4.57)
and the outgoing wave vector for the mode |n0 0i
sµ
¶2
2
a⊥ kn
kn0 =
+ n − n0
a⊥
2
(4.58)
from which the desired scattering probabilities can be computed. One can show
easily that the number ne of open excited transverse channels -the maximum
value of n0 for which kn0 in (4.58) is real- coincides with the integer part of the
dimensionless energy ε. For n > ne , Kn0 is purely imaginary number which
results in a decaying wave, thus this channels are asymptotically closed. The
interatomic interaction V (r) mixes different transversal channels and leads to
transitions n → n0 between the open channels n, n0 ≤ ne , i.e., to transverse
excitation and deexcitation processes during the collisions.
One can now easily compute the elastic and inelastic transition probabilities
for collisions under transverse harmonic confinement. We begin by considering
the asymptotic forms of the total wavefunction given by (4.56), which are given
by
ikn0 z
e
(4.59)
lim hn0 0z|ψ(ε)i = δn0 ,n eikz + Θ[ε − n0 ]fnn
0e
z→∞
ikn0 z
e
lim hn0 0z|ψ(ε)i = Θ[ε − n0 ]fnn
0e
z→−∞
(4.60)
where Θ(x) is the Heavyside step function. Because of energy conservation, an
inelastic collision results in a change in the longitudinal momentum kn → kn0 , so
that the introduction of inelastic transmission and reflection coefficients must
be based on conservation of total incident and outgoing probability current.
Using the asymptotic wavefunctions (4.59) and (4.60) and the total current
conservation one obtains
ne
X
(Tnn0 + Rnn0 − δnn0 ) = 0
(4.61)
n0 =0
for the inelastic transmission (reflection) coefficients Tnn0 (Rnn0 ). kn (kn0 ) is the
initial (final) relative wave vector and n (n0 ) the transverse excitation number
according to (4.58). We have [43]
Tnn0 = Θ[ε − n0 ]
kn0
2
e
|δn,n0 + fnn
0|
kn
(4.62)
kn0 e 2
|f 0 |
(4.63)
kn nn
The transition probability Wnn0 , characterizing the transverse excitation and
deexcitation, into a particular channel n0 from the initial state n is given by the
sum of the corresponding transmission and reflection coefficients
Rnn0 = Θ[ε − n0 ]
Wnn0 = Tnn0 + Rnn0
(4.64)
Due to the time reversal symmetry of the Hamiltonian we have Tnn0 = Tn0 n ,
0
0
Rnn0 P
= Rn0 n and Wnn
P = Wn n . The total transmission (reflection) coefficient
0
0
T = n0 Tnn (R = n0 Rnn ) is given by the sum of the transmission (reflection) coefficients of all the open channels. Equation (4.61) leads to T + R = 1.
56
4.2.4
Single-channel scattering and effective one-dimensional interaction
potential
Let us now consider the special case of a single-channel scattering
0 ≤ ε < 1,
ne = 0
(4.65)
In this case we have
·
¸
as
as
i
gχ(ε) = − ζ(1/2, −ε) = −
ζ(1/2, 1 − ε) + √
a⊥
a⊥
ε
Using an alternative representation for ζ(1/2, 1 − ε)
"N
#
X
√
1
√
ζ(1/2, 1 − ε) = lim
−2 N
↓
N →∞
n−ε
n=1
(4.66)
(4.67)
and making use of the expansion in powers of ε,
∞
√
X (2j − 1)!!
1
1
=√ +
εj
n j=1 2j j!nj+1/2
n−ε
|ε| < 1,
(4.68)
n>0
allows to write
ζ(1/2, 1 − ε) = ζ(1/2) + L(ε)
where
L(ε) =
∞
X
(2j − 1)!!ζ(j + 1/2)
2j j!
j=1
(4.69)
εj
(4.70)
which clearly separates the zero energy limit from the finite energy corrections.
According to (4.57) this leads to
1
h
i
³ 2 2 ´i
f e (k) = − h
a k
1 − i a2⊥ aa⊥s + ζ(1/2) k − i a⊥2 k L ⊥4
(4.71)
where f e (k) = f e (k0 ← k0 )0←0 is the even single-channel scattering amplitude
and ζ(1/2) = −1.4603 . . ..
It is now tempting to introduce an effective one-dimensional interaction potential in such a way that its scattering amplitude, introduced through the
one-dimensional scattering solution as
ψ1D (z) = eikz + [f e (k) + sgn(z)f o (k)] eik|z|
(4.72)
matches (4.71), i.e., solve the corresponding one-dimensional inverse scattering
problem. It turns out that this problem is ill-posed due to the presence of open
channels unaccessible within the one-dimensional model. Nevertheless one may
pose the following problem: find a one-dimensional potential, whose scattering
57
amplitude reproduces the exact one (4.71) with the relative error O(k 3 ). Such
an object does exist, and it is represented by a zero-range scatterer
v1D (z) = g1D δ(z)
(4.73)
A straightforward calculation shows that the scattering amplitude fδe (k) of the
one-dimensional potential (4.73) has a form
fδe (k) = −
1
1 + ika1D
(4.74)
where the one-dimensional scattering length a1D is related to the potential
strength by
h̄2
g1D = −
(4.75)
µa1D
Comparison of the scattering amplitudes (4.71) and (4.74) leads to a conclusion that the waveguide scattering amplitude (4.71) at low velocities can be
reproduced by a delta-potential (4.73) of a scattering length
·
¸
a⊥ a⊥
a1D = −
+ ζ(1/2)
(4.76)
2 as
It follows from (4.75) and (4.76) that
g1D =
2h̄2 as
1
µa2⊥ (1 − Cas /a⊥ )
(4.77)
which has a resonant form showing a Confined Induced Resonance (CIR) at
as = a⊥ /C, where C = −ζ(1/2) = 1.4603 . . . (Fig. 4.1). Hence the whole range
of 1D coupling constants from −∞ to +∞ is experimentally achievable by tuning
as over a narrow range in the neighborhood of the 1D resonance. This can result
in a total atom-atom reflection, thereby creating a gas of impenetrable bosons
(see Fig. 4.2 for the transmission coefficient as a function of the scattering
length). The effect was recently [28] interpreted in terms of Feshbach resonance
between ground and excited vibrational manifolds.
It was shown in [28] that at low longitudinal energies ka⊥ ¿ 1 the 1D
scattering amplitude generated by the interaction g1D δ(z) reproduces the exact
3D scattering amplitude in the waveguide to within a relative error O(k 3 ).
The relation between g1D and the 1D scattering amplitude may be found by
substituting (4.72) into 1D Schrödinger equation
·
which gives
¸
h̄2 ∂ 2
−
+ g1D δ(z) + h̄ω ψ1D (z) = Eψ1D (z)
2µ ∂z 2
h̄2 k Re[f e (k)]
k→0 µ Im[f e (k)]
g1D = lim
58
(4.78)
(4.79)
To establish the physical origin of the CIR, we follow [28]. We show that the
CIR is in fact a zero-energy Feshbach resonance, occurring when the energy of
a bound state of the asymptotically closed channels (i.e., the excited transverse
modes) coincides with the continuum threshold of the open channel (lowest
transverse mode).
This explanation is best verified by artificially serving the coupling between
the ground transverse mode and the manifold of excited modes. If a bound state
of the decoupled excited manifold exists, then a zero-energy Feshbach resonance
will occur when this bound-state energy coincides with the continuum threshold
of the ground transverse mode. To determine the relevant bound-state energies,
we assume the standard Huang-Fermi peudopotential
Vpseudo =
2πh̄2 as 3
∂
δ (x) (r.)
µ
∂r
(4.80)
which supports a single bound state in free space at E = h̄2 /µa2s for the case
as > 0. To determine the location of the CIR, we need to ask what happens
to the energy of this bound state if a single transverse mode is projected out of
the Hilbert space. We therefore proceed by formally splitting the Hamiltonian
onto “ground,” “excited,” and “ground-excited coupling” parts according to
H = Hg + He + Hg−e
= Pg HPg + Pe HPe + (Pe HPg + H.c.)
(4.81)
P∞
where Pg = |0ih0|, Pe =
n=1 |nihn|, are the corresponding projection operators, |ni being the eigenstate of the transverse two-dimensional harmonic
oscillator with radial quantum number n and zero axial angular momentum.
The ground Hamiltonian has a 1D coordinate representation of the form
Hg = −
h̄2 ∂ 2
+ g1D δ(z) + h̄ω
2µ ∂z 2
(4.82)
corresponding to the motion of a one-dimensional particle in the presence of a
δ barrier with the coupling constant g1D is defined via
Z
2πh̄2 as 3
g1D δ(z) = 2πρdρ|φ0 (ρ)|2
δ (r)
(4.83)
µ
which gives
g1D =
2h̄2 as
µa2⊥
(4.84)
The spectrum of Hg is continuous for energies above the threshold energy Ec,g =
h̄ω. Likewise, the spectrum of the excited Hamiltonian is clearly continuous for
energies Ec,e = 3h̄ω but, as we will see below, He supports one bound state of
energy Eb,e < Ec,e for all values of the 3D scattering length, as .
According to the Feshbach scheme, one would predict a resonance in the
renormalized g1D for a set of parameters, such that the energy of the bound
59
state of He coincides with the continuum threshold of Hg . Thus, the CIR
condition can be expressed as Eb,e = Ec,g . As we will see below, this scheme
indeed predicts a position of the CIR exactly.
The energy Eb,e of the bound state of He can be found using the following two
step procedure. First, we identify the bound-state energy of the full Hamiltonian
H as a pole of the scattering amplitude on the physical Riemann sheet. Second,
we make use of the peculiar property of the two-dimensional harmonic oscillator
that the excited Hamiltonian He and the full Hamiltonian H can be transformed
to each other via a simple unitary transformation. This leads to a simple relation
between their bound-state energies.
The even-wave one-dimensional scattering amplitude f e at an energy E
(Ec,g ≤ E < Ec,e ), is given by (4.71) which can be written as
f e (k) = −
h
ka⊥
2i
a⊥
as
i
+ ζ(1/2, −(ka⊥ /2)2 )
(4.85)
where the wave vector k is given by
E = Ec,g +
h̄2 k 2
2µ
(4.86)
The bound state energies of the full Hamiltonian H will be given by the poles,
k, on the positive imaginary axis of the analytical continuation of f e (k) :
2
Eb = −
h̄2 Im(k )
2µ
(4.87)
One can see that, in order to avoid crossing the branch cuts of the zeta function,
the continuation should be performed inside the 0 ≤ Arg(k) ≤ π/2 quadrant
of the complex plane. In the end, we find a single pole corresponding to the
following implicit equation for the bound-state energy
µ
¶
1
a⊥
ζ
, −ε = −
(4.88)
2
as
Eb
which we have expressed through the dimensionless energy εb = 2h̄ω
− 21 . Now
the full Hamiltonian H and the excited Hamiltonian He are connected via a
simple transformation
He = A† HA + 2h̄ωI
(4.89)
where
A† =
∞
X
|n + 1ihn|
n=0
This can be proved as follows:
̰
! Ã ∞
!†
X
X
†
0
0
A HA =
|n + 1ihn| H
|n + 1ihn |
=
n=0
∞
X
n0 =0
0
|n + 1ihn|Hz0 + H⊥
+ V |n0 ihn0 + 1|
n=0,n0 =0
60
(4.90)
2
2
2
h̄ ∂
h̄
1
0
2 2
Here Hz0 = − 2µ∂z
2 , H⊥ = − 2µ ∇⊥ + 2 µω ρ and V is given through (4.80). On
the other hand we have
0
hn|Hz0 + H⊥
+ V |n0 i = δnn0 Hz0 + δnn0 h̄ω(2n0 + 1) + hn|V |n0 i
0
= hn + 1|Hz0 |n0 + 1i + hn + 1|(H⊥
− 2h̄ω)|n0 + 1i
+hn + 1|V |n0 + 1i
For the last term we have used the fact that the m = 0 eigenfunctions of the
two-dimensional harmonic oscillator all have the same value at the origin. The
3D δ interaction thus has the same matrix elements between all the harmonic
oscillator states; hence, the interaction matrix is unaffected by the shift operator.
Now one can write
∞
X
A† HA =
0
|n + 1ihn + 1|Hz0 + H⊥
− 2h̄ω + V |n0 + 1ihn0 + 1|
n=0,n0 =0
∞
X
|nihn|(Hz0 + H⊥ − 2h̄ωI + V )
=
∞
X
|n0 ihn0 |
n0 =1
n=1
= Pe HPe − 2h̄ωI = He − 2h̄ωI
Here I is the unit operator in the Hilbert space of the excited Hamiltonian He .
From the above, we conclude that the rescaled bound-state energies εb,e of
the excited Hamiltonian and εb of the full Hamiltonian are related to each other
via
εb,e = εb + 1
(4.91)
and thus satisfies the equation
µ
¶
1
a⊥
ζ
, −εb,e + 1 = −
2
as
The CIR condition can now be explicitly formulated as
µ
¶
1
a⊥
ζ
, −εc,g + 1 = −
2
as
(4.92)
(4.93)
Using εc,g = 1, we finally arrive at the exact CIR condition a⊥ /a = −ζ(1/2, 1) =
−ζ(1/2, 0) = C. A similar effect is associated with resonance behavior in harmonically confined 2D scattering for a < 0 [47]. This resonance would most
likely be observed via changes in the macroscopic properties of the ground state
of a many-atom system, e.g., the density distribution, as described in detail in
[48].
We note that, While in free space a weakly bound state exists only for a > 0,
we see that in the waveguide such a state exists for all a. These bound states
may be of significant interest, allowing the formation of dimers via a modulation
of the waveguide potential at the frequency (Ec,g − Eb )/h̄. This may lead to an
atom-waveguide based scheme for forming ultracold molecules, as well as the
possibility to use molecular spectroscopy as a sensitive probe of the atomic field
inside the waveguide.
61
map
g1D
/
h
1
a⊥ k 2
i
5
0
−5
−10
CIR
−15
−0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1
ap /a⊥
mapp
Fig. 4.3: The mapped coupling constant g1D
as a function of the p-wave scattering
length ap for single-channel scattering of two fermions under a harmonic
confinement.
4.3 Remarks on p-wave Scattering
The results obtained for quasi-1D bosons leads us to the following question:
How do two spin-polarized fermions under quasi-1D confinement behave? Attempting to carry through a program analogous to the above for the
case of p-wave scattering between polarized fermions meet numerous (hopefully
technical) obstacles: Most of the limiting procedures become mutually nonuniformly convergent and no clear way to identify a correct order is visible. In
any case the closest candidate for the p-wave analog of the free-space threedimensional T -matrix is the pseudopotential introduced in [49]
hχ|TH,V (E)|ψi =
27πh̄2 Vp
∇χ∗ (0) · ∇ψ(0)
µ
(4.94)
that can be shown to reproduce correctly the low-energy behavior of the p-wave
scattering amplitude. Here Vp is the p-wave scattering volume, that defines the
low-energy behavior of the p-wave scattering phase via [50]
Vp = − lim tan δp (k)/k 3
k→0
(4.95)
An elegant way around these difficulties has been found recently by Granger
and Blume [32], who used a K-matrix technique that does not explicitly involve
any zero-range objects.
Application to identical fermions shows that confinement modifies the freespace scattering properties of two fermions significantly; i.e., spin-polarized
fermions can have infinitely strong effective interactions for a finite 3D scattering volume Vp when confined to a quasi-1D geometry.
Similar to bosons, we can drive an effective 1D Hamiltonian that describes
many of the low energy properties of two spin-polarized fermions in a waveguide.
62
Importantly, the 1D zero-range potential g1D δ(z), which has been very successful in treating bosons, cannot be used directly since it results in an unphysical
scattering amplitude for fermions. One way around this difficulty would be to
map the effective fermionic 1D K-matrix to a bosonic 1D K matrix (along with
the corresponding wave functions). Mappings between fermions and bosons are
important in theoretical treatments of 1D many body systems, as they allow one
to understand systems of strongly interacting 1D bosons (fermions) by mapping
them to weakly interacting systems of 1D fermions (bosons) [51, 52] (For a
briefly review of the mapping theorem see appendix D).
Application of mapping theorem to effective 1D two-fermion scattering , results in an equivalent system of two 1D bosons interacting through the potential
map
g1D
δ(z), with the “mapped coupling constant”
·
µ
¶¸
map
g1D
a3
1
Vp
= − ⊥ 1 − 12 3 ζ − , −ε + 1
(4.96)
a⊥ h̄ω
6Vp
a⊥
2
This remarkable result implies that two spin-polarized quasi-1D fermions with
infinitely strong interactions when the scattering volume Vp has the critical value
·
µ
¶¸−1
Vpcrit
1
= 12ζ − , −ε + 1
a3⊥
2
(4.97)
can be mapped to a system of noninteracting bosons (see Fig. 4.3); the converse
of the Bose-Fermi mapping [52].
4.4 Beyond the s-wave Approximation
So far we have studied scattering of two particles under 2D confining potential
by considering the lowest nonzero angular momentum component (i.e., s-wave
for bosons and p-wave for fermions) and neglecting the higher components. For
atomic systems, both the s and p waves can be brought into play, although
separately, by means of magnetic Feshbach resonances [53]. In the additional
presence of electrostatic fields, however, both component are predicted to arise
simultaneously, despite the low energies involved [54]. In this latter case, a
simultaneous resonance in the s- and p-wave scattering occurs such that the
reflection coefficient R ≈ 0. This so-called dual-CIR first observed numerically
in [38] predicts the possibility of suppressing the effective two-body interaction
in a quasi-1D cylindrical geometry, in spite of a strong interaction in free 3D
space. This effect arises from the quantum interference of even and odd contribution to the scattering amplitude, may provide effectively a quasi-1D gas of
noninteracting atoms or a free flow of carriers in a quantum wire. Recently this
low dimensionality effect has analytically been modelled by applying a previous
formalism [36] to quantum scattering processes under cylindrical confinement.
This analytic formalism has been improved in order to account colliding between
two distinguishable particles including both s and p waves [55].
63
Transmission Coeff. T
1
0.8
0.6
0.4
0.2
0
8
8.5
9
9.5
−V0
Fig. 4.4: Transmission coefficient T as a function of the depth −V0 of the potential
(4.111) for scattering of two distinguishable atoms under a harmonic confinement.
Following [55] the equation describing the relative motion between two distinguishable colliding particles of reduced mass µ and interacting through a
central potential V (r) can be written as
[∇2 − u(ρ) + k 2 ]ψ(x) = v(r)ψ(x)
(4.98)
Here E ≡ h̄2 k 2 /2µ is the total energy, v(r) ≡ 2µV (r)/h̄2 , u(ρ) ≡ 2µU (ρ)/h̄2
and U (ρ) = 12 µω 2 ρ2 is the harmonic confining potential. A general solution to
this equation can be written as
Z
ψ(x) = ψin (x) − d3 x0 Gu (x, x0 )v(r0 )ψ(x0 )
(4.99)
where ψin (x) describes an incident wave and Gu (x, x0 ) is the scattering Green
function under confinement satisfying
[∇2 − u(ρ) + k 2 ]Gu (x, x0 ) = −δ 3 (x − x0 )
(4.100)
The correct boundary condition on Gu (x, x0 ) should recover the form of ψ(x)
far from the scattering region where v(r) is negligible, i.e., for r À RV with RV
being the range of v(r), which can be assumed in many cases to be short-ranged
in the sense that RV ¿ RU , RU is the range of u(ρ).
This form can be deduced from an explicit expansion of ψ(x) in the orthonormalized basis {φn } of eigenstates of the confining potential U (ρ).
For solutions ψ(x) that are cylindrically symmetric, the ϕ0 integration over
Gu in (4.99) can be carried out and it suffices to work with Green function under
confinement
Z
2πGc (x, x0 ) ≡ dϕ0 Gu (x, x0 )
(4.101)
and with the basis φn = φn (ρ), n = 0, 1, . . . , having zero angular momentum
component along z and eigenvalues En ≡ h̄2 qn2 /2µ, with φ0 being the ground
64
state and qn , the respective transverse linear momenta.
By comparison with
P
(4.99) and with an expansion of the type ψ(x) = n ψn (z)φn (ρ), an expression
for Gc (x, x0 ) which holds for all x, x0 follows thus
0
Gc (x, x ) =
∞
X
φn (ρ)φ∗n (ρ0 )Gn (z − z 0 )
(4.102)
n=0
ikn |z|
where Gn (z) = −ep
/2ikn is the purely 1D Green function,
p with longitudinal
momentum kn = k 2 − qn2 for 0 ≤ n ≤ ne and kn = i qn2 − k 2 for n > ne ,
ne denoting the highest “open channel” for a given total momentum k, i.e., the
largest integer ne such that Ene < E < E1+ne . Although formally exact, this
decomposition in the basis {φn } is best suited for the region r, r0 À RV , where
the cylindrical symmetry prevails. Without loss of generality, we may assume
from now on that these cylindrically symmetric functions φn (ρ) are real.
In the region r ¿ RU where u(ρ) is negligible, however, a spherical basis
should better describe the solution ψ(x). In this regard, it has been discussed in
[36] that, when r, r0 ¿ RU (i.e., in the absence locally of any confinement), Gu
should approach a certain free space 3D Green function G. Using then (4.101),
it turns out that Gc should be well approximated by [36]
Z
dϕ0
0
Gc (x, x ) ≈
G(x, x0 ) + ∆c (x, x0 )
2π
Ã
!
Z
0
0
dϕ0
eik|x−x |
e−ik|x−x |
=
γ+
+ γ−
+ ∆c (x, x0 ) (4.103)
2π
4π|x − x0 |
4π|x − x0 |
where γ± ≡ (1 ± γ)/2, such that γ+ + γ− = 1, and γ is the prefactor of the
imaginary part of G. The function ∆c should stem from the homogeneous part
of the solution Gu , i.e., that without the delta function δ(x − x0 ). It should
account, together with the factor γ, for the correct boundary condition given by
(4.102) and relates only to the low lying levels n ≤ ne , for which their quantum
nature is most pronounced. The upper levels for n > ne , in turn, are summed
up to yield, in the region r, r0 ¿ RU , G itself. This indicates that the form
of (4.103) requires only that u(ρ) vanishes as ρ ¿ RU , being thus a property
qualitatively independent of the particular basis {φn }, i.e., of whether u(ρ) is
parabolic or not [36].
A first estimate of γ and ∆c [see Eq. (16) in [36]] reproduces previous
results on quasi-1D scattering that are obtained when v(r) is replaced by zeroranged and regularized pseudopotentials of the type δ(x)∂r (r.). This neglects
all angular momentum components except the first l = 0, the so-called s-wave
approximation. Other treatments to extend the scattering process (in free space)
to higher values of l but still using zero-ranged pseudopotentials can be found
in the Refs. [56, 57].
4.4.1 Effective quasi-1D scattering amplitudes
Far from the scattering region where the interaction potential v(r) is negligible, the degrees of freedom under geometrical confinement should be “frozen”
65
into the ground state of the confining potential u(ρ) (or at most into a few
excited states, depending on the total energy) and only the unconfined variable, z, should reflect the overall dynamics occurring in the region around v(r).
Indeed, substituting Gc in the form of (4.102) into (4.99) and neglecting the
exponentially decaying terms n > ne for large |z| yields
ψ(x) ≈
ne ³
X
´
bn eikn z + fn± eikn |z| φn (ρ)
(4.104)
n=0
The effective quasi-1D scattering amplitudes fn+ and fn− , for n ≤ ne and z > 0
and z < 0, respectively, are then given by [38]
Z
h
i∗
0
1
f± =
d3 x0 e±ikn z φn (ρ0 ) v(r0 )ψ(x0 )
2ikn
X
4π(2l + 1)αln
=
(±1)l
Tl
2ikn
l
≡ fng ± fnu
(4.105)
g(u)
where fn contains only the matrix elements Tl of even (odd) angular momenta
l. Here the incident state is assumed to populate only the open channels and to
be of the form
ne
X
ψin (x) =
bn eikn z φn (ρ)
(4.106)
n=0
l
for some weights bn . i (2l + 1)αln are the coefficients of the expansion of each
term eikn z φn (ρ) on the basis {jl , Pl }, for l = 0, 1, 2, . . . and the “scattering
matrix” element Tl is defined by
Z
l
Tl ≡ (1/i 4π) d3 x0 [jl (kr0 )Pl (cos θ0 )]v(r0 )ψ(x0 )
(4.107)
jl and Pl are the spherical Bessel functions and Legendre polynomials respectively.
4.4.2 Comparison with the unconfined 3D case. The inclusion of p-waves
As a definite example of the above improved formalism, it can be used, for example, to model analytically the remarkable effect of a dual-CIR, which was first
observed in [38] numerically by solving fully the Schrödinger equation (4.98). In
spite of a strong interaction from v(r), which shows a resonant 3D scattering in
both s and p waves with large scattering phase shifts δ0 and δ1 , respectively, the
confinement u(ρ) is able paradoxically to suppress the effective action of v(r)
on the unconfined z variable, as if it could pass freely through the scattering
region without any reflection.
Consider then a total energy E just below the first excited level, i.e., E0 =
h̄ω < E < E1 = 3h̄ω, such that ne = 0. In this single mode regime, only the
66
ground state is open with a longitudinal scattering energy E|| = h̄2 k02 /2µ and
momentum k0 along the z-axis such that E = E0 + E|| and the incident state
should be restricted to bn = δn0 . Supposing additionally that only the phase
shifts δl for l = 0 and l = 1 are large, e.g., due to shape resonances of v(r), by
neglecting the contribution of all terms with l > 1 one obtains [55]
1
£
¤
1 − i a⊥ /as − (C 2 − a2⊥ k02 )1/2 a⊥2k0
1
£ 3
¤
f0u = −
3
2
1 − i a⊥ /ap − (C − a2⊥ k02 )3/2 6a⊥1 k0
f0g = −
(4.108)
where C = 2. The quasi-1D scattering cross section can be obtained from the
reflection coefficient R, which in the present situation can be written as
R = |f0g − f0u |2
(4.109)
In the absence of any confinement, the equivalent 3D quantity describing the
scattering by the same potential is the standard total cross section
σ = σs + σp = 4π
a2s
(kap )4
+
12π
a2
1 + (kas )2
1 + (kap )6 p
(4.110)
Evidently, σ can only increase as ap is no longer negligible, particularly if kap ∼ 1
as ap becomes comparable to the confining length scale RU ∼ a⊥ due, e.g., to a
shape resonance in v(r). Under confinement, however, the reflection coefficient
R could be sharply decreased if the s and p waves interfere destructively. This
can results in an almost complete transmission in spite of the strong interatomic
interaction in free space. This has been shown [36, 38] for spherical square well
(i.e., V (r) = V0 , if 0 < r < r0 , and zero otherwise), for which the zero energy
scattering lengths as and ap can be found straightforwardly as functions of the
well depth V0 , as well as for a more realistic interaction, namely, an attractive
(V0 < 0) screened Coulomb interaction with a screening length r0
V (r) = V0
r0 −r/r0
e
r
(see Fig. 4.4).
67
(4.111)
68
5. NUMERICAL DESCRIPTION OF ATOMIC SCATTERING
AND CONFINEMENT-INDUCED RESONANCES IN
WAVEGUIDES
In the present work we develop a general grid method for multichannel scattering
of identical as well as distinguishable atoms confined by a transverse harmonic
trap. The method applies to arbitrary atomic interactions, permitting a rich
spectral structure where several different partial waves are participating in the
scattering process, or even the case of an anisotropy of the interaction. The
only limitation is that we consider harmonic traps with a single frequency for
every atom causing a separation of the c.m. and relative motion. With our
approach we analyze transverse excitations and deexcitations in the course of the
collisional process (distinguishable or identical atoms) including all important
partial waves and their couplings due to the broken spherical symmetry. Special
attention is paid to the analysis of the CIRs in the multimode regimes for nonzero collision energies, i.e., to suggest a non-trivial extension of the CIRs theory
developed so far only for the single-mode regime and zero-energy limit.
5.1 Hamiltonian and Two-body Scattering Problem in a
Waveguide
Let us consider collisions of two atoms under the action of the transverse harmonic confinement. We address both cases, distinguishable and indistinguishable atoms under the action of the same confining potential, i.e., the trap is
characterized by a single frequency ω for every atom. The corresponding Hamiltonian is given by
H=−
h̄2 2
h̄2 2 1
1
∇1 −
∇ + m1 ω 2 ρ21 + m2 ω 2 ρ22 + V (x1 − x2 )
2m1
2m2 2 2
2
(5.1)
where mi is the mass of the ith atom, V (x1 − x2 ) is the two-body potential
describing the interaction between two colliding atoms in free space, and xi =
(ρi , zi ) = (ri , θi , φi ) is the coordinate of the ith atom. As we have seen in chapter
4 the Hamiltonian is separable with respect to the relative/CM variables, the
problem, thus, reduces to scattering of a single effective particle with the reduced
mass µ and collision energy E > h̄ω, off a scatterer V (r) at the origin, under
transverse harmonic confinement with frequency ω
¸
·
1
h̄2
(5.2)
− ∇2x + µω 2 ρ2 + V (r) ψ(x) = Eψ(x)
2µ
2
For the two-body interaction we choose a screened Coulomb potential
V (r) = V0
r0 −r/r0
e
r
(5.3)
already employed in Ref. [38, 39, 40] for analyzing the ultracold scattering in
cylindrical waveguides in the single-mode regime as E⊥ = E − Ek . The chosen
potential (5.3) depends on two parameters - the depth V0 < 0 and the screening
length r0 > 0. Following the computational scheme already developed in Ref.
[40], we implement different spectral structures of the atomic interaction by
varying the single parameter V0 for a fixed length scale r0 . Obvious advantages
of the screened Coulomb potential (5.3) compared to the s-wave pseudopotential
used in ref. [43] that is devoted to the multi-channel scattering of bosons are
the following. First, by varying V0 one can vary the number of bound states of
s-wave character in the interaction potential, and second, one can create new
bound and resonant states of higher partial wave character. As a consequence
we can consider not only bosonic but also fermionic as well as mixed collisions
in a trap and including the case of higher energies. This will, as we shall see
below, permit us to investigate new regimes and effects of multi-channel confined
scattering.
Performing the scale transformation
r→
r
a0
,
E→
E
E0
, V0 →
V0
E0
and ω →
ω
ω0
(5.4)
with the units a0 = h̄2 /µV0 r0 , E0 = h̄2 /µa20 , and ω0 = E0 /h̄, it is convenient to
rewrite the equation (5.2) in the rescaled form
·
¸
1
1
− ∇2x + ω 2 ρ2 + V (r) ψ(x) = Eψ(x)
(5.5)
2
2
where V is now the correspondingly scaled potential and we fix µ = 1 and r0 = 1
in the subsequent consideration.
In the asymptotic region |z| → ∞, where the transverse trapping potential
dominates the interaction potential, the axial and transverse motions decouple
and the asymptotic wavefunction can be written as a product of the longitudinal
eikn z and transverse φn,m (ρ, ϕ) components with
φn,m (ρ, ϕ) = [
πa2⊥ (n + |m|)! −1/2 ρ |m| −ρ2 /(2a2⊥ ) |m| 2 2 imϕ
]
( ) e
Ln (ρ /a⊥ )e
n!
a⊥
(n = 0, 1, 2, . . . ∞ and m = 0, ±1, ±2, . . . ± ∞)
(5.6)
being the eigenfunctions of the transverse trapping Hamiltonian
1 ∂
1 ∂2
1
1 ∂2
+ 2
) + ω 2 ρ2 ,
H⊥ = − ( 2 +
2 ∂ρ
ρ ∂ρ ρ ∂ϕ2
2
(5.7)
with the corresponding eigenvalues E⊥ = ω(2n + |m| + 1) and the angular mom
mentum projection m onto
√ the z axis. Here Ln (x) are the generalized Laguerre
polynomials and a⊥ = 1/ ω, is the transverse oscillator length.
70
Due to the axial symmetry of the transverse confinement and the spherical
symmetry of the interatomic interaction (5.3) the angular momentum component along the z axis is conserved. Therefore, the problem (5.5) is reducible to
a 2D one by separating the ϕ-variable and can be solved for every m independently. The quantum number n is a good one only in the asymptotic region
|z| → ∞ and used for the definition of the initial (incident) asymptotic state,
i.e., channel
in
(x) = eikn z φn,m (ρ, ϕ)
ψn,m
(5.8)
of two infinitely separated atoms confined in a transverse state < ρϕ|nm >. In
addition to the quantum numbers n and m the asymptotic scattering state is
defined also by the momentum kn of the channel
r
2
|m|
kn =
(5.9)
ε−n−
a⊥
2
which we express through the dimensionless energy ε = E/(2ω)−1/2. It is clear
that the integer part of the dimensionless energy ε coincides with the number ne
of open excited transverse channels. For n ≤ ne we have (ε − n − |m|
2 ) > 0. The
spherically symmetric interatomic interaction (5.3) mixes different transversal
channels and leads to transitions n → n0 between the open channels n, n0 ≤ ne ,
i.e. to transverse excitation and deexcitation processes during the collisions.
Assuming the system to be initially in the channel n, the asymptotic wavefunction takes at | z |→ +∞ the form [43]
ψn,m (x) = eikn z φn,m (ρ, ϕ) +
ne
X
ikn0 |z|
o
e
φn0 ,m (ρ, ϕ) (5.10)
[fnn
0 + sgn(z)fnn0 ] e
n0 =0
e
o
where fnn
0 and fnn0 are the matrix elements of the inelastic scattering amplitudes for the even and odd partial waves, respectively, which describe transitions
between the channels n and n0 . For a bosonic (fermionic) collision just the symmetric (antisymmetric) part of (5.10) should be considered.
It is clear that the scattering amplitude depends also on the index m which,
however, remains unchanged during the collision due to the axial symmetry of
the problem. Hereafter we consider only the case m = 0 and the index is omitted
in the following.
Using the asymptotic wavefunction (5.10) and the total current conservation
one obtains
ne
X
(Tnn0 + Rnn0 − δnn0 ) = 0
(5.11)
n0 =0
for the inelastic transmission (reflection) coefficients Tnn0 (Rnn0 ). kn (kn0 ) is the
initial (final) relative wave vector and n (n0 ) the transverse excitation numbers
according to (5.9). We have
Tnn0 = Θ[ε − n0 ]
kn0
e
o
2
|δn,n0 + fnn
0 + fnn0 |
kn
71
(5.12)
Rnn0 = Θ[ε − n0 ]
kn0 e
o
2
|f 0 + fnn
0|
kn nn
(5.13)
where Θ(x) is the Heavyside step function. The transition probability Wnn0 ,
characterizing the transverse excitation/deexcitation, into a particular channel
n0 from the initial state n is given by the sum of the corresponding transmission
and reflection coefficients
Wnn0 = Tnn0 + Rnn0
(5.14)
Due to the time reversal symmetry of the Hamiltonian we have Tnn0 = Tn0 n ,
0 n , and Wnn0 = Wn0 n .
Rnn0 = RnP
The total transmission (reflection) coeffiP
cient T = n0 Tnn0 (R = n0 Rnn0 ) is given by the sum of the transmission
(reflection) coefficients of all the open channels. (5.11) leads to T + R = 1.
50
Scattering Parameter
40
30
20
10
0
−10
−20
−30
−40
−50
0
1
2
3
4
5
−V0
6
7
8
9
10
Fig. 5.1: s- and p-wave scattering parameters as (red) and Vp (black) as a function
of the potential depth V0 for the free-space effective potential V (r) + l(l +
1)/(2r2 ). Divergences correspond to the appearance of new bound (s-wave
scattering) or shape resonant (p-wave scattering) states in the effective potential. All quantities are given in the units (5.4).
5.2 Numerical approach
To obtain the observable quantities Tnn0 , Rnn0 , and Wnn0 of the scattering process, we have to calculate the matrix elements fnn0 of the scattering amplitude
fˆ by matching the numerical solution of the Schrödinger equation (5.5) with
the scattering asymptotics (5.10). To integrate this multi-channel scattering
problem in two dimensions r and θ (z and ρ) we adopt the discrete-variable
method suggested in Ref. [58] for solving the nonseparable 2D scattering problems. This approach was applied in Ref. [59] to the case of a 3D anisotropic
scattering problem of ultracold atoms in external laser fields.
72
First, we discretize the 2D Schrödinger equation (5.5) on a 2D grid of angular
N
θ
{θj }N
j=1 and radial {rj }j=1 variables. The angular grid points θj are defined as
the zeroes of the Legendre polynomial PNθ (cos θ) of the order Nθ . Using the
completeness property of the normalized Legendre polynomials which remains
valid also on the chosen angular grid
NX
θ −1
p
Pl (cos θj )Pl (cos θj 0 ) λj λj 0 = δjj 0
(5.15)
l=0
where λj are the weights of the Gauss quadrature, we expand the solution of
PNθ −1
equation (5.5) in the basis fj (θ) = l=0
Pl (cos θ)(P−1 )lj according to
N
ψ(r, θ) =
θ
1X
fj (θ)uj (r)
r j=1
(5.16)
Here P−1 is the inverse
of the Nθ × Nθ matrix P with the matrix elements
p
defined as Pjl = λj Pl (cos θj ). Due to this definition one can use the completeness relation
(5.15) in order to determine the matrix elements (P−1 )lj as
p
−1
(P )lj = λj Pl (cos θj ). It is clear from (5.16) that the unknown coefficients
uj (r) in the expansion are the values ψ(r, θj ) of the
p two-dimensional wave function ψ(r, θ) at the grid points θj multiplied by λj r. Near the origin r → 0
we have uj (r) ' r → 0 due to the definition (5.16) and the demand for the
probability distribution | ψ(r, θj ) |2 to be bounded. Substituting (5.16) into
(5.5) results a system of Nθ Schrödinger-like coupled equations with respect to
1/2
θ
the Nθ -dimensional unknown vector u(r) = {λj uj (r)}N
1
[H(0) (r) + 2(EI − V(r))]u(r) = 0
where
(0)
Hjj 0 (r) =
Nθ −1
d2
1 X
0 −
δ
Pjl l(l + 1)(P−1 )lj 0
jj
dr2
r2
(5.17)
(5.18)
l=0
1
Vjj 0 (r) = V (r, θj )δjj 0 = {V (r) + ω 2 ρ2j }δjj 0 , ρj = r sin θj
(5.19)
2
and I is the unit matrix. We solve the system of equations (5.17) on the quasiuniform radial grid [60]
rj = R
eγxj − 1
, j = 1, 2, ..., N
eγ − 1
(5.20)
of N grid points {rj } defined by mapping rj ∈ (0, R → +∞] onto the uniform
grid xj ∈ (0, 1] with the equidistant distribution xj − xj−1 = 1/N . By varying
N and the parameter γ > 0 one can choose more adequate distributions of the
grid points for specific interatomic and confining potentials.
By mapping the initial variable r in (5.17) onto x we obtain
[H (0) (x) + 2{EI − V(r(x))}]u(r(x)) = 0
73
(5.21)
with
µ
(0)
Hjj 0 (x) = f 2 (x)δjj 0
d2
d
−γ
dx2
dx
¶
−
Nθ −1
1 X
Pjl l(l + 1)(P−1 )lj 0
r2 (x)
(5.22)
eγ − 1
Reγx γ
(5.23)
l=0
where
f (x) =
The uniform grid with respect to x gives six-order accuracy for applying a
seven-point finite-difference approximation of the derivatives in the equation
(5.21). Thus, after the finite-difference approximation the initial 2D Schrödinger
equation (5.5) is reduced to the system of N algebraic matrix equations
3
X
Ajj−p uj−p + [Ajj + 2{EI − Vj }]uj +
p=1
3
X
Ajj+p uj+p = 0
p=1
for
uj +
(1)
αj uj−1
+
(2)
αj uj−2
+
j = 1, 2, ..., N − 3
(3)
αj uj−3
for
(4)
+ αj uj−4 = gj
j = N − 2, N − 1, N
(5.24)
where each coefficient Ajj 0 is a Nθ × Nθ matrix, each αj is a diagonal Nθ × Nθ
matrix, and each gj is a Nθ -dimensional vector. Here the functions u−3 , u−2 ,
u−1 , and u0 in the first three equations of the system (for j = 1, 2, and 3) are
eliminated by using the “left-side” boundary conditions: u0 = 0 and u−j = uj
(j = 1, 2, 3). The last three equations in this system for j = N, N − 1, and
N −2 are the “right-side” boundary conditions approximating at the edge points
rN −2 , rN −1 , and rN = R of the radial grid, the scattering asymptotics (5.10) for
the desired wave function u(rj ). In order to construct the “right-side” boundary
conditions (5.24) at j = N − 2, N − 1, and N we used an idea of Ref. [59], i.e.,
the asymptotic behavior (5.10) at the edge points rN −2 , rN −1 , and rN = R are
considered as a system of vector equations with respect to the unknown vector
fnn0 of the scattering amplitude for a fixed n. By eliminating the unknowns fnn0
from this system we implement the “right-side” boundary conditions defined by
(5.24) at j = N − 2, N − 1, and N (see Appendix E).
The reduction of the 2D multi-channel scattering problem to the finitedifference boundary value problem (5.24) permits one to apply efficient computational methods. Here we use, in the spirit of the LU decomposition [61],
and the sweep method [62] (or the Thomas algorithm [63]), a fast implicit matrix algorithm which is briefly described in Appendix B. The block-diagonal
structure of the matrix of the coefficients in the system of equations (5.24) with
the width of the diagonal band equal to 7×Nθ makes this computational scheme
an efficient one.
Solving the problem (5.24) for the defined initial vector kn and a fixed n from
the possible set 0 ≤ n ≤ ne we first calculate the vector function ψ(kn , r, θj ).
74
Then, by matching the calculated vector ψ(kn , R, θj ) with the asymptotic behavior (5.10) at r = R, we calculate the nth row of the scattering amplitude
matrix fnn0 describing all possible transitions n → n0 = 0, 1, ..., ne . This procedure is repeated for the next n from 0 ≤ n ≤ ne . After calculating all the
elements fnn0 of the scattering amplitude we obtain any desired scattering parameter T , R, or W .
5.3 Results and Discussion
With the above-described method being implemented we have analyzed the
two-body scattering under the transverse harmonic confinement for both cases
of identical and distinguishable colliding atoms. For confined scattering of identical atoms one has to distinguish the bosonic and fermionic cases. In the case
of two colliding bosons the two-body wave function must be symmetric and only
even scattering amplitude provides us with a nonzero contribution. First, we
show that our result for the special case ε < 1 of a single-channel scattering is in
agreement at ε → 0 with the s-wave pseudopotential approach [27], and, particularly, reproduces s-wave CIR predicted and analyzed in Refs. [27, 28, 43, 38, 40]
for bosons. Then we extend our consideration to the multi-channel scattering
ε > 1. We demonstrate that our results are in a good agreement in the limit
of a long-wavelength trap ω = 2πc/λ → 0 with the analytical expression given
in [43] which has been obtained in the s-wave pseudopotential approach for the
zero-energy limit. The range of validity of the analytical investigation in Ref.
[43] is explored. Next we present results for multi-channel scattering of two
fermions under transverse harmonic confinement. For a fermionic collision the
two-body wave function is antisymmetric, i.e. only the odd scattering amplitude is nonzero. In the special case of single-channel scattering we reproduce
the p-wave CIR for fermions [32]. These results are also in agreement with our
previous investigations in Refs. [38, 39, 40] performed within a wave-packet
propagation method [40]. Finally we consider the confined multi-channel scattering of two distinguishable atoms. In this case both even and odd amplitudes
contribute to the scattering process.
For modelling different interatomic interactions in the subsequent sections
we vary the depth V0 of the potential (5.3) in the wide range −10 < V0 < 0 for
a fixed width r0 = 1. In free space, this potential being superimposed with the
centrifugal term l(l + 1)/(2r2 ) makes an effective potential which may support
some even or odd bound states depending on the value of the quantum number
l and the parameter V0 . For l 6= 0 there might be also some shape resonances
for certain relative energies. These shape resonances may enhance strongly the
contribution of l 6= 0 partial waves in the energy domain where one would have
expected a pure l = 0 scattering.
It is known that in the zero-energy limit, when the scattering process does
not depend on the details of the potential, the collision can be described by a single parameter: the s-wave scattering length as = − limk→0 tan δs (k)/k for l = 0
(bosonic collision) and the p-wave scattering volume Vp = − limk→0 tan δp (k)/k 3
75
10
8
(a)
6
g1D /[ a1⊥ ]
4
2
0
−2
−4
−6
−8
−10
−3
−2
−1
0
as /a⊥
1
2
3
−2
−1
0
as /a⊥
1
2
3
1
0.9
(b)
Transmission Coef. T
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−3
1
0.8
(c)
0.6
Im(f e )
amplitude
0.4
00
0.2
1 + Re(f e )
00
0
−0.2
−0.4
−0.6
−0.8
−1
−3
−2
−1
0
as /a⊥
1
2
3
Fig. 5.2: (a) The effective quasi-1D coupling constant g1D , (b) the transmission coefe
ficient T , and (c) the scattering amplitude f00
as a function of the scattering
length as for one-channel scattering of two bosons under a harmonic confining potential with ω = 0.002 for a longitudinal relative energy Ek = 0.0002
together with the analytical results obtained for the s-wave pseudo-potential
zero-energy limit (solid curves) [27, 28, 43]. The constant V0 is varied in the
region −2.30 < V0 < −0.32.
76
for l = 1 (fermionic collision). For sufficiently low collision energy the contribution of the partial waves with larger l can be neglected. In Fig. 5.1 we have
plotted as and Vp in the region −10 < V0 < 0. Fig. 5.1 demonstrates the
rich spectral structure of the chosen form of the interatomic interaction: the
scattering parameters as and Vp can be positive or negative and they diverge
for the values of V0 corresponding to the appearance of new bound states. In
the case of p-wave scattering the increase of the depth V0 of the potential first
leads to a shape resonance which approaches zero energy and finally transforms
to a p-wave bound state.
5.3.1 Multichannel scattering of bosons
For two bosons colliding in a transverse harmonic confinement the scattering
o
wave function is symmetric with respect to the exchange z → −z, i.e. fnn
0 = 0
in (5.10). We consider multichannel scattering with the dimensionless energy
ε = E/(2ω) − 1/2 < 4 (5.9) permitting the collisional transverse excitations and
deexcitations n → n0 < 4 up to four open channels (four-mode regime). For
comparison with analytical results [28, 43] obtained in the s-wave pseudopotential approach, we have extracted the effective quasi-1D coupling constant
e
e
(k)}k/µ as well as the transmission coeffi(k)}/Im{f00
g1D = limk→0 Re{f00
e
(5.10), as a function
cient T = T00 (5.12) and the scattering amplitude f00
of the scattering length as in the single-mode regime (0 < ε < 1). The calculated parameters are presented along with the analytical results in Fig. 5.2
for ω = 0.002 and the longitudinal relative energy Ek = E − E⊥ = 0.0002.
Fig. 5.2(a) shows the coupling constant g1D as a function of the scattering
length as . Our numerical result clearly exhibit a singularity at as /a⊥ ≈ 1/C
with C = −ζ(1/2) = 1.4603.., which corresponds to the well-known s-wave CIR
[27, 28]. In Fig. 5.2(b) we present the transmission coefficient T versus as .
The transmission coefficient T goes to unity (total transmission) when as tends
to zero (i.e. no interaction between the atoms), while at the CIR position, it
exhibits the well-known minimum (blocking of the atomic current by the CIR).
e
as a function of as . The ampliFig. 5.2(c) shows the scattering amplitude f00
tude approaches zero at as /a⊥ = 0 and −1 at the CIR position , which results
in total transmission and total reflection, respectively. In general the presented
e
values of f00
are in very good agreement with the analytical results.
In the multimode regime our results are in good agreement with the analytical ones obtained within the s-wave pseudopotential approach in the zero-energy
limit [43] for long-wavelength traps (tight confinement: ω = 2πc/λ → 0). In
e
Fig. 5.3 we present our scattering amplitude f00
along with the analytical results as a function of the dimensionless energy ε for ω = 0.0002 and V0 = −3.0.
Apart from energies close to the channel thresholds, the real and imaginary
e
e
part of the scattering amplitude f00
show a monotonous behavior: Re(f00
) is
e
monotonically increasing and approaching zero asymptotically whereas Im(f00
)
decays monotonically and also approaches zero for large values of ε. The peak
structure located at integer values of ε is due to the resonant scattering once
a new previously closed channel opens with increasing energy. There is a good
77
agreement between our results and the analytical ones given by (6.9) in Ref.
[43] for the complete range 0 < ε < 4. However, we encounter major deviations
with increasing ω except for narrow regions close to the channel thresholds (as
ε approaches integer values) for the real parts of the scattering amplitudes (see
Fig. 5.4). These deviations are most presumably due to the energy dependence
of the s-wave scattering length which is neglected in Ref. [43].
0
0.5
0.4
−0.2
0.3
e
)
00
Im(f
Re(f
e
)
00
0.2
−0.4
−0.6
0.1
0
−0.1
−0.2
−0.3
−0.8
−0.4
−1
0
1
2
ε
3
−0.5
0
4
1
2
ε
3
4
e
Fig. 5.3: The scattering amplitude f00
(dots) as a function of the dimensionless energy
ε along with the analytical results -solid curves- for ω = 0.0002 and V0 = −3.0
(i.e., as = −8.95).
For the multichannel regime (ε > 1) we find a strong dependence of the total
transmission coefficient T on the population of the initial state n, except for the
case as /a⊥ → 0 of noninteracting bosons in free space. Fig. 5.5 shows the
calculated transmission coefficients (5.12) as a function of as /a⊥ for ω = 0.002
for the (a) two-mode regime with ε = 1.05 and (b) the three-mode regime
with ε = 2.05. Note, that all these cases correspond to near-threshold collision
energies if the maximal integer is subtracted from ε. Similar to the single-mode
regime, T goes to unity (total transmission) when as tends to zero. We also
encounter a minimum. However, the value of the transmission at the minimum
is not zero anymore in the multi-mode regime, the larger the number of open
channels, the position of the minima will be more shifted to the left. For a fixed
value of the ratio as /a⊥ 6= 0, a lower initially populated transverse level n, leads
to a larger total transmission.
In Fig. 5.6 we present the transmission as a function of the dimensionless
energy ε for the two cases of the system being initially in the ground n = 0
(a) or first excited n = 1 (b) transverse states, for different values of the ratio
as /a⊥ . With increasing collision energy the transmission coefficient exhibits
a nonmonotonic behavior: we observe a sequence of minima and peaks. For
0 < ε < 1 the value of T at the minimum is zero. As the number of open
channels increases with increasing collision energy E, the transmission values
at the corresponding minima increase strongly. The corresponding transmission
peaks T = 1 are located at the channel thresholds. The shape of the transmis78
−0.2
0.3
−0.4
0.1
e
)
00
0.5
Im(f
e
)
00
Re(f
0
−0.6
ω = 0.02
ω = 0.04
ω = 0.06
−0.8
−1
0
1
2
ε
3
−0.1
ω = 0.02
ω = 0.04
ω = 0.06
−0.3
−0.5
0
4
1
2
ε
3
4
e
Fig. 5.4: The scattering amplitude f00
(dots) as a function of the dimensionless energy
ε along with the analytical results (solid curves) for ω = 0.02 (red), ω = 0.04
(blue) and ω = 0.06 (black). The amplitudes have been calculated for V0 =
−3.0 (as = −8.95).
1
1
(a)
0.8
Transmission Coeff. T
Transmission Coeff. T
0.8
0.6
0.4
0.6
0.4
0.2
0.2
(b)
0
−3
−2
−1
0
as /a⊥
1
2
0
−3
3
−2
−1
0
as /a⊥
1
2
3
Fig. 5.5: The total transmission coefficient T for a bosonic collision in a harmonic
confinement ω = 0.002 as a function of as /a⊥ for (a) two-open channels with
ε = 1.05 being initially in the transverse ground state n = 0 (red) and first
excited state n = 1 (black), and (b) three-open channels with ε = 2.05 being
initially in the ground state n = 0 (red), first excited state n = 1 (black) and
second excited state n = 2 (blue).
79
1
1
0.8
0.8
Transmission Coeff. T
Transmission Coeff. T
0.9
(a)
0.6
as /a⊥ =
−30.08
−0.54
0.13
0.52
0.68
1.72
0.4
0.2
0
0
1
2
ε
3
0.7
(b)
0.6
as /a⊥ =
0.5
−30.08
−0.54
0.13
0.52
0.68
1.72
0.4
0.3
0.2
0.1
0
1
4
2
ε
3
4
Fig. 5.6: The total transmission coefficients T for bosonic collisions as a function of
the dimensionless energy ε for the two cases of the system being initially in
the ground n = 0 (a) and first excited n = 1 (b) transverse states, for several
ratios of as /a⊥ and ω = 0.002. The black curve corresponds to as /a⊥ = 1/C
for which the zero-energy CIR in the single-mode regime is encountered.
sion “valleys” in between two integer values of ε as well as the positions of the
minima strongly depend on the ratio as /a⊥ which can be changed by varying
the strength of the interatomic interaction V0 or the trap frequency ω. Both,
changing V0 and/or ω leads to energetic shifts of the bound states (in particular
for the excited transversal channels) of the atoms in the presence of the confinement. If the collision energy coincides with a bound state of the corresponding
closed channel we encounter an occupation of the closed channel in the course
of the scattering process, i.e. a Feshbach resonance occurs. The interpretation
of the minimum of the transmission T in terms of a Feshbach resonance at the
point as /a⊥ = 1/C for the zero-energy limit was provided in Refs. [27, 28],
where it was shown that the origin of the CIR is an intermediate occupation of
a bound state belonging to an excited transverse (closed) channel.
To demonstrate that the above-discussed behavior (minima) of the transmission coefficient T (ε) in certain regions of ε is due to Feshbach resonances
we have analyzed the probability density of the scattering wave function of the
atoms in the trap. In the single-mode regime and in the zero-energy limit we
encounter the well-known CIR: Fig. 5.7 shows the corresponding probability
density |ψ(x, z)|2 for an initial transverse ground state n = 0 and ε = 0.05 as
well as as /a⊥ = 0.68. For small |z| one observes additional two pronounced
peaks along the transverse (x-) direction corresponding to the occupation of
the bound state (with the binding energy εB
n=1 ∼ 0) in the first excited closed
channel of the transverse potential. The probability density tends to zero as
z → +∞. This leads to a zero of the transmission T (ε) for ε → 0 [see Fig.
5.6(a)] corresponding to the zero-energy CIR.
In Fig. 5.8 we show the probability densities |ψ(x, z)|2 at as /a⊥ = +4.39
for several values of the dimensionless energy ε for (a) the single- and (b) two-
80
T=0.0
|\(x,z)|
-40
-20
x 0
20
40
20
10
2
-10
0 z
-200
-100
-100
-50
x 0
0 z
100
50
100
200
Fig. 5.7: The probability density |ψ(x, z)|2 for bosonic collisions as a function of x
and z at as /a⊥ = 0.68 for ε = 0.05 - zero-energy CIR . The corresponding
transmission values are also indicated. The result has been obtained for
ω = 0.002.
mode regimes for collisions with initial transverse state n = 0. The corresponding probability density exhibits for small values of |z| additional two (for
the single-mode regime) and four (for the two-mode regime) pronounced peaks
with respect to the transverse (x) direction as ε approaches the CIR position.
This demonstrates the occupation of bound states of higher, namely first exB
cited (with binding energy εB
n=1 ) and second excited (with binding energy εn=2 )
channels in the course of the scattering process. The corresponding transmission values are also indicated in Fig. 5.8 . Zero transmission is observed also
for as /a⊥ = +4.39 and ε = 0.75 with no probability density being present for
large positive values of z, see Fig. 5.8(a).
The energy of the bound state εB
n=1 , that leads to a minimal transmission
due to a resonant scattering process, changes with varying as /a⊥ as follows.
It is below ε = 0 for 0 < as /a⊥ < 0.68 and consequently no minimum is
encountered for T (ε) in the range 0 < ε < 1 (see Fig. 5.6). At the position of
the zero-energy CIR it is located just above the threshold ε = 0 leading to a zero
transmission for ε → 0. For as /a⊥ > 0.68 the bound state energy is somewhere
in between the channel thresholds ε = 0 and ε = 1 whereas for as /a⊥ < 0
it is below but close to ε = 1 leading again to a corresponding minimum of
T . For both cases, as /a⊥ < 0 and as /a⊥ > 0.68, an increase of as /a⊥ leads
to a narrow transmission well and the corresponding transmission minimum is
shifted towards the next higher channel threshold. The dependence of εB
n=1 on
the parameter as /a⊥ is in agreement with the pseudopotential analysis given in
ref.[28] (see Fig. 2 of this reference).
In Fig. 5.9(a) we show the transition probabilities Wnn0 as a function of
as /a⊥ for ω = 0.002 and ε = 3.05 corresponding to four open channels. We
81
T=0.24
T=0.67
|\(x,z)|
-40
-20
0
x 20
-150
-100
40 30
-50
x
20
0
100
150
2
H=1.625
H = 0.25
-10
0
10 z
50
|\(x,z)|
2
-300
-200
-100
0 z
100
200
300
-150
-100
-50
x
0
50
100
150
-300
-200
-100
0 z
100
200
300
T=0.0
T=0.59
|\(x,z)|
|\(x,z)|
2
H=1.813 (CIR)
H=0.75 (CIR)
-150
-100
-300
-200
-100
-50
x
0
50
100
150
2
0 z
100
200
300
-150
-100
-50
x
0
50
100
150
-300
-200
-100
0 z
100
200
300
T=0.69
T=0.45
|\(x,z)|
2
|\(x,z)|
H=0.875
-150
-100
(a)
-50
0
x
H=1.900
-300
-200
-100
0
100 z
50
100
150
2
-150
-100
200
300
-50
x
(b)
0
50
100
150
-300
-200
-100
0 z
100
200
300
Fig. 5.8: The probability density |ψ(x, z)|2 for bosonic collisions as a function of x and
z at as /a⊥ = +4.39 for two cases of the single-mode regime (a) and two-mode
regime (b) with different values of ε. The corresponding transmission values
are also indicated. All subfigures are for ω = 0.002 and n = 0.
82
1
1
W00
(b)
W11
0.8
Transition Probability Wnn’
Transition Probability Wnn’
(a)
W22
0.6
W33
0.4
W23 W13 W03
0.2
W12
0
−3
−2
−1
0
as /a⊥
W
02
1
0.8
W00
W11
0.6
0.4
W22
W33
W01
W12
0.2
W03 W13
W02
W23
W
01
2
0
1
3
2
ε
3
4
Fig. 5.9: (a) The transition probabilities Wnn0 as a function of as /a⊥ in the four mode
regime for ε = 3.05. (b) The calculated transition probabilities Wnn0 as a
function of ε up to four open channels for as /a⊥ = −30.08. ω = 0.002 for
both subfigures.
observe that the probability of remaining at the same initial state Wnn (i.e.
elastic scattering) is in the complete range of the ratio as /a⊥ much larger than
the probability of a transition into a different state Wnn0 (i.e. inelastic scattering). With increasing n or n0 the inelastic transition probabilities Wnn0 increase
but the elastic probabilities Wnn decrease. In an inelastic (elastic) collision
Wnn0 (Wnn ) goes to zero (unity) as as tends to zero. Wnn0 (Wnn ) possess a
maximum (minimum) at the resonance position as /a⊥ ≈ 0.35 consistent with
the minimum of T (ε) at ε = 3.05. It is instructive to see how the distribution
of the initial flux among the open channels changes due to pair collisions as a
function of the collision energy. Fig. 5.9(b) shows the transition probabilities
Wnn0 as a function of the dimensionless energy up to four open channels. The
probability of elastic scattering remains larger than that of inelastic scattering
in the complete range of the energy. For two open channels the elastic collision
probability Wnn is independent of the initial state (W00 = W11 ). For a higher
number of open channels Wnn is decreasing with increasing initial value of n.
Near the thresholds, the probabilities of the inelastic (elastic) transitions Wnn0
(Wnn ) go to zero (unity).
5.3.2
Multichannel scattering of fermions
In this section we focus on fermionic collisions in harmonic traps. In this
case the interatomic wave function is antisymmetric with respect to the ine
terchange of the two fermions and the even amplitude fnn
0 in (5.10) is zero. We
have analyzed the multichannel scattering of fermions up to four open transverse channels for different interatomic interactions, by varying the potential
strength V0 in the vicinity of the value V0 = −4.54 (see Fig. 5.1) generating
a resonant p-wave state in free space. Fig. 5.10 shows corresponding results
for the single-mode regime. Fig. 5.10(a) shows the mapped coupling constant
83
(a)
0
map
/
g1D
h
1
a⊥ k 2
i
5
−5
−10
−15
−0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1
ap /a⊥
Transmission Coeff. T
1
(b)
0.8
0.6
0.4
0.2
0
−0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1
ap /a⊥
1
0.8
(c)
1+Re(fo )
00
Amplitude
0.6
0.4
Im(fo00)
0.2
0
−0.2
−0.4
−0.6
−0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1
ap /a⊥
map
Fig. 5.10: (a),(b) The mapped coupling constant g1D
and the transmission coefficient
T as a function of the scattering length ap for single-channel scattering of two
fermions under a harmonic confinement ω = 0.002 and for the longitudinal
energy Ek = 0.0002 (red), ω = 0.02, Ek = 0.002 (black), and ω = 0.06,
o
Ek = 0.002 (blue). (c) The scattering amplitude f00
as a function of the
scattering length ap for ω = 0.002 and Ek = 0.0002. The corresponding
constant V0 is varied in the region −4.54 < V0 < −4.47.
84
map
o
g1D
= limk→0 Im{f o (k)}/Re{f
(k)}µk [32, 40] as a function of the p-wave
p
map
3
scattering length ap = Vp . The mapped coupling constant g1D
goes to zero
at the position of the mapped CIR [32], which for ω = 0.002 and Ek = 0.0002
is equal to ap /a⊥ = −0.31. The position of the mapped CIR obviously depends
on the values of ω and ε. In Fig. 5.10(b) we have plotted the transmission coefficient T as a function of ap . The transmission coefficient exhibits a minimum
and an accompanying well (i.e. the blocking due to the resonance) at the position of the CIR, and tends to unity (i.e. total transmission) for ap far from the
CIR position. For larger ω the well becomes wider, and its minimum is shifted
o
to the left, see also Ref. [38]. Fig. 5.10(c) shows the scattering amplitude f00
as a function of ap for ω = 0.002 and Ek = 0.0002. The amplitude approaches
zero far from the CIR position and −1 at the CIR position , which results in
o 2
total transmission T = |1 + f00
| → 1 (i.e. the fermions do not scatter each
o 2
other) and total reflection T = |1 + f00
| → 0 (i.e. strongly interacting and
impenetrable fermions), respectively.
In Fig. 5.11 we present the total transmission coefficient as a function of ap
for ω = 0.002 in (a) two-mode regime for ε = 1.05 and (b) three-mode regime
for ε = 2.05. Similar to the single-mode regime, T exhibits a minimum and
accompanying well, however, the position of the minimum is shifted to the right
and the value at the minimum is nonzero. For the lower degree of transversal
excitation, we observe a deeper and narrower transmission well. With increasing
energy the transmission well becomes wider and more shallow and its position
is shifted to larger values of ap /a⊥ . This is also demonstrated in Fig. 5.12,
where the transmission coefficient is plotted as a function of ε for several values
of ap /a⊥ for the two cases of being initially in the ground n = 0 (a) and the
first excited n = 1 (b) transverse states for ω = 0.002. We observe that for
any number of open channels, T exhibits a minimum for some value of ap /a⊥ .
With increasing ap /a⊥ the transmission well becomes more shallow (except for
the single-mode regime) and wider. In contrast to the bosonic case, there is no
specific threshold behavior. This is a consequence of the fact that the relative
motion does not feel the interatomic interaction in the closed channels which are
strongly screened by the centrifugal repulsion playing a dominant role for nearthreshold collision energies or in other words, we encounter a weak coupling of
the different scattering channels.
In Fig. 5.13 we show the transition probabilities Wnn0 as a function of ap /a⊥
for four open channels. We see that the probability of an elastic scattering
process (i.e. to remain in the same transversal state) is much larger than that
of an inelastic collision (i.e. the transition to a different transversal state). For
an elastic collision the probability Wnn shows a minimum and corresponding
well which becomes wider and more shallow with increasing initial quantum
number n (i.e., with the population of a higher excited initial transversal state).
For an inelastic collision Wnn0 (n 6= n0 ) exhibits a peak which becomes less
pronounced as the quantum numbers n or n0 increase.
85
1
Transmission Coeff. T
Transmission Coeff. T
1
0.8
0.6
0.4
0.2
−0.4
−0.2
ap /a⊥
0
0.8
0.7
0.6
0.5
(b)
0.4
−0.25 −0.23 −0.21 −0.19 −0.17 −0.15
ap /a⊥
(a)
−0.6
0.9
0.2
Fig. 5.11: Total transmission coefficient T in a fermionic collision as a function of ap
for ω = 0.002 for (a) two-open channels with ε = 1.05 being initially in
the ground state n = 0 (black) and first excited state n = 1 (red), and (b)
three-open channel with ε = 2.05 being initially in the ground state n = 0
(black), first excited state n = 1 (red) and second excited state n = 2 (blue).
ap /a⊥ = −0.362
0.8
−0.320
0.6
−0.180
−0.314
0.4
−0.186
−0.202
0.2
−0.213
0 −0.293
0
2
ε
3
0.8
−0.227
0.7
−0.166
0.6
0.4
1
4
−0.246
0.9
0.5
(a)
−0.246
1
ap /a⊥ =
1
Transmission Coeff. T
Transmission Coeff. T
1
−0.213
−0.202
−0.180
−0.170
−0.186
(b)
2
ε
3
4
Fig. 5.12: Total transmission coefficient T for a fermionic collision as a function of the
dimensionless energy ε for the two cases of the system being initially in the
ground n = 0 (a) and first excited n = 1 (b) transverse states, for several
ratios of ap /a⊥ . We have used ω = 0.002.
86
1
W
Transition Probability W
nn’
33
0.8
W22
0.6
W11
W
00
0.4
W01
W02
W12
0.2
W03
W
13
W23
0
−0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05
ap /a⊥
Fig. 5.13: Transition probabilities Wnn0 in fermionic collisions as a function of ap /a⊥
for four open channels. ω = 0.002 and E = 0.0142 are employed.
Transmission Coeff. T
1
0.8
0.6
0.4
0.2
0
1
2
3
4
5 6
−V0
7
8
9 10
Fig. 5.14: Transmission T as a function of the depth −V0 of the potential (5.3) for
two distinguishable atomic species for ω = 0.002 for Ek = 0.0002. Units
according to (5.4).
5.3.3
Multichannel scattering of distinguishable particles
In this section we analyze the multichannel scattering of two distinguishable
particles in a harmonic trap with the same trap frequency ω1 = ω2 = ω allowing
the separation of the CM motion. Such a case corresponds to the different
atomic species confined by the same potential as it is the case, e.g., for two
different isotopes in the same optical dipole trap. The two-body scattering wave
function of the distinguishable atoms does not possess a well-defined symmetry
with respect to the reflection z → −z, i.e. both s- and p-state contributions
(f e and f o ) must be taken into account for the scattering amplitude. Fig. 5.14
shows our results for the transmission coefficient T in the single-mode regime
(0 < ε < 1), which is plotted as a function of the tuning parameter −V0 of the
interparticle interaction. In general the scattering process can not be described
by a single scattering length as or ap for this case. In regions with a negligible
p-wave contribution (see the regions in Fig. 5.1 with Vp → 0), T exhibits a
87
Scattering Amplitude
1
1+Re(fe00+fo00)
0.5
0
Im(fe00+fo00)
−0.5
−1
8.7 8.75 8.8 8.85 8.9 8.95
−V0
9
Fig. 5.15: Scattering amplitude as a function of the depth −V0 of the potential (5.3)
for two distinguishable atomic species for ω = 0.002 and Ek = 0.0002. Units
according to (5.4).
behavior similar to bosonic scattering. We observe the well-known s-wave CIRs
which lead to zeros of the transmission T at the positions as /a⊥ = 1/C and
tend to unity when as goes to zero (together with ap ). In regions, where as and
ap are comparable we observe the effect reported in Ref. [38]: remarkable peaks
o 2
e
| → 1, i.e., almost complete transmission
+ f00
of the transmission T = |1 + f00
in spite of the strong interatomic interaction in free space. This is the so-called
dual CIR: Quantum suppression of scattering in the presence of confinement
due to destructive interference of odd and even scattering amplitudes. Equally
o 2
e
| → 0 due to the interference of even and odd
+ f00
minima of T = |1 + f00
scattering amplitudes under the action of the transverse confinement can occur
o
e
= −1 − i while
+ f00
(see Fig. 5.14). Complete transmission corresponds to f00
o
e
total reflection corresponds to f00 +f00 = −1. Fig. 5.15 shows the corresponding
o
e
as a function of −V0 .
+ f00
amplitudes f00
Fig. 5.16 shows the total transmission coefficient versus −V0 in the threemode regime for ω = 0.002 and ε = 2.05. Similar to the single-mode regime,
when ap is negligible compared to as , T behaves analogously to the case of a
bosonic collision, and tends to unity (complete transmission) when as tends to
zero, while at the s-wave CIR position, it exhibits a minimum with a nonzero
value. For a lower degree of transversal excitation of the initial state, we encounter a larger transmission coefficient. For the same reasons as in the singlemode regime, we observe in the regions of V0 where the p-wave scattering length
ap is comparable to as , sharp peaks of T . However, in contrast to the singlemode regime we do not observe complete transmission, i.e., T 6= 1 (see, e.g., T
at V0 = −8.85 in Fig. 5.16 ).
Figure 5.17 presents the transmission coefficients as a function of the dimensionless energy ε for ω = 0.002. Figure 5.17(a) shows the results for several
ratios of as /a⊥ if |ap | is small compare to |as |. Here the system is initially
in the transversal ground state. The behavior of the transmission coefficient
is similar to the case of two-boson scattering [see Fig. 5.6(a)]. Figure 5.17(b)
88
Transmissin Coeff. T
1
0.8
0.6
0.4
1
2
3
4
5 6
−V
7
8
9 10
0
Fig. 5.16: Transmission coefficient T as a function of the depth −V0 of the potential
(5.3) for two distinguishable atomic species for ω = 0.002 and ε = 2.05
in the three-mode regime, being initially in transverse ground state (blue),
first excited state (red) and second excited state (black). Units according
to (5.4).
shows the transmission for the scattering of two distinguishable atoms together
with the results for bosonic-collision (blue) and fermionic-collisions (black), for
−V0 = −4.505 when both scattering length as and ap are large. In the limit
ε → 0 the behavior of the transmission for the distinguishable and the bosonic
case are very similar while in the vicinity of the energy of the p-wave shape
resonance ε ∼ 0.7, apart from a small shift to larger energies, we can find a
complete coincidence of the transmission behavior for the case of distinguishable and fermionic scattering. At the threshold energy ε = 1 the latter two
transmission curves cross. Figure 5.17(c) shows the transmission coefficient versus ε for −V0 = −4.505 initially occupying the ground state n = 0 (blue), the
first excited state n = 1 (black), the second excited state n = 2 (red), and the
third excited state n = 3 (green). In the limit εk → 0, T drops rapidly to zero.
Apart from the single mode regime, it is for εk >> 0 approximately constant
with T ≈ 1.
In Fig. 5.18 we present the transition probabilities Wnn0 as a function of −V0
for four open channels. For the regions of V0 where ap is negligible compared to
as , a very good agreement of the transmission behavior for distinguishable and
bosonic atoms is observed. In regions where as and ap are comparable, similar
to the fermionic case, we observe narrow and deep wells (for elastic collision)
and narrow as well as strongly pronounced peaks (for inelastic collision).
5.4 Summary and conclusions
We have analyzed atomic multichannel scattering in a 2D harmonic confinement.
Identical bosonic and fermionic scattering as well as scattering of distinguishable
atoms in traps with the same frequency for the different species have been
explored. Equal frequencies allows us to separate the c.m. and relative motion.
89
1
Transmission Coeff. T
(a)
0.8
0.6
ap /a⊥ < 0.1
0.4
as /a⊥ =
0
+0.52
+0.68
+1.72
−30.08
−0.54
−0.50
0.2
0
1
2
ε
3
4
1
(b)
Transmission Coeff. T
0.8
0.6
0.4
as /a⊥ = +0.131
0.2
ap /a⊥ = −0.246
0
0
1
2
ε
3
4
1
2
ε
3
4
1
(c)
Transmission Coeff. T
0.8
0.6
0.4
0.2
0
0
Fig. 5.17: Transmission coefficient T as a function of the dimensionless energy ε for
scattering of two distinguishable particles in a harmonic confinement ω =
0.002 (a) for several ratios as /a⊥ in the case where ap is negligible compared
to as , (b) for −V0 = −4.505 when both scattering lengths as and ap are
comparable [here the transmission coefficients for bosonic-collision (blue)
and fermionic-collision (black) are also provided], and (c) for −V0 = −4.505
when the system is initially in the ground transversal state (blue), in the
first excited state (black), in the second excited state (red), and in the third
excited state (green).
90
1
W00
W
Transition Probability Wnn’
11
0.8
W22
0.6
W
33
0.4
W03
W02
0.2
W13
0
0
1
2
W23
W01
W12
3
4
5
−V
6
7
8
9
10
0
Fig. 5.18: Transition probabilities Wnn0 for quasi-1D scattering of distinguishable particles in a harmonic trap ω = 0.002, as a function of −V0 for four open
channels and ε = 3.05. V0 is given in units of (5.4).
First we reproduced the well-known s-wave CIR for bosonic collision in the
single-mode regime [27, 28, 43, 38, 39, 40]. Next bosonic collisions in the multichannel regime including elastic and inelastic processes, i.e., transverse excitations and deexcitations have been investigated. Transmission coefficients as
well as transition probabilities for energies covering up to four open channels
are reported on. It is shown that the transmission coefficient as a function
of the scattering length exhibits a minimum at the CIR position. Except for
0 < as /a⊥ < 1/C in the first channel, the transmission curves show, with
varying energy, a minimum and accompanying well. For a single open channel
the value of the minimum of the transmission is zero. Increasing the degree of
transverse excitation leads to an increase of the transmission minima. A distinct
threshold behavior is observed at every ε = n.
For as /a⊥ < 0 the accompanying transmission well is more pronounced
compared to the case as /a⊥ > 0 and it is also deeper in the case of several open
channels. The position of the minimum is closer to the upper threshold again
when compared to the case as /a⊥ > 0. With increasing ratio as /a⊥ for either
as /a⊥ > 0 or as /a⊥ < 0, the corresponding transmission well becomes also
more pronounced, i.e., we encounter a deeper and narrower well for more open
channels and the energetic position of the minimum moves to the neighboring
upper threshold. Our results for the transition probabilities Wnn0 show that
the probability of remaining in the same state (elastic collision) is larger than
transitions into a different transversal state (inelastic collision). In the case of
elastic collisions (inelastic collisions) it has to go to unity (zero) as as tends to
zero. The inelastic transition probability Wnn0 increases as the numbers n or n0
increase while the elastic transition probability Wnn0 decreases.
Our next focus has been the multichannel scattering of fermions address91
ing up to four open channels in the waveguide. In the regime of a single open
channel we reproduced the well-known p-wave mapped CIR in the zero-energy
limit [32, 40]. We have analyzed the p-wave CIR dependence on the collision
energy ² and the trapping frequency ω. Consequently the multimode regime has
been explored by determining the behavior of the transmission as well as the
transition probabilities. The dependence of the transmission on the p-wave scattering length exhibits a minimum with zero transmission and an accompanying
well for the single-mode regime. Increasing the energy, this well becomes more
shallow (except for the single-mode regime),wider and its position is shifted to
the region of higher energies. In the multimode regime the transmission well
depends also on the degree of the initial transverse excitation. Increasing the
degree of initial transverse excitation the transmission well becomes shallower
and wider and the transmission is overall increased. For a fixed number of open
channels the transmission exhibits as a function of the energy a minimum for
some value of the p-wave scattering length. Increasing ap /a⊥ leads to a shift
of the transmission minimum to higher energies and an increasingly shallower
well. This holds except for the single mode regime in which the minimal value
is zero.
In contrast to the bosonic case, we do not observe a distinct threshold behavior. The transition probabilities Wnn0 for fermionic collisions show that elastic
collisions are more probable than inelastic ones. The probability Wnn for an
elastic process shows a well which becomes wider and shallower as the initial
population of the excited states n increases, while for an inelastic collision it
exhibits a peak which becomes smaller as the channel numbers n or n0 increase.
By varying the potential parameter V0 in the vicinity of the value V0 = −4.54,
we found a set of shape-resonances.
Finally we analyzed the multichannel-1D scattering of distinguishable atoms
in the trap. Here both s- and p-wave contributions have been taken into account. We have studied the transmission coefficient in the single-mode regime
and reproduced the CIR [27, 28, 43, 32] as well as the dual-CIR [38, 39, 40] corresponding to total reflection and transmission, respectively. In the multimode
regime the transmission versus −V0 shows except for the regions where ap is
comparable with as a behavior analogous to the situation of bosonic scattering.
We encounter T → 0 for as → 0. At the position of the CIR, T exhibits a
minimum with a nonzero value. The lower is the transverse excitation of the
initial state, the larger is the transmission coefficient. In regions of V0 where
the p-wave scattering length ap is comparable to as , T exhibits sharp peaks or
dips near the dual-CIRs due to comparable contributions of both s and p waves.
In contrast to the single-mode regime there is not a complete transmission at
the dual CIR points. Our results for the transmission coefficient versus energy
show that in the limit of zero longitudinal relative energy the distinguishable
atoms behave similar to bosons, while in the vicinity of the position of a shape
resonance, they behave similar to fermions. For larger energies ε >> 1 T is
close to unity. Finally we have analyzed the transition probability as a function
of −V0 . For values of V0 where ap is negligible compared to as we find an excellent agreement with the results for bosonic collisions. In regions where these
92
two scattering parameters are comparable, we observe, similar to the fermionic
case, sharp downward (for elastic collision) and upward (for inelastic collision)
peaks but with different values. We conclude with the general statement that
our multichannel scattering results in waveguides are of immediate relevance to
cold or ultracold atomic collisions in atomic waveguides or impurity scattering
in quantum wires.
93
94
Part III
ATOMIC RESONANCES IN A QUADRUPOLE
MAGNETIC TRAP
96
6. ATOMIC RESONANCES IN MAGNETIC QUADRUPOLE
FIELDS: AN OVERVIEW
Ultracold atomic gases offer a wealth of opportunities for studying quantum
phenomena at mesoscopic and macroscopic scales [64, 65, 66, 67, 68]. The interaction among the atoms as well as their behavior in external fields depends
crucially on their electronic and to some extent also on their nuclear properties. The majority of experiments in the ultra-low temperature regime have
been performed with alkali atoms. Their ground-state electronic structure is
characterized by the fact that all electrons but one occupy closed shells. The
latter, the so-called valence electron, is situated in the s-orbital of the outermost
shell. The coupling of the nuclear spin I to the total angular momentum of the
active electron via the hyperfine interaction causes a splitting of the electronic
energy levels into several branches which can be characterized by the quantum
number of the total spin F. Due to a vanishing orbital angular momentum this
hyperfine interaction consist exclusively of the coupling of the electronic spin
S = 12 , to the nuclear spin I. This leads to the two possibilities F = I ± S.
Inhomogeneous magnetic fields represent a very powerful and versatile tool
for trapping and confining of ultracold atoms. They allow to influence and control the dynamical properties of atoms and molecules, which result in a wealth
of new and often surprising phenomenon. In particular they are very well suited
for miniaturization via, e.g., so-called atom-chips [69, 70, 71]. Atom-chips are
surface mounted wires with a typical width of about a few micrometers, which
create a miniaturized landscape of inhomogeneous magnetic field. Depending
on the particular structure setup a wide variety of magnetic field landscape can
be created, enabling to trap ensemble of atoms very close to the chip surface
(∼ 10µm) where the magnetic field gradient is extremely large.
The basic idea behind the control of ultracold atoms via inhomogeneous
magnetic fields is very simple. The neutral atoms couple to the magnetic field
via their magnetic moment. In the case of enough weak gradient field, when the
spatial variations of the field is small enough, it can be considered homogeneous
over the size of an atom, therefore the atom can be assumed as a point-like
particle. For ground state atoms, this assumption is in general justified.
Assuming strong hyperfine interactions (compared to the magnetic interactions) the atomic magnetic moment is due to the total angular momentum
being composed of the electronic and nuclear angular momentum. Traditionally
an adiabatic approximation, which reduces the vector coupling of the magnetic
moment to the field to a potential term proportional to the magnitude of the
field, is then employed. However, in order to understand the basic physics of
(individual) atoms in magnetic traps it is likewise necessary to study the quantum dynamics of the Hamiltonian for the case of the vector coupling. A number
of investigations in this direction have been performed in the past.
Firstly fundamental configurations of static fields for the trapping of ground
state atoms have been explored in the late eighties and early nineties [72, 73].
Because of their pioneering work in trapping of ultracold atoms by magnetic
traps, S. Chu, C.N. Cohen-Tannoudji and W.D. Phillips were awarded the Nobel
prize of physics in 1997. Planar current geometries for microscopic magnetic
traps were investigated in Ref. [74]. Specifically the quadrupole field [75, 76], the
wire trap [77, 78, 79, 80] and the magnetic guide and Ioffe trap [81, 82, 83, 84, 85]
have been addressed. In these traps we typically encounter quantum resonances
with a certain lifetime instead of stationary stable quantum states. For example,
in Refs. [81, 82] the resonances of particles with spin 12 and 1 in a magnetic guide
have been determined via the phase shift of scattered waves. Alternatively the
complex scaling method can be used to study the widths and positions of the
resonances [83]. More recently [76] an extensive study of the quantum dynamics
of neutral spin 12 and spin 1 particles in a 3D magnetic quadrupole field has been
performed, including the derivation of an effective scalar Schrödinger equation
to describe and understand long-lived states possessing large angular momenta.
Very recently [86] a new class of short-lived resonances possessing negative
energies for the case of the 3d magnetic quadrupole field was found. These resonances, which have been overlooked in previous investigations [75, 76], originate
from a fundamental symmetry of the underlying Hamiltonian. In contrast to the
positive energy resonances, they are characterized by the fact that the atomic
magnetic moment is aligned antiparallel to the local direction of the magnetic
field, and their lifetimes decrease with increasing total magnetic quantum number. A mapping of the two branches of positive and negative energy states has
been derived [86].
In the above works, the hyperfine interaction has been assumed to be much
stronger than the interaction with the external inhomogeneous magnetic field
and therefore the electronic and nuclear angular momenta firstly provide a total
angular momentum which then interacts with the magnetic field. This physical picture truly holds for (alkali) atoms in their electronic ground state and
macroscopic as well as microscopic (atom chip) gradient fields. The atom is
consequently treated as a point particle with the total angular momentum F.
However in the case of small hyperfine interactions and/or strong gradient fields
(e.g., for electronically excited atoms) this simplification fails and an admixture
of different hyperfine states F due to the field interaction has to be expected.
In Ref. [87], the resonance of alkali-atoms in a magnetic quadrupole field have
been analyzed in the case for which both the hyperfine interaction and the field
interaction, have to be taken into account on equal level for the description of
the neutral atoms in the field.
In this part firstly we describe the magnetic quadrupole trap in details and
then we study resonances of neutral particles in this kind of trap. Finally we
study atomic hyperfine resonances of alkali atoms in a magnetic quadrupole
field.
98
6.1 The Magnetic Quadrupole Field
In ultra-cold atom experiment, the Zeeman interaction between magnetic moment of the atom and magnetic field is used to confine atoms to certain spatial
regions. Depending on the direction of the magnetic moment with respect to
the magnetic field this gives rise to either confining or nonconfining potentials.
Maxwell’s equation do not permit the occurrence of magnetic field maxima but
only minima in a source-free region. Therefore in order to ensure trapping one
has to focus on such atomic states for which the underlying potentials grows
with increasing magnetic field strength. Atoms in such states are referred to as
low-field seeker [69]. Several magnetic field configuration exhibit a local field
minimum, from which, the 3D quadrupole field is probably the most prominent.
Such field finds its application in virtually any ultra-cold atom experiment as
it is a main element of a magneto-optical trap MOT [88]. It is generated by
a so-called anti-Helmholtz configuration, which consists of a pair of axially assembled coils of radius R, separated by a distance 2D (Fig. 6.1). The two
coils are flown through by counter-propagating currents of equal magnitude I.
According to Biot and Savart’s law, the magnetic field B(r) resulting from the
infinitely thin coils is given by
Z
µ0 I
dr0 × (r − r0 )
B(r) =
4π
|r − r0 |3
Z
Z
∂r01 (s)
∂r02 (s)
0
0
µ0 I
µ0 I
∂s × (r − r1 (s))
∂s × (r − r2 (s))
+
=
ds
ds
4π r01 (s)
|r − r01 (s)|3
4π r02 (s)
|r − r02 (s)|3
(6.1)
In the second step, the current loops have been parameterized by the curves
r01 (s) and r02 (s). A suitable parametrization is given by




R cos(s)
R cos(s)
(6.2)
r01 (s) =  −R sin(s)  and r02 (s) =  R sin(s) 
−D
D
with 0 ≤ s ≤ 2π. The z−axis has been chosen to point along the symmetry
axis of the setup. In the vicinity of the origin we can expand the√integrand
in equation (6.1) in terms of the small dimensionless vector e
r = r/ D2 + R2
around r = 0. Carrying out the integration one obtains


x
e
2
3µ0 IDR 
ye  + o(|e
B(e
r) =
r2 |).
(6.3)
2(D2 + R2 )2
−2e
z
or


x
B(r) = b  y 
−2z
with
99
b=
3µ0 IDR2
2(D2 + R2 )5/2
(6.4)
cooled
atom cloud
vacuum
chamber
I
B
2D
R
magnetic coils
I
laser beam
Fig. 6.1: Experimental setup of the MOT.
The magnetic field gradient b is the only variable parameter characterizing the
quadrupole
√ field.
Since D2 + R2 is of macroscopic size (typically ∼ 10−1 m), it should be
clear from (6.3) that for regions of atomic or mesoscopic size (|r| ≤ 10−5 m)
around the trap center, Eq. (6.4) will be a fairly accurate description of the
actual geometry of the magnetic field B created by a pair of anti-Helmholtz
coils.
z
(b)
y
(a)
x
y
Fig. 6.2: The intersection of the magnetic field given in (6.4) along the (a) x − y- and
(b) y − z-plane.
The macroscopic setup which has been considered here is bulky and therefore
inflexible. As a result of the growing efforts for miniaturization and integration
the so-called atom chips have been developed. Atom chips are surface mounted
wires with a typical width of about a few micrometers, which create a miniatur100
ized landscape of inhomogeneous magnetic field. Depending on the particular
structure setup a wide variety of magnetic field landscape can be created, enabling to trap ensemble of atoms very close to the chip surface (∼ 10µm) where
the magnetic field gradient is extremely large. A 3D quadrupole field is achieved
by using a U-shaped wire. An expression for magnetic field configurations and
atom chips in more details can be found in Ref. [89].
6.1.1 Symmetry properties of the quadrupole magnetic field
The quadrupole magnetic field given in (6.4) in cylindrical polar coordinates
ρ, ϕ, and z reads
B(ρ, ϕ, z) = b(ρb
eρ − 2zb
ez )
(6.5)
Since B does not depend on the azimuthal angle ϕ, it is symmetric with respect to continuous rotation around the z-axis. Apart from this, B exhibits a
discrete mirror symmetry with respect to the x − y-plane. Figure 6.2 shows the
intersection of the magnetic field given in (6.4) along the x − y- and y − z-plane.
101
102
7. SCATTERING RESONANCES OF SPIN-1 PARTICLES IN A
MAGNETIC QUADRUPOLE FIELD
In this chapter, we investigate resonances of spin 1 bosons in a three-dimensional
magnetic quadrupole field. Complementary to the well-known positive energy
resonances it is shown that there exist short-lived, i.e., broad, negative energy
resonances. The latter originate from a fundamental symmetry of the underlying
Hamiltonian and are characterized by an atomic spin that is aligned antiparallel
to the local magnetic field direction. In contrast to the positive energy resonances the lifetimes of the negative energy resonances decreases with increasing
total magnetic quantum number. We derive a mapping of the two branches of
the spectrum.
7.1 The Hamiltonian
The Hamiltonian describing the motion of a point-like particle of mass M and
magnetic moment µ in a 3D magnetic quadrupole field B = b(x, y, −2z) reads
H=
p2
−µ·B
2M
(7.1)
For a spin-S-particle with magnetic moment µ = − g2 S, one obtains in atomic
units
1
H=
[p2 + bgM (xSx + ySy − 2zSz )]
(7.2)
2M
Here g is the g-factor of the particle. Performing the scale transformation
p̄i = (2M bµB gs )−1/3 pi
and x̄i = (2M bµB gs )1/3 xi
and thereupon omitting the bars one obtains
M (bgM )−2/3 H → H =
1 2
(p + xSx + ySy − 2zSz )
2
Therefore the energy level spacing scales according to
Hamiltonian (h̄ = 1) in spherical coordinates reads
H=
1
2/3
.
M (bgM )
1
∂2
2 ∂
L2
(− 2 −
+ 2 + r sin θK − 2r cos θSz )
2 ∂r
r ∂r
r
(7.3)
The scaled
(7.4)
Σx = U1 Py Pz
Px Py Pz Ixy U4
T U1 P z
T Px Pz Ixy U4
Σy = Px U2 Pz
Pz Ixy U4∗
T Px Py U2 Pz
T Py Pz Ixy U4∗
Σz = Px Py U3
Py Ixy U5
T Px U3
T Ixy U5
1
Px Ixy U5∗
T Py
T Px Py Ixy U5∗
Tab. 7.1: Discrete symmetries of the Hamiltonian (7.3). Each symmetry is composed
of a number of elementary symmetries which are listed in table II
Operator
T
Pxi
Ixy
i
U1 = e h̄ πSx
i
U2 = e h̄ πSy
i
U3 = e h̄ πSz
i π
U4 = U1 e− h̄ 2 Sz
i π
U5 = e h̄ 2 Sz
Operation
A → A∗
xi → −xi
x→y y→x z→z
Sx → Sx Sy → −Sy Sz → −Sz
Sx → −Sx Sy → Sy Sz → −Sz
Sx → −Sx Sy → −Sy Sz → Sz
Sx → −Sy Sy → −Sx Sz → −Sz
Sx → −Sy Sy → Sx Sz → Sz
Tab. 7.2: Set of discrete operations out of which all discrete operations of the Hamiltonian (7.3) can be composed.
Here K = cos ϕSx + sin ϕSy . For a spin-1-particle, which is the case we focus
on in the following except explicitly stated otherwise, we have




0 e−iϕ
0
1 0 0
1
0
e−iϕ  and Sz =  0 0 0 
K = √  eiϕ
2
iϕ
0
e
0
0 0 −1
7.2 Symmetries and Degeneracies
By analyzing the symmetry properties of the resulting Hamiltonian one obtains
16 discrete symmetries, which are listed in table 7.1. Each symmetry operation
is composed of a number of elementary operations given in table 7.2.
The discrete operation Ui , acts on the spin space only and their appearance
depends on the choice of the representation of the spin operators. General representations of Ui operations are given in table 7.2. The operations Pxi and Ixy
act on coordinate space only. These operations are unitary. The operation T
acts on both, the coordinate and the spin space, and is an antiunitary operation.
It is the conventional time reversal operation for spinless particles, and corresponds to complex conjugation. Its action on the coordinate xi , the momentum
Pi , the angular momentum Li , and the spin operators Si is given by
T Xi T −1 : Xi → Xi
T Pi T −1 : Pi → −Pi
T Li T −1 : Li → −Li
104
T Si T −1 : Sx → Sx
Sy → −Sy
Sz → Sz
(7.5)
The operations discussed above satisfy the relations
[Pxi , T ] = [T, Ui ] = [Ui , Pxi ] = [Pxi , Pxj ] = [Ui , Uj ] = 0
(7.6)
Ui2 = Px2i = T 2 = 1
(7.7)
and
The symmetry operations listed in table 1 are either unitary or anti-unitary.
The anti-unitary operations, involve the conventional time reversal operator T .
The symmetry operations shown in the table 7.1, formally possess the same
decomposition in terms of elementary operations for both fermions and bosons,
but however their algebraic structure are different. For instance for bosons we
have [Σi , Σj ] = 0, while in the case of fermions one finds {Σi , Σj } = 0
Apart from the discrete symmetries there is a continuous symmetry generated by Jz = Lz + Sz which is the z-component of the total angular momentum.
This is a consequence of the rotational invariance of the system around the
z-axis of the coordinate system. Due to [Jz , H] = 0 one can find energy eigenfunctions which are simultaneously eigenfunctions of Jz . For a spin-S-particle
they read in the spatial representation
|mi
(S)
=
S
X
CmS ei(m−mS )ϕ |mS i
(7.8)
mS =−S
P
with mS |CmS |2 = 1 where |mS i are the spin eigenfunctions with respect to Sz .
Exploiting the discrete and continuous symmetries of the system one can show
that there exists a twofold degeneracy in the energy spectrum of the Hamiltonian
(7.3) for m 6= 0. Let us consider | E, mi to be an energy eigenstate which is
simultaneously an eigenstate of Jz . Using the anti-commutator {Jz , Σx } = 0 we
have
Jz Σx | E, mi = −Σx Jz | E, mi = −mΣx | E, mi
(7.9)
Therefore the state Σx | E, mi is an eigenstate of Jz with eigenvalue −m. On
the other hand, because of the symmetry of the Hamiltonian Σx | E, mi is an
energy eigenstate with energy E and thus it can be identified with | E, −mi.
For m = 0 the states | E, mi and Σx | E, mi form no degenerate pair.
7.3 Numerical Approach
We are interested in the resonances of the Hamiltonian (7.4). In order to investigate the resonances we employ the method of complex scaling theory together
with linear variational method. To this end we employ a so-called Sturmian
basis set of the form
| n, l, mS im = Rn(ζ) (r)Ylm−mS (θ, ϕ)χS (mS ),
105
(7.10)
here the functions Ylm are the spherical harmonics. For fixed m the linear
m
variational combination of the basis functions ψn,l,m
, yields, per construction,
S
eigenstates of the Hamiltonian and Jz simultaneously. For expanding the radial
part we take the non-orthogonal set of functions
Rn(ζ) (r) = e−
ζr
2
Ln (ζr)
(7.11)
where Ln (x) are the Laguerre polynomials. The free parameter, ζ, has the
dimension of an inverse length and can be tuned to improve the convergence
behavior in different regions of the energy spectrum. It should be chosen such
that 1/ζ corresponds to the typical length scale of the states to be approximated.
7.4 Results
We shall see below that the spectrum consists of two well-separated parts: One
set of resonances localized in the negative energy region with short lifetimes
and a second set localized in the positive energy domain where the lifetimes are
much larger and can, at least in principle, become arbitrarily long. The positive
energy resonances have already been investigated in several works (see Refs.
[75, 76] and references therein) whereas, to our knowledge, no negative energy
resonances have been reported up to date.
7.4.1 Positive-energy resonances
In Fig. 7.1 we present the energies and decay widths for positive energy resonances for a spin-1-particle. For low total angular momentum [see figure 7.1(a)(b)] one observes the resonances to cover the area of a right triangle in the ² − Γ
plane. They are located on lines with similar negative slopes. This pattern becomes increasingly disturbed when considering resonances with higher energies
and small decay widths. For sufficiently large values of the angular momentum
m we observe a very regular pattern for their distribution in the ² − Γ plane,
still with a triangular boundary, but now the resonances are located on lines
possessing a positive slope [see figure 7.1(c)-(d)].
The decay width decreases exponentially with increasing value m for the angular momentum. The larger the angular momentum of a state the farther away
it is located from the center of the trap which is the zero of the magnetic field.
In the vicinity of the center spin-flip transition from, e.g., bound to unbound
states take place. With increasing angular momentum the resonances probe less
and less of this central region and become therefore increasingly stable.
For sufficiently large values of the angular momentum the wave functions
become localized far away from the center of the trap. They form concentric
circles around the z-axis. Since transitions to continuum states mainly occur
in the center this results in a significant increase of the lifetimes of the states.
Thus one could expect such states to be approximately describable as bound
states of a certain effective Schrödinger equation. Such an equation has been
106
0.8
0.3
0.6
Γ [a.u.]
Γ [a.u.]
0.4
0.2
0.1
0.4
0.2
(a)
0
2
(b)
3
4
ε [a.u.]
0
2
5
3
4
5
ε [a.u.]
6
7
−3
0.025
1.2
(c)
1
0.02
0.015
Γ [a.u.]
Γ [a.u.]
x 10
(d)
0.01
0.8
0.6
0.4
0.005
0
4
0.2
5
6
ε [a.u.]
7
0
7
8
8
9
ε [a.u.]
10
Fig. 7.1: Decay width and energies of the resonances possessing positive energies for
m = 0 (a), m = 1 (b), m = 10 (c) and m = 20 (d).
obtained in [76] by applying the unitary transformation
U1 = e−iSz ϕ
(7.12)
to the Hamiltonian (7.3) which in cylindrical coordinates yields
e = U † HU1
H
1
·
¸
1
∂2
1 ∂
1
∂2
2
=
− 2−
+ (Lz − Sz ) − 2 + ρSx − 2zSz
2
∂ρ
ρ ∂ρ ρ2
∂z
(7.13)
In this frame any explicit dependence on the azimuthal angle ϕ is removed.
Hence now Lz instead of Jz is conserved which is because of
U1† Jz U1 = Lz
(7.14)
Therefore we can replace Lz by its quantum number m and consider each Lz
subspace separately. The spatially dependent unitary transformation
U2 = e−iSy β
with sin β = √
ρ
ρ2 +4z 2
and cos β = √ −2z
2
ρ +4z 2
(7.15)
diagonalizes the spin-field interac-
tion term of (7.13), i.e.,
U2† (ρSx − 2zSz )U2 =
107
p
ρ2 + 4z 2 Sz
(7.16)
The derivatives result in additional terms, which gives rise to additional offdiagonal couplings. However, these terms are proportional to powers of z −1 as
ρ → 0 and ρ−1 as z → 0. Therefore they can be neglected far from the center
e 2 which allows for bound
of the trap. Considering only the component of U2† HU
solutions that we denote as |ψqb i we obtain the Schrödinger equation
"
#
∂2
1
∂2
m2
2mz
ρ2 + z 2
1p 2
− 2− 2+ 2 + p
+ 2
+
ρ + 4z 2 |φqb i
2
∂ρ
∂z
ρ
(ρ + 4z 2 )2
2
ρ2 ρ2 + 4z 2
= Eqb |φqb i
(7.17)
1
Here we have introduced the wavefunction |φqb i = ρ 2 |ψqb i. For all m there is
a remarkably good agreement between the exact resonance energies E and the
energies of the quasi-bound states Eqb [76]. The approaching performs better
and better with increasing values of the quantum number m. The angular
momentum term m2 /ρ2 together with the two consecutive terms in equation
(7.17) constitute a potential barrier which prevents a particle from entering the
vicinity around the center of the trap. We have already stated the off-diagonal
e 2 involve inverse powers of the spatial coordinates. Thus the
elements of U2† HU
matrix element for transition between bound and unbound solutions becomes
only significant if there is a sufficiently large overlap of the wave function with
the central region of the trap. This explain why the approximation performs
better for larger values of m.
We can now discuss the origin of the regular pattern formed by the resonance
positions which emerges for high values of the quantum number m. For larger m
the minimum and the associate well of the effective potential in the Schrödinger
equation (7.17) becomes more and more pronounced. The minimum is approximately located at the position (z0 = 0, ρ0 = (2m)2/3 ). The system now
becomes almost integrable, and consequently the states can be characterized by
their number of nodes in ρ- and z-direction which we designate by nρ and nz ,
respectively. States with higher number of nodes in ρ-direction posses larger
widths (figure 7.2). This is easily understood considering that such states are
elongated in the ρ-direction and the particles therefore posses higher oscillation
frequencies in that direction. Hence these states posses a higher probability to
penetrate the angular momentum barrier and undergo a transition to unbound
solutions (which takes place in the vicinity of the center of the trap). Unlike
that, states with small nρ are mainly elongated along the z-direction and thus
avoid contact with the trap center [76].
7.4.2 Negative-energy resonances
Figure 7.3 shows the energies and decay widths for the negative energy resonances. One immediately notices that they are arranged somewhat similarly
to the ones with positive energy but on lines with a negative slope. The pattern of their distribution becomes more regular for higher m values. Unlike the
positive energy resonances their decay width increases with increasing angular momentum. States with a larger angular momentum experience a stronger
108
−3
10
m = 20
(4,0)
Γ [a.u.]
(nρ, nz)
(3,0)
(2,0)
−4
10
(1,1)
(1,0)
−5
(0,0)
(0,1)
(0,2)
(2,1)
(1,2)
(0,3)
(3,1)
(2,2)
(1,3)
(0,4)
10
7
7.5
8
ε [a.u.]
8.5
9
Fig. 7.2: Resonances in the m = 20 subspace. The resonance states can be characterized by the two quantum numbers (nρ , nz ) which denote the number of
nodes in the ρ- and z-direction, respectively.
magnetic field and, as we shall show below, the magnetic moment of the atom
is parallel to the local direction of the field in case of a negative energy resonance. Consequently the atom feels an increasingly repelling force with respect
to the trap center if the total angular momentum increases. The latter leads to
correspondingly broader resonances.
7.4.3 Comparing the two classes of resonances
In Fig. 7.4 we present the expectation value Re(hS · B/|B|i) of the spin component which points along the local direction of the field as a function of the
energy for the two cases m = 0 and m = 10. For the negative energy resonances
this value is approximately −1 indicating that the spin is aligned opposite to
the local direction of the magnetic field and the magnetic moment is in the same
direction as the field (we have assumed g > 0). For positive energy resonances
the spin is parallel to the local field and the magnetic moment is antiparallel.
It is instructive to consider the average value of the radial coordinate r as
a function of the energy (see Fig. 7.5) . The resonances are distributed in an
area possessing the shape of a triangle. Again this pattern becomes increasingly
regular, when considering higher angular momenta [see Fig. 7.5(b)]. Re(hri)
increases with increasing absolute value of the energy which is in agreement with
our above argumentation with respect to the localization of the resonances.
7.4.4 Mapping among the two classes of resonances
The results on the negative and positive energy resonances suggest an intimate
relationship among the two. Indeed due to a complex symmetry, which we shall
derive in the following, there exists a mapping of the two classes of resonances.
The Schrödinger equation belonging to the Hamiltonian which results when
109
12
10
(a)
(b)
10
Γ [a.u.]
Γ [a.u.]
8
6
6
4
2
−3
−2.5
−2
−1.5
ε [a.u.]
−1
13
4
−3
−0.5
−2
−1.5
ε [a.u.]
−1
−0.5
(d)
15
11
Γ [a.u.]
Γ [a.u.]
−2.5
16
(c)
12
10
14
13
9
8
−3.6
8
−3.3
−3
ε [ a.u.]
−2.7
−2.4
12
−4.7 −4.5 −4.3 −4.1 −3.9 −3.7 −3.5
ε [a.u.]
Fig. 7.3: Decay width and energies of the resonances possessing negative energies for
m = 0 (a), m = 1 (b), m = 10 (c), and m = 20 (d).
applying the complex scaling to the Hamiltonian (7.4) reads
1
∂2
2 ∂
m2
(−e−i2η 2 − e−i2η
+ e−i2η 2 + eiη r sin θK − 2eiη r cos θSz )ψE,m
2
∂r
r ∂r
r
= EψE,m
(7.18)
Taking the complex conjugate and multiplying by e−i
2π
3
one obtains
π
π
π
π
∂2
2 ∂
m2
1
(−e−i2( 3 −η) 2 − e−i2( 3 −η)
+ e−i2( 3 −η) 2 + ei( 3 −η) e−iπ r sin θK ∗
2
∂r
r ∂r
r
π
2π
∗
∗
−2ei( 3 −η) e−iπ r cos θSz∗ )ψE,m
= e−i 3 E ∗ ψE,m
(7.19)
By transforming
Si → S̄i = e−iπ Si∗ = −Si∗
π
η → η̄ = − η
3
(7.20)
one can show that
1
∂2
2 ∂
m2
(−e−i2η̄ 2 − e−i2η̄
+ e−i2η̄ 2 + eiη̄ r sin θK̄ − 2eiη̄ r cos θS̄z )ψ̄Ē,m
2
∂r
r ∂r
r
= Ē ψ̄Ē,m
(7.21)
110
/ /
2
2
(a)
1
Re(〈S⋅B/|B|〉)
Re(〈S⋅B/|B|〉)
1
0
−1
−2
−3
/ /
(b)
0
−1
−2
−1
/ /
2
3
ε [a.u.]
4
5
−2
−4
6
−3
−2
/ /
5
ε [a.u.]
6
7
8
Fig. 7.4: Expectation values of the spin component along the local direction of the
magnetic field for the positive and negative energy resonances for m = 0 (a)
and m = 10 (b).
which is unitarily equivalent to (7.18) with the transformed energies and resonance wave functions
2π
ψE,m
E → Ē = e−i 3 E ∗
iη̄
∗
(eiη r, θ, ϕ, −ms )
→ ψ̄Ē,m (e r, θ, ϕ, ms ) = ψE,m
(7.22)
Therefore each resonance belonging to (7.18) with eigenvalue E possesses
a counterpart, i.e., a partner resonance with the eigenvalue Ē = e−i2π/3 E ∗ .
We therefore have a mirror symmetry of the underlying Hamiltonian the corresponding mirror line being placed at −π/3. While the positive energy continuum
rotates clockwise around its zero energy threshold by 2η its image rotates anticlockwise by 2η starting from the line defined by the polar angle − 2π
3 . When
a resonance is revealed at the angle −2η0 in the positive energy domain, its
image resonance for negative energies is revealed at the polar angle position
−2π/3 + 2η0 (figure 7.6).
/ /
(b)
(a)
Re(〈 r〉) [a.u.]
Re(〈r〉) [a.u.]
6
10
/ /
7
5
4
3
2
8
6
4
1
0
−3
−2
−1
/ /
2
3
ε [a.u.]
4
2
−4
5
−3
−2
/ /
5
6
ε [a.u.]
7
8
Fig. 7.5: Average of the radial coordinate r as a function of the energy for both positive
and negative energy resonances for m = 0 (a) and m = 10 (b).
111
Im(E)
Positive Energy
Resonances
Negative Energy
Continuum
...
Re(E)
0
2π
3
2η
π
3
...
2η
Psitive Energy
Continuum
Partner Resonances
Symmetry Line
Fig. 7.6: Mapping among the two classes of resonances. Each resonance possesses a
partner with the eigenvalue given by (7.22)
Employing the transformation (7.20), the expectation values of r and of the
spin-component along the local direction of the magnetic field S·B
|B| transforms
as follows
hri → eiπ/3 hri∗
¿
S·B
|B|
À
¿
iπ
→e
S·B
|B|
À∗
(7.23)
¿
=−
S·B
|B|
À∗
(7.24)
which are in good agreement with the results of our numerical calculations.
Finally we remark that although our study was performed for the specific case
of a spin-1-particle our conclusions hold for any boson with non-zero spin and
also for fermions.
7.5 Summary and Concludes
In summary we observe, to our knowledge, for the first time negative energy
resonances of spin-1-bosons in a 3D quadrupole magnetic trap. The overall
spectrum is arranged in two disconnected parts each of which contains exclusively resonances with positive and negative energies. The latter are exclusively
of short-lived character and the spin is antiparallel to the local direction of the
magnetic field while for the former the spin is parallel to the field. As the total
angular momentum of the boson increases, the decay width of a negative energy
resonance state increases while for a positive energy resonance it decreases (assuming a typical laboratory gradient field 5T /m and the species 87 Rb the order
of magnitude for the lifetimes of the negative energy resonances is microseconds). A property of the complex scaled Hamiltonian has been established
which allows to map the two branches of the spectrum.
112
8. ATOMIC HYPERFINE RESONANCES IN A MAGNETIC
QUADRUPOLE FIELD
In the previous chapter we assumed that the hyperfine interaction is much
stronger than the interaction with the external inhomogeneous magnetic field
and therefore the electronic and nuclear angular momenta firstly provide a total
angular momentum which then interacts with the magnetic field. This physical picture truly holds for (alkali) atoms in their electronic ground state and
macroscopic as well as microscopic (atom chip) gradient fields. The atom is
consequently treated as a point particle with the total angular momentum F.
In the present work we study the case for which both interactions, the hyperfine
and the field interaction, have to be taken into account on equal level for the
description of the neutral atoms in the field. This case is primarily of principal
interest but is expected to describe the magnetized hyperfine properties of systems with very small hyperfine interactions and/or strong gradient fields, such
as electronically excited atoms. In the latter case an admixture of different hyperfine states F due to the field interaction has to be expected. In Ref. [90] the
possibility to tune the hyperfine splitting by dressing the electronic energy levels
by a microwave has been demonstrated. Hence, by this method it is thinkable to
achieve a scenario in which the magnetic and the hyperfine-interaction become
comparable even for ground state atoms.
We focus on atoms possessing a single active valence electron with spin S = 12
and a nucleus with spin I = 32 in a three dimensional magnetic quadrupole field
and describe, as indicated, the combined influence of the hyperfine interaction
and the Zeeman effect on the quantum resonances.
8.1 Hamiltonian
Taking into account the hyperfine interaction, the Hamiltonian describing the
motion of an atom with mass M and electronic and nuclear spin S and I,
Σx = U1 Py Pz
Px Py Pz Ixy U4
T U1 P z
T Px Pz Ixy U4
Σy = Px U2 Pz
Pz Ixy U4∗
T Px Py U2 Pz
T Py Pz Ixy U4∗
Σz = Px Py U3
Py Ixy U5
T Px U3
T Ixy U5
1
Px Ixy U5∗
T Py
T Px Py Ixy U5∗
Tab. 8.1: Discrete symmetries of the Hamiltonian (8.9). Each symmetry is composed
of a number of elementary operations which are listed in table 8.2.
Operator
T
Pxi
Ixy
i
U1 = e h̄ π(Sx +Ix )
Operation
A → A∗
xi → −xi
x→y y→x z→z
Sx → Sx Sy → −Sy Sz → −Sz
Ix → Ix Iy → −Iy Iz → −Iz
Sx → −Sx Sy → Sy Sz → −Sz
Ix → −Ix Iy → Iy Iz → −Iz
Sx → −Sx Sy → −Sy Sz → Sz
Ix → −Ix Iy → −Iy Iz → Iz
Sx → −Sy Sy → −Sx Sz → −Sz
Ix → −Iy Iy → −Ix Iz → −Iz
Sx → −Sy Sy → Sx Sz → Sz
Ix → −Iy Iy → Ix Iz → Iz
i
U2 = e h̄ π(Sy +Iy )
i
U3 = e h̄ π(Sz +Iz )
i π
U4 = U1 e− h̄ 2 (Sz +Iz )
i π
U5 = e h̄ 2 (Sz +Iz )
Tab. 8.2: Set of discrete operations out of which all discrete symmetry operations of
the Hamiltonian (8.9) can be composed.
respectively, reads in a magnetic quadrupole field
H=
p2
+ HHF + HB
2M
(8.1)
where HHF describes the hyperfine interaction between the outermost single
valence electron and the nucleus
HHF = AI · S
(8.2)
Here A is a constant which for an s-electron is given by [93]
A=
2 µ0
gS gI µB µN |ψs (0)|2
3 h̄2
(8.3)
gS and gI are the g-factors of the electron and the nucleus, respectively, and
ψs (0) is the value of the valence s-electron wave function at the nucleus. µB =
eh̄
eh̄
2me and µN = 2mp are the Bohr magneton for the electron and the proton,
respectively. HB accounts for the interaction of the magnetic moment of the
electron and the nucleus with the magnetic field
HB = −µe · B − µn · B
(8.4)
where µe = −µB gS S and µn = µN gI I. Conventionally one introduces the ratio
α=−
µN gI
me
≈−
µB gS
mp
(8.5)
and writes
HB = µB gS (S + αI) · B
114
(8.6)
The vector of the three-dimensional quadrupole magnetic field is given by
B(r) = b(x, y, −2z)
(8.7)
Consequently the total Hamiltonian becomes
1
[p2 + 2M bµB gS (x(Sx + αIx ) + y(Sy + αIy ) − 2z(Sz + αIz ))
2M
+2M AI · S]
H=
(8.8)
Performing the scale transformation
p̄i = (2M bµB gS )−1/3 pi
and x̄i = (2M bµB gS )1/3 xi
and thereupon omitting the bars one obtains
H=
1 2
[p + x(Sx + αIx ) + y(Sy + αIy ) − 2z(Sz + αIz ) + βI · S]
2
(8.9)
1
with β = bµBAgS . The energy is now measured in units of M
(2M bµB gS )2/3 . For
convenience and in anticipation of the forthcoming discussion we transform the
Hamiltonian to a spherical coordinate system, i.e., (x, y, z) → (r, θ, φ). Writing
the momentum operator explicitly and using atomic units we obtain
H
1
∂2
2 ∂
L2
[− 2 −
+ 2 + r sin θ cos ϕ(Sx + αIx) + r sin θ sin ϕ(Sy + αIy )
2 ∂r
r ∂r
r
− 2r cos θ(Sz + αIz ) + βI · S]
(8.10)
=
8.2 Symmetries and Degeneracies
Let us now study the symmetry properties of the Hamiltonian (8.9) in order
to gain first insights into the resonance spectrum of the system. We obtain
16 discrete symmetry operations which are listed in table 8.1. Each symmetry
operation is composed of a number of elementary operations given in table 8.2
along with the general representation of Ui operations.
Apart from the discrete symmetries given in table 8.1 there is a continuous
symmetry group generated by Jz = Lz + Fz which is the z−component of the
total angular momentum of the atom. Here Lz and Fz are the z−component of
the orbital angular momentum and the total spin, respectively. With the same
argument as the previous chapter one can find a two degeneracy in the energy
spectrum of the Hamiltonian (8.9) for m 6= 0. The general result for an atom
with electron spin S and nuclear spin I reads
| mi(I,S) =
S,I
X
m
Cm
ei(m−mI −mS )ϕ | mI i⊗ | mS i
I ,mS
mS =−S
mI =−I
with
P
mI ,mS
m
| Cm
|2 = 1.
I ,mS
115
(8.11)
(a)
Γ [a.u.]
Γ [a.u.]
5
4
3
−626.5
−626.2
ε [a.u.]
(b)
1
0.6
0.2
−625.5
−625.9
−625
ε [a.u.]
−624.5
x 10
Γ [a.u.]
(c)
5
−622.5
ε [a.u.]
−621.5
0
Γ [a.u.]
10
−2
10
−4
10
−6
10
0
//
(e)
10
(d)
1
0
−1
//
−626 −625 −623 −622
ε [a.u.]
Re(〈 F2 〉) [a.u.]
0
−623.5
Re(〈 F⋅ B / |B|〉) [a.u.]
−3
9
20
//
3
2
(f)
1
//
−626 −625 −623 −622
ε [a.u.]
m
Fig. 8.1: Decay width and energies of the resonances for β = 1000, m = 10 for mF = 1
(a), mF = 0 (b) and mF = −1 (c). (d) Expectation values of the total spin
component along the local direction of the magnetic field for the three sets
of resonances. (e) Decay width as a function of the quantum number m
for the energetically lowest state with mF = −1 (square), the energetically
highest state with mF = 0 (triangle) and the energetically highest state with
mF = +1 (star). (f) Expectation value of the total spin squared as a function
of the energy.
116
2
10
1
T [s]
10
0
10
−1
10
−2
10
2
10
4
10
6
β [a.u.]
10
8
10
Fig. 8.2: The lifetime of the energetically lowest long-lived resonance state for m = 25
as a function of the parameter β (Zeeman regime).
8.3 Numerical Approach
The Hamiltonian (8.9) does not support bound states. In order to calculate
the energies and decay widths of the scattering wave functions we employ the
complex scaling method in conjunction with the linear variational method. To
this end we employ a so-called Sturmian basis set of the form
| n, l, mI , mS im = Rn(ζ) (r)Ylm−mI −mS (θ, ϕ)χI (mI )χS (mS ),
(8.12)
here the functions Ylm are the spherical harmonics. For fixed m the linear varim
ational combination of the basis functions ψn,l,m
, yields, per construction,
I ,mS
eigenstates of the Hamiltonian and Jz simultaneously. For expanding the radial
part we take the non-orthogonal set of functions Rnζ (r) which has been introduced in the previous chapter. Using the basis set (8.12) all matrix elements of
the Hamiltonian (8.10) can be calculated analytically.
8.4 Results
Let us now discuss the results we obtained while studying an atom with hyperfine interaction in a magnetic quadrupole field. We present the resonance
energies and decay widths for different values of the field gradient. The resulting spectrum consist of several well-separated parts. Concerning the resonance
positions one can distinguish three regimes, each of which reveals individual
characteristics: the weak, the intermediate, and the strong gradient regimes. In
the weak gradient regime, the Zeeman term HB is very small compared with
the hyperfine interaction HHF and only slightly perturbs the zero-field eigenstates of H. In this case the atom, being primarily in its hyperfine ground state
117
3
(a)
5
Γ [a.u.]
Γ [a.u.]
5.5
4.5
3.5
−32.6
−3
−32.4
ε [a.u.]
1
−31.265 −31.26 −31.255 −31.25
−32.2
ε [a.u.]
Re(〈F⋅B/|B|〉) [a.u.]
x 10
4
Γ [a.u.]
2
1.5
4
5
(b)
2.5
(c)
3
2
1
0
−29.5
−29
ε [a.u.]
(d)
1
0
−1
−2
−33
−28.5
//
2
−32
Γ [a.u.]
Re(〈F2〉) [a.u.]
10
−10
10
0
(e)
5
10
15
20
m
−29
−28
(f)
2
1
−33
25
ε [a.u.]
//
3
0
//
−31
−32
//
−31
ε [a.u.]
−29
−28
Fig. 8.3: Decay width and energies of the resonances for β = 50, m = 10 for mF = 1
(a), mF = 0 (b) and mF = −1 (c). (d) Expectation values of the total
spin component along the local direction of the magnetic field for the three
sets of resonances. (e) The decay width of the energetically lowest state
with mF = −1 (square), the decay width of the energetically highest state
with mF = 0 (triangle), the decay width of the energetically highest state
with mF = +1 (star), as a function of the quantum number m. (f) The
expectation value of the total spin squared as a function of the energy.
118
2
10
0
10
−2
T [s]
10
−4
10
−6
10
−8
10
30
50
70
90
β [a.u.]
110
150
190
Fig. 8.4: The lifetime of the energetically lowest long-lived resonance state for m = 25
as a function of the parameter β (intermediate regime).
F = I − 12 , remains in this manifold and behaves approximately like a neutral
particle of spin F with the g-factor
gF ' gS
F (F + 1) − I(I + 1) + S(S + 1)
2F (F + 1)
(8.13)
This regime is also called the Zeeman regime. In the intermediate gradient regime the Zeeman and the hyperfine interactions are of the same order of
magnitude, and the atom in the ground state may represent a significant admixture of different hyperfine states F . Finally in the strong gradient regime
the Zeeman term dominates the hyperfine energies at least for sufficiently large
distances from the coordinate origin. In this case the spin component of the
electron along the local direction of the magnetic field is almost conserved and
as we will see, the resonance positions in the complex energy plane are grouped
according to different values of its quantum number mS . This regime is also
called the Paschen-Back Regime.
We will analyze the average values of the components of the spins which
point along the local direction of the field as a function of the energy. This
enable us to explain different sets of resonances in each regime. We also discuss
in this section the dependence of the decay width of a resonance state on its
angular momentum as well as on the field gradient.
8.4.1
Resonance positions in the Zeeman regime
For large values of β (β > 200) we are in the Zeeman regime. In Fig. 8.1(a)-(c)
we present the energies and decay widths for β = 1000 and m = 10, for an atom
with nuclear spin I = 23 being in its hyperfine ground state. The resonances are
localized in the negative energy region and their distribution consists of three
well-separated parts. They can be classified according to the expectation value
119
mF = Re(hF · B/|B|i) which is the projection of the total spin onto the local
direction of the magnetic field. There are long-lived resonances which can be
identified with mF = −1, and two sets of resonances with shorter lifetimes whose
spin projections are mF = 0, +1. The values of mF are shown in the panel d.
For mF = −1 one observes the resonances to be located on lines with similar
slopes covering the area of a right triangle in the ² − Γ plane. For mF = 0 and
mF = +1 one immediately notices that both sets are arranged on lines with an
infinity and a negative slope respectively. In Fig. 8.1(e) we present decay width
of the energetically lowest state of the long-lived resonances (the resonances
with mF = −1) as a function of the angular momentum m. The decay width
decreases exponentially with increasing value of m. We also present the decay
width of the energetically highest state of the two sets of short-lived resonances
as a function of the quantum number m. Unlike the case mF = −1, the decay
width of resonances with mF = +1 and mF = 0 increases with increasing
angular momentum (for an explanation of this behavior see [76]). Fig. 8.1(f)
shows the expectation value of the total spin squared, Re(hF2 i) as a function
of the energy for the same parameter values. For all resonance states, the value
is approximately +2 which corresponds to F = 1, i.e., the atom behaves like a
spin-1 particle.
In Fig. 8.2 we present the lifetime of the energetically lowest long-lived
resonance state (mF = −1) for m = 25 as a function of the parameter β.
The lifetime decreases as the hyperfine parameter β decreases, i.e., the gradient
field b increases. Performing a line fit we find the dependence T (s) ≈ 8.3 ×
10−4 β 2/3 [a.u.].
8.4.2
Resonance positions in the intermediate regime
Let us now focus on the intermediate regime covering the values 0.2 < β < 200,
where the Zeeman and hyperfine interactions become comparable. We observe
three different types of behavior. For 40 < β < 200, the resonance spectrum for
an atom being in the hyperfine ground state, is localized in the negative energy
domain and consists of three well-separated parts to which we can assign the
values mF = −1, 0, +1. (see Fig. 8.3 for m = 10 and β = 50). In Fig.
8.3(c) we present the decay width of the lowest energy state of the long lived
resonances as well as the decay width of the highest energy state of the two
sets of short-lived resonances as a function of the quantum number m. For
the two short-lived states, following the same reasoning as above, the width
increases when m increases, while for the long-lived states it decreases. Fig.
8.3(f) shows the expectation value of the total spin squared, Re(hF2 i) of the
three sets of resonances as a function of energy. Note that the atom still behaves
approximately like a spin-1 particle. However, in contrast to the Zeeman regime
(see Fig. 8.2), the lifetime of the long-lived states (mF = −1) increases when
the hyperfine parameter β decreases, i.e., the field gradient b increases (see Fig.
8.4).
For β < 40 this value decreases very rapidly when β decreases. In this
case the resonance states which correspond to higher hyperfine levels F = 2
120
0
Re(〈 F ⋅ F〉) [a.u.]
10
Γ [a.u.]
(a)
−5
10
−10
10
−15
6
8
10
12
ε [a.u.]
Re(〈F⋅ B /|B|〉) [a.u.]
10
14
6.5
(b)
6
5.5
5
4.5
6
8
10
ε [a.u.]
12
14
2 (c)
1
0
−1
6
8
10
ε [a.u.]
12
14
Fig. 8.5: (a) Decay width and energies of the long-lived resonances with higher hyperfine quantum number F = 2 for β = 19 and m = 25. (b) The expectation
value of the total spin squared as a function of the energy. (c) Expectation
values of the total spin component along the local direction of the magnetic
field as a function of the energy.
are more stable, and are localized in the positive region of the spectrum.
Fig. 8.5(a) shows the decay widths and energies of the resonances possessing
positive energies for β = 19. We observe four different curved and triangular
shaped regions with distinct classes of resonances. In Fig. 8.5(b) we present
the expectation values of the total spin squared, Re(hF2 i), of these resonances
as a function of the energy. The values range from approximately 5.5 to 6.0
indicating a substantial conservation of F . Fig. 8.5(c) shows the expectation
values of the total spin component along the local direction of the field for the
resonances of Fig. 8.5(a). The resonances are divided into four well-separated
parts corresponding to mF = +2, +1, 0, −1. Resonances with mF = −2 have
very short lifetime and are not shown. For β < 3 the non-conservation of F
becomes even more explicit [see Fig. 8.6(a) for β = 1]. In Fig. 8.6(b)-(c) we
present the respective decay widths and energies. The resonances which are
localized in the negative energy region have short lifetimes and are divided into
four subgroups. Each group lies on a line with negative slope, the four slopes
121
/ /
Re(〈F 〉) [a.u.]
7
(a)
2
5
4
7
6
3
2
−2
/ /
2
ε [a.u.]
4
5
−3
6
−2
Re(〈S⋅B/|B|〉) [a.u.]
Γ [a.u.]
10
−4
10
ε [a.u.]
5
6
−1.5
−1
/ /
1
0.5
−2
ε [a.u.]
(d)
0
−1
−2
/ /
2
ε [a.u.]
4
6
−1
10
(e)
1
(f)
T [s]
Re(〈I⋅B/|B|〉) [a.u.]
4
/ /
2
−2.5
−0.5
(c)
3
(b)
8
Γ [a.u.]
6
−2
10
0
−1
−2
−2
/ /
−3
2
ε [a.u.]
4
6
10
0
1
2
β [a.u.]
3
Fig. 8.6: (a) Expectation value of the total spin squared as a function of the energy.
Decay width and energies of the resonances with mS = − 21 (b), and mS = + 12
(c). (d) Expectation values of the electronic spin component along the local
direction of the magnetic field. (e) Expectation values of the nuclear spin
component along the local direction of the magnetic field. (f) Lifetime of
the energetically lowest resonance state of the positive energy domain of the
spectrum as a function of β. The data have been calculated for m = 10 and
β = 1.0.
122
being very similar. Moreover, the resonances investigated form subgroups on
these lines. The positive energy resonances have much longer lifetimes and cover
an area of approximately triangular shape in an irregular manner, i.e., no pattern
is visible. In Fig. 8.6(d) the corresponding expectation values of the electronic
spin component along the local direction of the magnetic field, Re(hS · B/|B|i)
is shown. For the negative energy resonances this value is approximately -0.5
indicating that the spin is aligned opposite to the local direction of the magnetic
field, while for positive energies the spin is parallel to the magnetic field. Fig.
8.6(e) shows the expectation values of the nuclear spin component along the
local direction of the field. In the negative energy domain of the spectrum the
pattern divides into four parts with values mI = − 32 , − 12 , + 12 , + 23 , while in the
positive energy region, the pattern is strongly disturbed and mI is not conserved.
Fig. 8.6(f) presents the lifetime of the energetically lowest long-lived state as
a function of β. This value increases when β decreases, i.e., the gradient field
b increases. Our results show that for the short-lived state, i.e., the negative
energy resonance state, the width increases when m increases, while for the
long-lived state, i.e., the positive energy resonance state it decreases.
8.4.3 Resonance positions in the hyperfine Paschen-Back regime
The hyperfine Paschen-Back regime includes β < 0.2. In case the Zeeman term
dominates the hyperfine energy, it is natural to decompose the Hamiltonian
according to H = H0 + H1 where H0 = 12 (p2 + xSx + ySy − 2zSz ) describes the
motion of a spin 21 particle in the magnetic field, and H1 = α(xIx +yIy −2zIz )+
βI · S perturbs the eigenstates of H0 . The spectrum consists of two parts: again
we have one set of resonances localized in the negative energy region with short
lifetimes and a second set localized in the positive energy domain possesses
much larger lifetimes. In Fig. 8.7(a)-(b) we present the energies and decay
widths of resonances for β = 0.1 and m = 10. The negative energy resonances
are arranged on lines with a negative slope. The positive energy resonances
cover an area of triangular shape, some of them being located on straight lines.
Fig. 8.7(c) shows the expectation value Re(hF2 i) of the square of the total
spin for m = 10 and β = 0.1. This value is almost 4.5 which again indicates
that the total spin quantum number F is not conserved. In Fig. 8.7(d) the
respective expectation value Re(hS · B/|B|i) of the electronic spin component
which points along the local direction of the magnetic field is presented. For
the positive energy resonances this value is approximately +0.5 indicating that
the spin is aligned parallel to the local direction of the magnetic field while for
negative energy resonances it is antiparallel. The nuclear spin component along
the local field is not conserved, however it is instructive to consider its average
value, Re(hI · B/|B|i), as a function of the energy. For large values of m [see
Fig. 8.7(e) for m = 10] the resonances are grouped into four regular subparts
for both the positive and negative energy domain.
We remark that the decay widths of the energetically lowest states of the
positive energy resonances [see Fig. 8.7(b)] show again an exponential decaying
behavior as a function of the quantum number m whereas the decay widths of the
123
−3
x 10
8
3
Γ [a.u.]
Γ [a.u.]
(a)
7
6
(b)
2
1
5
−2
Re(〈F2〉) [a.u.]
5
ε [a.u.]
//
(c)
4.5
3
1
0.5
3.5
ε [a.u.]
//
4
4.5
3.5
4.5
(d)
0
−0.5
//
4
−2.5 −1.5 2.5
Re(〈I⋅B/|B|〉) [a.u.]
0
2.5
−1.8 −1.6 −1.4
Re(〈S⋅B/|B|〉) [a.u.]
−2.2
0.2
ε [a.u.]
//
3.5
4.5
//
−1
−2.5 −1.5
2.5
ε [a.u.]
(e)
0.1
0
−0.1
−0.2
−2.5 −1.5
//
2.5
ε [a.u.]
3.5
4.5
Fig. 8.7: Decay width and energies of the resonances for β = 0.1, m = 10 with mS =
− 21 (a) and mS = + 12 (b). (c) The corresponding expectation value of the
total spin squared as a function of the energy. (d) Expectation values of
the electronic spin component along the local direction of the magnetic field
for the two sets of resonances. (e) Expectation values of the nuclear spin
component along the local direction of the magnetic field.
124
energetically highest negative energy resonance states increases with increasing
m. The lifetime of the energetically lowest long-lived resonances for m = 25 as
a function of β can be fitted according to T (s) ≈ 1.1 × 10−1 β 2/3 [a.u.].
8.5 Summary
We have investigated the resonant quantum properties of an atom with a single
active valence electron taking into account its hyperfine structure (I = 23 ) in a
3D magnetic quadrupole field. The underlying Hamiltonian possesses 16 discrete
symmetry operations as well as a continuous unitary symmetry generated by
the conserved quantity Jz = Lz + Fz . Exploiting these symmetries we have
found a twofold degeneracy of any energy level with m 6= 0.
We have calculated and analyzed the energies and decay widths of the resonance states of the Hamiltonian employing the complex scaling approach and a
Sturmian basis set. With respect to the resonance position one can distinguish
essentially three regimes. In the weak gradient regime, the Zeeman term is very
small compared with the hyperfine interaction and only slightly perturbs the
zero-field eigenstates of the Hamiltonian. In this case the atom in the hyperfine
ground state behaves approximately like a particle of spin 1, and the resonance
states are grouped into three well-separated parts, corresponding to three different directions of the total spin with respect to the local direction of the field.
The resonance states with the total spin being perpendicular (mF = 0) or parallel (mF = 1) to the local field possess short lifetimes, while the resonances for
which the total spin is antiparallel to the field (mF = −1) possess much more
longer lifetimes. As the total angular momentum of the atom increases, the
decay width of a short-lived resonance state increases while for a long-lived resonance state it decreases. We have calculated the lifetimes of some low energy
state belonging to the class of long-lived resonances as a function of the scaled
hyperfine parameter β, showing that if β increases so does the lifetime.
In the intermediate regime the Zeeman and hyperfine interactions are comparable. For a sufficiently weak field gradient the atom in the hyperfine ground
state behaves approximately like a spin-1 particle, and the resonances are arranged somehow similar to the Zeeman regime but in contrast to the Zeeman
regime its long-lived resonance states become more stable when the gradient
field increases and/or β decreases. For stronger field gradients and/or smaller
values of the parameter β the lifetime decreases rapidly. In this case resonances
which correspond to the higher hyperfine level, F = 2, are more stable. For even
smaller values of β, the non-conservation of F becomes remarkable. In this case
the electronic spin component along the local direction of the magnetic field is
almost conserved. A resonance which is localized in the negative energy region
of the spectrum possesses a short lifetime. In such a state the electronic spin is
aligned opposite to the local direction of the magnetic field. The positive energy
resonances possess a much longer lifetime with their electronic spin being parallel to the magnetic field. We have studied the dependence of the resonances
on the total angular momentum of the atom. For a short-lived state the width
125
increases with increasing m, while for a long-lived state it decreases. For the
latter we also investigated the dependence on the parameter β. Our results
show that the lifetime of this state increases with decreasing values of β.
In the strong gradient regime, the Zeeman term dominates the hyperfine
energy. Here the component of the electronic spin along the local direction of the
field is almost conserved. The energy spectrum is arranged into two disconnected
parts each of which contains exclusively resonance states with negative and
positive energies. The former are of short-lived character and the electronic
spin is antiparallel to the local direction of the magnetic field. The latter possess
much longer lifetimes and the spin of the electron is parallel to the field. As the
total angular momentum of the atom increases, the decay width of a negative
energy resonance state increases while for a positive energy resonance state it
decreases. We have calculated the lifetimes of the energetically lowest resonance
states possessing a positive energy as a function of the parameter β. Similar to
the weak gradient regime, it decreases with decreasing β.
126
APPENDIX
128
A. SCATTERING THEORY IN FREE SPACE: SINGLE-MODE
REGIME
Let’s consider a single spinless particle in a fixed potential. Thus H is the space
L2 (R3 ), with wave functions ψ(x) ≡ hx|ψi depending on a single variable x;
and the Hamiltonian has the form H = H 0 + V , where H 0 = p2 /2m, is the
Hamiltonian of a free particle, and V is the potential. We shall suppose that
the potential is local; that is, that V is a function of the particle’s position only.
Let us suppose that the orbit U (t)|ψi describes the evolution of some scattering experiment. This means that when followed back to a time well before
the collision, U (t)|ψi represents a wave packet that is localized far away from
the scattering center and, therefore, behaves like a free wave packet. Now, the
0
motion of a free particle is given by the free evolution operator U 0 (t) ≡ e−iH t ,
and we therefore expect that as t → −∞,
U (t)|ψi−−−−−−−−−→ U 0 (t)|ψin i
t → −∞
(A.1)
for some vector |ψin i. Similarly, after the collision the particle moves away again
and we expect that
U (t)|ψi−−−−−−−−−→ U 0 (t)|ψout i
t → +∞
(A.2)
for some vector |ψout i. This two limits are analogous to the classical limits (2.1)
and in analogy with the classical terminology we shall call the asymptotic free
orbits of (A.1) and (A.2) the in and out asymptotes of the actual orbit U (t)|ψi.
The results of scattering theory will certainly not hold for all possible potentials. For example, if V (r) does not fall off sufficiently fast as r → ∞, the
particle will not behave like a free particle as it moves far away.
Unfortunately, the conditions under which some of the principal results have
been proved are quite complicated; different proofs use different conditions.
Further, a set of conditions that is both sufficient and necessary for all results is
not known. However, the principal results of scattering theory hold for a wide
class of “reasonable” potentials, but definitely excluding the Coulomb and any
attractive potentials more singular than r−2 at the origin. In the continue we
assume that the potential satisfies all the conditions which are necessary for the
results which we discuss.
Orbit U(t)|ψ〉
Orbit U(t)|ψ〉
|ψ〉=Ω+|ψin〉
|ψ〉=Ω−|ψout〉
Out Asymptote U0(t)|ψ 〉
out
In Asymptote U0(t)|ψ 〉
in
Fig. A.1: Classical representation of roles of the Møller operators.
A.1 The Asymptotic Condition
Every vector in H (labelled by |ψin i or |ψout i as appropriate) does represent
the asymptote of some actual orbit; that is , for every |ψin i in H there is a |ψi
such that
lim U (t)|ψi − U 0 (t)|ψin i = 0
t→−∞
and likewise for every |ψout i in H as t → +∞.
The asymptotic condition guaranties that any |ψin i in H is in fact the in
asymptote of some actual orbit U (t)|ψi. The actual state |ψi of the system at
t = 0 is linearly related to the in asymptote |ψin i, specifically,
|ψi = lim U (t)† U 0 (t)|ψin i ≡ Ω+ |ψin i
t→−∞
(A.3)
Similarly the actual state |ψi at t = 0 that will evolve into the out asymptote
labelled by |ψout i is
|ψi = lim U (t)† U 0 (t)|ψout i ≡ Ω− |ψout i
t→+∞
(A.4)
The two operators Ω± , defined as the limits
Ω± = lim U (t)† U 0 (t)
t→∓∞
(A.5)
are called the Møller wave operators. They are the limits of a unitary operator
and so, are isometric. Their significance should be clear: Acting on |ψin i (or
|ψout i) in H, they give the actual state at t = 0 that would evolve from (or
to) the asymptote represented by this vector. This is illustrated symbolically in
Fig. A.1.
130
A.2 Orthogonality and Asymptotic Completeness
We have seen that every vector in H labels the in or out asymptote of some
actual orbit U (t)|ψi. We must now consider the converse question: Does every
|ψi in H define an orbit U (t)|ψi that has in and out asymptote? Just
as in the classical case the answer to this question is, in general, no. The
Hamiltonian H = H 0 + V will usually have some bound states; and if |φi is a
bound state, then the orbit U (t)|φi describes a stationary state in which the
particle remains localized close to the potential and, hence, never behaves freely.
The situation is analogous to that of the classical problem. We do not
expect that every orbit U (t)|ψi will have asymptotes. We expect rather that
there will be certain scattering orbits that do have asymptotes, and that there
may be some bound states. Under some special conditions, there might be also
some resonances. The scattering states together with the bound states will span
the space H of all normalizable states. The resonance states however are not
normalizable and thus, they are not in the Hilbert space H.
Let us denote by B the subspace spanned by the bound states. We next note
that any state with an in asymptote is given by |ψi = Ω+ |ψin i. These vectors
make up the range of Ω+ . We denote this range by R+ . Similarly, R− will
denote the range of Ω− which is the set of all states with out asymptote.
The following theorems hold for a wide class of potentials
1. Orthogonality Theorem. The subspaces R± are orthogonal to the subspace B:
R+ ⊥B
R− ⊥B
2.Asymptotic Completeness. The subspaces R+ and R− are the same:
R+ = R− = R
If the scattering theory holds the asymptotic completeness, it is called asymptotically complete.
Due to above theorems, the space H can be written as direct sum of R and
B:
H=R⊕B
(A.6)
Our description of the scattering process can be summarized as follows: As
far as the actual orbits of the system are concerned, the Hilbert space H is
divided into two orthogonal parts; the subspace B spanned by the bound states,
and the subspace R of scattering states. For every |ψi in R, the orbit U (t)|ψi
describes a scattering process with in and out asymptote,
U (t)|ψi−−−−−−−→ U 0 (t)|ψin i
t → −∞
−−−−−−−→ U 0 (t)|ψout i
t → +∞
(A.7)
131
Every |ψin i (or |ψout i) in H labels the in (or out) asymptote of a unique
actual orbit U (t)|ψi and the Møller operators Ω± map each |ψin i (or |ψout i) in
H onto the corresponding scattering state |ψi in R,
|ψi = Ω+ |ψin i = Ω− |ψout i
(A.8)
As we have already noted, the Møller operators are isometric. This means
that for each normalized |ψin i or |ψout i in H, there is a unique corresponding
normalized |ψi in R, and conversely, for each normalized |ψi in R there are
unique normalized asymptotes |ψin i and |ψout i. This situation is summarized
schematically in Fig. A.2
H =B⊕R
H
Ω+
|ψ 〉
in
H
|ψ〉
R
Ω−
Ω†−
|ψ
B
〉
out
Fig. A.2: The Møller operators Ω± map the in and out asymptotes, represented by
|ψin i and |ψout i, onto the actual orbit labelled by |ψi.
One question which may arises here, is that how a bound state vector |φi
can possibly represent a free asymptote? To answer this, one has only to recall
the significance of the asymptotes. The statement that |φi represents a possible
in asymptote (for example) means that if we multiply |φi by the free evolution operator U 0 (t) and take t large and negative, then U 0 (t)|φi will look very
like some actual scattering orbit. Now, the free evolution operator spreads all
states; in particular, it will take no notice of the fact that |φi is an eigenstate
of the full Hamiltonian H 0 + V . Thus, at the times in question (t large and
negative) U 0 (t)|φi does indeed represent the free motion of a particle far from
the potential; and as such is a legitimate in asymptote. It is for this reason that
all vectors in H can represent in or out asymptote.
A.3 The Scattering Operator
So far we have expressed the actual scattering state at t = 0 in terms of either of
its two asymptotes. Our ultimate goal is to express the out asymptote in terms
132
of the in asymptote without reference to the experimentally uninteresting actual
orbit. Since Ω− is isometric, the relation |ψi = Ω− |ψout i can be inverted. In
fact, since Ω†− Ω− = 1, we simply multiply on the left by Ω†− to give
|ψout i = Ω†− |ψi = Ω†− Ω+ |ψin i
(A.9)
If we define the scattering operator as
S = Ω†− Ω+
(A.10)
|ψout i = S|ψin i
(A.11)
this equation becomes
which is the desired result. The scattering operator S gives |ψout i directly in
terms of |ψin i. Since only the asymptotic free motion is observed in practice,
the single operator S contains all information of experimental interest. The
probability that a particle that entered the collision with in asymptote |ψin i =
|φi will be observed to emerge with out asymptote |ψout i = |χi is given by
w(χ ← φ) = |hχ|Ω†− Ω+ |φi|2 = |hχ|S|φi|2
(A.12)
The probability amplitude for the process (χ ← φ) is just the S−matrix element
hχ|S|φi.
In practice even the quantity w(χ ← φ) is not directly observable; this is
because one can not actually produce or identify uniquely defined wave packets
|φi and |χi (even tough in principle they can). However, the quantity that
is experimentally observable, the differential cross-section, can be expressed
directly in terms of the matrix elements of S. In practice, concerning the initial
state |φi we know only that it is a wave packet whose position and momentum
are both reasonably well defined. Concerning the final state, we measure only
wether or not the outgoing direction of motion lies in some element of solid angle
dΩ. The nature of the measurement on the outgoing particle is easily allowed
for, instead of w(χ ← φ) we have only to calculate the probability w(dΩ ← φ)
that the direction of motion of the out state lie inside the element of solid angle
dΩ. Our ignorance of the precise in asymptote |φi means that we must then
average this probability over all relevant states |φi. It is this averaging process
that leads to the notion of the differential cross section.
Before we calculate the cross section, it is convenient to establish two important properties of the S operator, conservation of energy and the decomposition
of S in terms of the scattering amplitude.
A.4 Conservation of Energy
One of the most important properties of the S operator is that it conserves the
energy. Because the Hamiltonian H is independent of time, energy is, of course,
conserved, and the expectation value of H for any actual orbit is constant.
However, the S operator is mapping of the asymptotic free orbits, which label
133
the particle’s state only when it is far away from the scatterer and does not feel
the potential. As far as the asymptotic states are concerned, the actual energy is
simply the kinetic energy and we need therefore to prove that S commutes with
the kinetic energy operator H 0 . The essential step in the proof is the so-called
intertwining relation for the Møller operators
HΩ± = Ω± H 0
(A.13)
Which can be proved by the following manipulation (h̄ = 1)
0
eiHτ Ω± = eiHτ [ lim eiHt e−iH t ]
t→∓∞
0
= lim [eiH(τ +t) e−iH t ]
t→∓∞
= lim [eiH(τ +t) e−iH
t→∓∞
= Ω± eiH
0
0
(τ +t)
]eiH
0
τ
τ
(A.14)
Differentiating with respect to τ and setting τ = 0, we obtain the desired result.
Now we have
SH 0 = Ω†− Ω+ H 0 = Ω†− HΩ+ = H 0 Ω†− Ω+ = H 0 S
(A.15)
It is convenient to write the matrix elements of S in the momentum representation. We often refer to {hp0 |S|pi} as the S matrix. The physical significance
of the improper matrix elements hp0 |S|pi is in the corresponding expansion of
the out wave function ψout (p) in terms of ψin (p),
Z
ψout (p) = d3 p0 hp|S|p0 iψin (p0 )
(A.16)
So it is convenient to visualize hp0 |S|pi as the probability amplitude that an in
state of momentum p lead to an out state of momentum p0 .
Since S commutes with H 0 , its momentum-space matrix elements satisfy
0 = hp0 |[H 0 , S]|pi = (Ep0 − Ep )hp0 |S|pi
(A.17)
This implies that hp0 |S|pi is zero except when Ep0 = Ep and hence it has the
form
hp0 |S|pi = δ(Ep0 − Ep ) × remainder
(A.18)
To explore further the structure of the momentum-space S matrix it is convenient to write it as
hp0 |S|pi = δ3 (p0 − p) +
i
δ(Ep0 − Ep )f (p0 ← p)
2πµ
(A.19)
here f (p0 ← p) is called the scattering amplitude. The significance of the two
terms in this decomposition of the S matrix is easily understood. The first term,
δ3 (p0 − p) is obviously the amplitude for the particle to pass the force center
134
without being scattered. The second is therefore the amplitude that it actually
is scattered. When the particle is scattered its momentum changes, while its
energy stays the same. Thus the second term should conserve energy, but not
the individual components of momenta.
Replacing hp|S|p0 i in (A.16) by its decomposition (A.19) one obtains
Z
i
ψout (p) = ψin (p) +
d3 p0 δ(Ep − Ep0 )f (p ← p0 )ψin (p0 )
(A.20)
2πµ
Here the first term is the unscattered wave and second the scattered wave.
A.5 Scattering of two spinless particles
The quantum scattering of two distinct spinless particles, can be reduced to the
scattering of a single spinless particle with a fixed target.
The states of the two particles system are represented by function ψ(x1 , x2 )
depending on the positions x1 and x2 of the two particles. The vector |ψi
belongs to the two-particle Hilbert space H, which can be written as
H = H1 ⊗ H2
(A.21)
where H1 and H2 are the one-particle space of the first particle and the second,
respectively. The space H can be spanned by product vectors; that is, every
vector can be expressed as a sum of product vectors like |φ1 i ⊗ |φ2 i. Here
|φ1 i is in the one-particle space H1 and |φ2 i is in space H2 . Two important
properties of product vectors are: first, if {|n1 i} and {|n2 i} are orthonormal
bases of the one-particle spaces H1 and H2 , then the products, |n1 i ⊗ |n2 i form
an orthonormal basis of the two-particle space H. Second, the scaler products
satisfy
(A.22)
(hφ01 | ⊗ hφ02 |)(|φ1 i ⊗ |φ2 i) = hφ01 |φ1 ihφ02 |φ2 i
Similar to vectors, for any operators A1 and A2 , acting on H1 and H2 respectively, we can define an operator (A1 ⊗ A2 ) acting on H by the relation
(A1 ⊗ A2 )(|φ1 i ⊗ |φ2 i) = (A1 |φ1 i) ⊗ (A2 |φ2 i)
(A.23)
In particular, the basic dynamical variables of a two particles system are all
operators of the form A1 ⊗ 1 or 1 ⊗ A2 .
It is usually unnecessary to distinguish A1 ⊗ 1 from A1 and we shall write
just A1 for either, similarly 1 ⊗ A2 will be abbreviated to A2 , and hence A1 ⊗ A2
which is the same as (A1 ⊗ 1)(1 ⊗ A2 ), to A1 A2 .
The two particle Hamiltonian will be of the form
H=
p21
p2
+ 2 + V (x)
2m1
2m2
(A.24)
where V is a function of the relative coordinate x1 − x2 = x only (we have
assumed a local and translationally invariant interaction), and the momentum
135
operators defined as follows
p1 = p1 (on H1 ) ⊗ 1(on H2 )
p2 = 1(on H1 ) ⊗ p2 (on H2 )
(A.25)
For any purposes, it is convenient to use center of mass and relative coordinates,
defined by
X=
m1 x1 + m2 x2
(on HCM )
m1 + m2
x = x1 − x1 (on Hrel )
(A.26)
The two-particles Hilbert space H can be defined as
H = HCM ⊗ Hrel .
(A.27)
By defining the total and relative momenta as
P = p1 + p2 (on HCM )
m1 p1 + m2 p2
p=
(on Hrel )
m1 + m2
(A.28)
(which are, of course, the momenta conjugate to the CM and relative position
operators), the Hamiltonian reads
· 2
¸
P2
p
H=
+
+ V (x)
2M
2µ
≡ HCM + Hrel
(A.29)
Where M = m1 + m2 and µ = m1 m2 /m1 + m2 are the total and reduced masses
respectively. Since HCM and Hrel act on different spaces, they commute, and
so the evolution operator can be factored as follows
U (t) = e−iHt = e−i(HCM +Hrel )t
= e−iHCM t e−iHrel
= e−iHCM t ⊗ e−iHrel t
(A.30)
Here the last expression serves to emphasize that U (t) is a product of two
evolution operators, one for HCM and the other for Hrel . This result means that
the motion of the center of mass and the relative coordinates are independent,
and the two-particle scattering problem, reduces to two single-particle scattering
problems. In particular, since HCM is just P2 /2M , the center of mass moves
like a free particle of mass M . Hrel describes the scattering of a single-particle
of mass µ, off a fixed potential V .
The free evolution operator can also be factored in the same way,
U 0 (t) ≡ e−iH
0
t
0
= e−iHCM t ⊗ e−iHrel t
136
(A.31)
of course U 0 (t) can also be factored as
U 0 (t) = exp(−i
p21
p2
t) ⊗ exp(−i 2 t)
2m1
2m2
(A.32)
corresponding to the factoring of H as H1 ⊗H2 . This second result expresses the
obvious fact that two noninteracting particles move independently. The same
result obviously does not hold for the full evolution operator U (t).
The general orbit of our two-particle system is U (t)|ψi where |ψi is any
vector in the two-particle space H. Just as in the one-particle case we expect
there to be certain scattering orbits for which the two particles move far apart
as t → ∓∞ and U (t)|ψi behaves like some freely moving U 0 (t)|ψin/out i. The
actual orbit is given by
|ψi = Ω+ |ψin i = Ω− |ψout i
(A.33)
where
Ω± ≡ lim U (t)† U 0 (t) = (1 ⊗ Ω± )
t→∓∞
(A.34)
The single-particle Møller operators Ω± act on Hrel = L2 (R3 ) and are given by
0
Ω± = lim eiHrel t e−iHrel t
t→∓∞
(A.35)
The factor 1CM in Ω± reflects the fact that the center of mass moves like a free
particle and is not scattered.
The two-particle operator S which maps any |ψin i in H onto the corresponding |ψout i = S|ψin i, simply is given by
S = Ω†− Ω+ = 1CM ⊗ S
(A.36)
the single-particle operator, S = Ω†− Ω+ , acts on Hrel and can be computed
from the Hamiltonian Hrel .
A.5.1
Conservation of energy-momentum and the scattering amplitudes
From the expression S = 1CM ⊗ S it is immediately clear that S commutes with
P (which acts only on HCM ), and hence, the total momentum is conserved.
Just as in the one-particle case, S commutes with H 0 and energy is conserved.
From conservation of energy and momentum it follows that the matrix elements,
hp01 , p02 |S|p1 , p2 i
contain the factors
δ(E10 + E20 − E1 − E2 )δ3 (p01 + p02 − p1 − p2 )
137
where E1 = p21 /2m1 and so on. As before it is convenient to decompose this
matrix element in terms of scattering amplitudes f
hp01 , p02 |S|p1 , p2 i = δ3 (p01 − p1 )δ3 (p02 − p2 )
X
X
X
X
i
δ(
Ei0 −
+
Ei )δ3 (
p0i −
pi )f (p0 ← p) (A.37)
2πµ
The first term is the amplitude that each particle passes through unscattered;
the second is the amplitude that the two particles actually scatter. The later
conserves total energy and total momentum but not, of course the individual
components of the relative momentum.
A.6 Invariance Principles and Conservation Laws
In this section we shall discuss the application of invariance principles to the
scattering theory. We shall find that invariance of the system under any of the
possible symmetry operations (rotation, parity, time reversal, etc.) implies sever
restrictions on the possible form of the scattering amplitude.
A.6.1 Translational invariance and conservation of momentum
The effect of a rigid translation through a vector a on any system is given by
the unitary translation operator
D(a) = e−ia·P
(A.38)
where P is the total momentum operator of the system. This means simply
that if the system occupies any state |ψi and is then rigidly displaced through
a, then the resulting state is D(a)|ψi.
Coming back to our two-particle system with Hamiltonian H = H 0 + V ,
we know that H 0 automatically commutes with D(a). Thus, if the system is
translationally invariant, the translation operators D(a) commutes with both H
and H 0 , and from this it follows that they commute with the Møller operators
Ω± , as well as with the scattering operator S. Since S commutes with D(a) for
any a, it must commute with P, the total momentum, [P, S] = 0. In particular
if we take momentum-space matrix elements of this equation we find that the
momentum-space matrix element of S is zero unless the initial and final total
momenta are equal, that is the total momentum is conserved.
A.6.2 Rotational invariance and the conservation of the angular momentum
The effect of any rotation on a quantum mechanic system through an angle α
clockwise about the direction u
b is given by the unitary rotation operator
u·J
b ) = e−iαb
R(α, u
where J is the total angular momentum operator.
138
(A.39)
Using the same arguments as above for our two-particle system and assume
b ) commutes with Ω± and S which
rotational invariance, we can show that R(α, u
results in the conservation of total angular momentum J. For two particles total
angular momentum is
J = J1 + J2
= X × P + Jint
(A.40)
Here the operator X × P is the angular momentum of the center of mass and
acts only on the space HCM of CM motion; while Jint is the internal angular
momentum and acts only on the relative space Hrel . Clearly S = 1⊗S commutes
with the total angular momentum J if and only if S commutes with the internal
angular momentum Jint .
In the special case where both particles are spinless, the angular momentum
Jint is simply L (the relative orbital angular momentum of the two particles).
In this case the rotationally invariance requires simply that V be spherically
symmetric; that is, a function of r only, V (x) = V (r). If V is spherically
symmetric, then S commutes with all rotations R and we have
S = R† SR
(A.41)
Taking momentum-space matrix elements of this equation we find
hp0 |S|pi = hp0R |S|pR i
(A.42)
where pR denotes the momentum obtained from p by the rotation R. From
this we fined the same result for the scattering amplitude f
f (p0 ← p) = f (p0R ← pR )
(A.43)
This result means that the amplitude f (p0 ← p), which is a priori a function of
five variables, p and the direction of p0 (recall that |p0 | = |p|), is in fact only a
function of two variables, which we take to be the energy Ep and the scattering
angle θ between p and p0
f (p0 ← p) = f (Ep , θ)
(A.44)
For a single spinless particle (or equivalently for relative motion of two spinless particles), the three operators H 0 , L2 , and Lz form a complete set of commuting observables. We will denote the spherical wave basis vectors of this representation by |E, l, mi where E, l(l + 1), and m are the eigenvalues of H 0 , L2 ,
and Lz respectively. The corresponding spatial wave functions hx|E, l, mi are
products of the spherical harmonics Ylm (θ, ϕ) and functions of r. If we write
these products in the form (1/r)u(r)Ylm (θ, ϕ), then the radial wave function
u(r) satisfies the free radial Schrödinger equation
· 2
¸
d
l(l + 1)
2
−
+
p
u(r) = 0
(A.45)
dr2
r2
139
This is an ordinary, second-order, linear, differential equation and so has
two linearly independent solutions, of which the physically relevant solution is
that which vanishes at the origin. As r → 0, the centrifugal term l(l + 1)/r2
dominates the energy term p2 , and the solutions behave like combination of rl+1
and r−l . The physically acceptable wave function is the Riccati-Bessel function
b
jl (z) = zjl (z)
where jl (z) is the Spherical Bessel function. A convenient choice for the second
solution is the Riccati-Neumann function
n
bl (z) = znl (z)
where nl (z) is the spherical Bessel function of the second kind. Now the spatial
wave functions look like
hx|E, l, mi = il (
2µ 1/2 1 b
)
jl (pr)Ylm (θ, ϕ)
πp
r
[p ≡ (2µE)1/2 ]
(A.46)
The normalization is
hE 0 , l0 , m0 |E, l, mi = δ(E 0 − E)δl0 l δm0 m .
(A.47)
Because S commutes with H 0 and L, the S matrix in this representation is
diagonal and has the form
hE 0 , l0 , m0 |S|E, l, mi = δ(E 0 − E)δl0 l δm0 m sl (E).
(A.48)
Here the number sl (E) is the eigenvalue of S corresponding to the eigenvector
|E, l, mi, and is actually independent of m (for a prove see Ref. [1] section 6.c).
Since S is unitary, each of its eigenvalue has modulus 1 and can be written
as the exponent of a purely imaging number
sl (E) = e2iδl (E)
(A.49)
where δl (E) is just the conventional phase shift, up to addition of an arbitrary
multiple of π.
Using the transformation matrix
hp|E, l, mi = (µp)−1/2 δ(Ep − E)Ylm (b
p)
with Ep = p2 /2µ
(A.50)
one can pass from the angular-momentum to the momentum basis. In particular,
from (A.19) we find
i
δ(Ep0 − Ep )f (p0 ← p) = hp0 |(S − 1)|pi
2πµ
Z
X
= dE
hp0 |(S − 1)|E, l, mihE, l, m|pi (A.51)
l,m
140
Now, |E, l, mi is an eigenvector of (S − 1), which can therefore be replaced by
the corresponding eigenvalue (sl − 1)
Z
X
= dE
[sl (Ep ) − 1]hp0 |E, l, mihE, l, m|pi
l,m
X
1
=
δ(Ep0 − Ep )
[sl (Ep ) − 1]Ylm (b
p0 )Ylm (b
p)∗
µp
(A.52)
l,m
Now we find for the amplitude f (p0 ← p) that
f (Ep , θ) ≡ f (p0 ← p) =
∞
X
(2l + 1)fl (E)Pl (cos θ)
(A.53)
l=0
where fl (E) is the partial-wave amplitude,
fl (E) =
eiδl (E) sin δl (E)
sl (E) − 1
=
2ip
p
and
µ
Pl (cos θ) =
4π
2l + 1
(A.54)
¶1/2
Yl0 (θ, ϕ)
(A.55)
is the Legendre polynomial. The expansion (A.53) is the so-called partial-wave
expansion for the full amplitude f (E, θ) in terms of the partial-wave amplitude
fl (E). It can of course be inverted. Using the well-known orthogonality of the
Legendre polynomials we find
Z
1 1
fl (E) =
d(cos θ)f (E, θ)Pl (cos θ)
(A.56)
2 −1
A.6.3 Parity
In addition to rotational invariance, many systems satisfy invariance under parity. Parity is defined as a reversal of the three spatial axes. More precisely, the
parity operator P changes the signs of all positions and momenta but leaves all
angular momenta (and the nature of the system itself) unchanged. Here the
system of interest contains two spinless particles, for which (as was the case
with rotations) we need only consider the space of the relative motion. For the
wave function we have
hx|P |ψi = ψP (x) = ηψ(−x)
(A.57)
and likewise for the momentum-space wave function. Here η is an arbitrary
number of modulus one.
The dynamics are invariant under parity if and only if P commutes with
H; or equivalently , V (x) = V (−x). This condition is automatically satisfied if
V (x) is spherically symmetric. It follows exactly as in our discussion of rotations
141
that invariance under parity implies that P commutes with S, and hence, that
S = P † SP . If |φi and |φ0 i are two states of definite parity,
P |φi = p|φi and
P |φ0 i = p0 |φ0 i
(A.58)
then hφ0 |S|φi = 0 unless p = p0 ; that is, parity is conserved. More generally,
for any initial and final states hφ0 |S|φi = hφ0p |S|φp i. In particular hp0 |S|pi = hp0 |S| − pi and, hence, for the amplitude we have the very natural result
f (p0 ← p) = f (−p0 ← −p)
(A.59)
A.6.4 Time reversal
So far we have discussed three types of symmetry operations, all of which are
represented by unitary operators. We now discuss a symmetry that is not given
by a unitary operator: time reversal. The time reversal operator T, is defined
to change the sign of the momenta and spins of all particles but to leave their
position unchanged.
The time reversal operator, T, is anti-unitary, that is T is a one-to-one map
of H onto H that is norm-preserving and anti-linear such that
T (a|ψi + b|φi) = a∗ T |ψi + b∗ T |φi.
(A.60)
This differs from the definition of a unitary operator only in that T is anti-linear.
The adjoint of the anti-linear operator T is defined by the relation
hφ|(T |ψi) = hT † φ|ψi∗
(A.61)
Using this definition one can show that
T †T = T T † = 1
(A.62)
For a single spinless particle after suitable adjustment of the arbitrary overall
phase of T , we will have
T |xi = T † |xi = |xi
T |pi = T † |pi = | − pi
If we expand any |ψi as
(A.63)
Z
|ψi =
d3 xψ(x)|xi
(A.64)
we find for T |ψi that
ψT (x) ≡ hx|T |ψi = hT † x|ψi∗
= hx|ψi∗
= ψ ∗ (x)
142
(A.65)
ψT (p) ≡ hp|T |ψi = hT † p|ψi∗
= h−p|ψi∗
= ψ ∗ (−p)
(A.66)
The effect of T on the spatial wave function is simply to replace it by its complex
conjugate, while for the momentum-space wave function it should be replaced
by its complex conjugate plus a simultaneously change in the sign of p.
Invariance under time reversal means that T commutes with H (and automatically with H 0 ), T H = HT . (For a spinless particle in a local potential,
or equivalently the relative motion of two spinless particles, this requires simply that V be real, which it has to be anyway in order that H be Hermitian.)
Because T is anti-unitary, this means that
T eiHt = e−iHt T
(A.67)
(The change of sign, arises because T i = −iT .) It follows that
0
T Ω± = T [ lim eiHt e−iH t ]
t→∓∞
0
= [ lim e−iHt eiH t ]T
t→∓∞
= Ω∓ T
or since T † T = 1,
(A.68)
Ω± = T † Ω∓ T
(A.69)
Thus, the effect of T on the Møller operators is to interchange Ω+ and Ω− .
From (A.68) it follows that
or
T S = T Ω†− Ω+ = Ω†+ T Ω+ = Ω†+ Ω− T = S † T
(A.70)
S = T †S†T
(A.71)
Taking matrix elements of this equation we find
hχ|(S|φi) = hχ|(T † S † T |φi)
¡
¢∗
= hT χ|(S † T |φi)
¡
¢∗
= hχT |S † |φT i
(A.72)
or
hχ|S|φi = hφT |S|χT i
(A.73)
This result shows that, as one would expect, time reversal invariance implies
that the probability W (χ ← φ) is the same as the probability W (φT ← χT )
for the process in which initial and final states are time reversed and their roles
exchanged.
143
In particular, for one-particle scattering (or equivalently the relative motion
of two-particle scattering), the result (A.73) - when written in the momentum
representation - gives
hp0 |S|pi = h−p|S| − p0 i
(A.74)
or equivalently
f (p0 ← p) = f (−p ← −p0 )
(A.75)
for the amplitude.
A.7 Scattering of the Two Particles With Spin
In this section we shall extend our scattering formalism to include particles
with spin. In moving from spinless particles to particles with spin we must
anticipate two obvious complications. First, the Hilbert space for particles with
spin is more complicated, because it must describe spin as well as spatial degrees
of freedom. Second, the Hamiltonian may be spin-dependent and contain terms
such as the spin-orbit interaction of an electron with a nucleus or the tensor
interaction between two nucleons.
The Hilbert space appropriate to single particle of spin s is the tensor product,
H = Hspace ⊗ Hspin
(A.76)
where Hspace is the space L2 (R3 ) of ordering wave functions and Hspin is the
(2s + 1)-dimensional spin space. As a basis for Hspin it is usual to use the
eigenvectors |mi of the third component of the spin operator,
Sz |mi = m|mi
(A.77)
A basis for the space H can be constructed from any bases of Hspace and
Hspin . One convenient basis is given by the eigenvectors of p and Sz ,
|p, mi = |pi ⊗ |mi
(A.78)
which are products of the momentum eigenvectors |pi in Hspace and the Sz
eigenvectors |mi in Hspin .
The Hilbert space for two distinct particles with spins s1 , and s2 is of course
the product H = H1 ⊗ H2 of the two one-particle spaces, each of which is itself
a product of the type just described. The spatial wave functions have the form
ψm1 m2 (x1 , x2 ) and just as in the spinless case it is convenient to rewrite these
as functions ψm1 m2 (X, x) of the CM and relative positions, X and x. Because
these can clearly be spanned by products of the form φ(X)χm1 m2 (x), we can
regard the space H as H = HCM ⊗ Hrel where HCM describes the motion of
the CM position only, while Hrel describes the relative motion, including both
spins. The space Hrel can itself be regarded as a product of one space for the
relative coordinate x and another for both spins.
144
As a basis for the spin space we can use either the eigenvectors |m1 , m2 i
of the two z components, or the eigenvectors |s, mi of the total spin and its z
component. The relation between these is
X
|s, mi =
|m1 , m2 ihs1 s2 m1 m2 |smi
(A.79)
m1 ,m2
where hs1 s2 m1 m2 |smi is the usual Clebsh-Gordan coefficient. When we do not
wish to commit ourselves to a particular basis we shall use the notation |ξi to
label any convenient choice. In practice, ξ usually stands for either (m1 , m2 ) or
(s, m), and in any case is a label taking on (2s1 + 1)(2s2 + 1) distinct values.
The general spin state of the two particles can be expanded as
X
|χi =
χξ |ξi
(A.80)
ξ
and is completely identified by the numbers χξ , which can be grouped into a
column spinor χ of (2s1 + 1)(2s2 + 1) components.
For any basis {|ξi} of the spin space there are several corresponding bases
of the complete space H and the space of the relative motion Hrel . The most
important basis of H consists of the momentum eigenvectors, which we write
(without serious danger of confusion) in either of the forms
|p1 , p2 , ξi ≡ |P, p, ξi
(A.81)
where P and p are the total and relative momenta as usual. The corresponding
basis vectors of Hrel are just |p, ξi in terms of which we can write |P, p, ξi as
|Pi ⊗ |p, ξi.
A.7.1 The S operator for particles with spin
Apart from the fact that the Hilbert space is a little more complicated than
before, we can now set up a scattering formalism exactly as for the spinless case.
The orbits have the usual form U (t)|ψi where |ψi is any vector in the space H
just described. The evolution operator U (t) is determined by the Hamiltonian
H = H0 + V
(A.82)
where H 0 is the same as for the spinless case,
H0 =
p21
p2
P2
p2
+ 2 =
+
2m1
2m2
2M
2µ
(A.83)
Typical examples of interaction V would be a nucleon-nucleon interaction of the
form
V = V1 (r) + S1 · S2 V2 (r) + S1 · xS2 · xV3 (r)
(A.84)
(Remember that x = x1 − x2 and r = |x|) or the spin-orbit interaction of an
electron in an atom or nucleon in a nucleus,
V = V1 (r) + L · SV2 (r)
145
(A.85)
In all cases, whether or not V depends on the spins, one expects that V will go
to zero as the two particles move apart. Thus, with the potentials (A.84) and
(A.85) we expect that the coefficients Vi (r) will go to zero suitably rapidly as
r → ∞.
Using the same arguments as in the spinless case one can prove the asymptotic condition, which asserts that every |ψin i in H labels the in asymptote of
some actual orbit U (t)|ψi,
U (t)|ψi
−−−−−−−−−→
t → −∞
U 0 (t)|ψin i
(A.86)
Exactly as in the spinless case, the asymptotic condition and asymptotic completeness both hold, for all “reasonable” potentials and shall confine attention
to these from now on. (As usual this excludes the Coulomb potential.) Thus
the Møller operators Ω± exist as the limits of U † (t)U 0 (t) and map each in or
out asymptote |ψin i or |ψout i onto the corresponding actual state |ψi at t = 0,
here as before we have
U (t)|ψi −−−−−−−−−→ U 0 (t)|ψin i
t → −∞
U (t)|ψi −−−−−−−−−→ U 0 (t)|ψout i
t → +∞
(A.87)
The operator S = Ω†− Ω+ is unitary and maps each |ψin i directly onto corresponding |ψout i.
Exactly as in the spinless case the S operator has the structure
S = 1CM ⊗ S
(A.88)
where of course, 1CM refers to the motion of the CM position only, while S acts
on the space of the relative motion, including both spins. This result can be
regarded as reducing the problem of two particles with spin to an equivalent
quasi-one-particle problem, namely the scattering of a single particle with spin
s1 , off a fixed target of spin s2 .
A.7.2
The amplitudes and amplitude matrix
For the same reasons as before S conserves energy and momentum and its matrix
elements can be decomposed into two terms,
hp01 , p02 , ξ 0 |S|p1 , p2 , ξi = δ3 (p01 − p1 )δ3 (p02 − p2 )δξ0 ξ
X
X
X
X
i
δ(
Ei0 −
Ei )δ3 (
p0i −
pi )f (p0 , ξ 0 ← p, ξ) (A.89)
−
2πµ
In (A.89) the first term is the amplitude for no scattering and leaves both momenta and spins unchanged. The second conserves energy and total momentum
but can in general connect states of different relative momenta and different
spins. Just as before, the scattering amplitude is related directly to the operator S of the relative motion; specifically, if we insert (A.88) into (A.89) and
146
factor out the total-momentum delta function, we find
hp0 , ξ 0 |S|p, ξi = δ3 (p0 − p)δξ0 ξ −
i
δ(Ep0 − Ep )f (p0 , ξ 0 ← p, ξ)
2πµ
(A.90)
Exactly as in the spinless case we can calculate the CM cross sections in
terms of the amplitude. However, in place of the single differential cross section
of the spinless case, we now find an infinite number of differential cross section
because the particles can enter the collision in any spin state |χi and one can
(in principle at least) measure the number of particles emerging into dΩ with
any given spin |χ0 i. We consider first the case where the particles enter in one
of the basis spin states |ξi and we count the number emerging into dΩ in the
basis spin state |ξ 0 i. For this case we obtain
dσ 0 0
(p , ξ ← p, ξ) = |f (p0 , ξ 0 ← p, ξ)|2
dΩ
(A.91)
which is the CM differential cross section for observation of the final particles
in the direction of p0 with spin given by |ξ 0 i if the initial particles had relative
momentum p and spin |ξi.
The cross section for observing an arbitrary final spin state |χ0 i coming from
any initial spin state |χi can be evaluated in the same way. If
X
χξ |ξi
(A.92)
|χi =
ξ
and similarly |χ0 i, we can calculate the relevant S-matrix element from (A.89)
and, hence, the corresponding CM cross section, which is easily seen to be
X
dσ 0 0
∗
(p , χ ← p, χ) = |
χ0ξ0 f (p0 , ξ 0 ← p, ξ)χξ |2
dΩ
0
(A.93)
ξ ,ξ
This result expresses the cross section for arbitrary spins (|χ0 i ← |χi) in terms
of the amplitude for the [(2s1 + 1)(2s2 + 1)]2 basis processes (|ξ 0 i ← |ξi). Its
form suggests that we rewrite the basic amplitudes as
f (p0 , ξ 0 ← p, ξ) = fξ0 ξ (p0 ← p)
(A.94)
and then regard them as the elements of an amplitude matrix
F (p0 ← p) = {fξ0 ξ (p0 ← p)}
(A.95)
Our result can then be written in the compact form
dσ 0 0
(p , χ ← p, χ) = |χ0† F (p0 ← p)χ|2
dΩ
(A.96)
from this it is clear that all information relevant to the scattering of the two
particles is contained in the matrix F (p0 ← p), just as, in the spinless case, all
information was contained in the single amplitude f (p0 ← p).
147
As a simple example, let’s consider a spin-half particle scattering off a spinless target. This example includes such important processes as the scattering of
electrons off a spin-zero atom, of nucleons off a spinless nucleus, and a number
of elementary particle processes, of which the most important is pion-nucleon
scattering. Because one particle is spinless, the spin space of the whole system
is just the two-dimensional spin space of the spin-half projectile. We use the
usual Sz basis with basis vectors |+i and |−i corresponding to the eigenvalues
m = ±1/2. According to (A.96), the scattering is determined by a (2 × 2)
amplitude matrix
µ
¶
f++ (p0 ← p) f+− (p0 ← p)
0
F (p ← p) =
(A.97)
f−+ (p0 ← p) f−− (p0 ← p)
The element fm0 m (p0 ← p) is the amplitude for an initial particle with momentum p and z component of spin m to be scattered into the direction p0 and
observed with z component of spin m0 . (For obvious reasons f+− and f−+ are
referred to as spin-flip amplitudes and f++ and f−− as spin-nonflip amplitudes.)
The most general initial and final spin states are given by two component spinors
χ and χ0 and, according to (A.96), the amplitude for observing the corresponding process (p0 , χ0 ← p, χ) is just the number χ0† F (p0 ← p)χ.
A.7.3 The In and Out spinors
In this section we give a useful alternative interpretation of the amplitude matrix
F (p0 ← p). To this end we consider particles incident in some definite spin state,
which we now label by the normalized spinor χin ; and in terms of χin we define
a second spinor,
χout = F (p0 ← p)χin
(A.98)
What we shall show is that χout is precisely the actual (unnormalized) spinor of
those particles which emerge with momentum p0 if the initial momentum and
spins were p and χin . To see this we focus attention on those particles emerging
with momentum p0 . The result (A.96) for the cross section (p0 , χ0 ← p, χin )
can be rewritten as
dσ 0 0
(p , χ ← p, χin ) = |χ0† F (p0 ← p)χin |2
dΩ
= |χ0† χout |2
(A.99)
which shows that the probability of those particles emerging in the direction p0
being found with spins χ0 is proportional to |χ0† χout |2 , for any χ0 . But according
to the elementary principles of quantum mechanics this means simply that χout
is the actual spin state of these particles.
The spinor χout is not normalized; on the contrary,
X X
X
2
2
|χout
|
fξ0 ξ (p0 ← p)χin
kχout k2 =
ξ0 | =
ξ |
ξ0
ξ0
ξ
X dσ
=
(p0 , ξ 0 ← p, χin )
dΩ
0
ξ
148
(A.100)
This sum will be recognized as the cross section (p0 ← p, χin ) that is measured
if we use spin-insensitive counters, which accept all particles irrespective of their
spin state. Thus the result is simply that
kχout k2 =
dσ 0
(p ← p, χin ) [out spins not monitored]
dΩ
(A.101)
A.8 Time-Independent Formulation of Quantum Scattering
So far we have set up a time-dependent description of collisions in which two
particles scatter elastically, the so called single-channel processes. The collisions
were first described in terms of the scattering operator S. The matrix-elements
of S were then decomposed in terms of the scattering amplitude, and the differential cross section was expressed in terms of the amplitude.
Our principal remaining problem in one-channel scattering is to setup methods for the actual computation of the amplitude in terms of a given interaction.
All such methods, fall within the so-called time-independent scattering theory.
This formalism is built around the so-called stationary scattering states and two
operators, the Green’s operator G(z) and the T operator T (z).
In the time-independent formulation of the quantum scattering process we
solve the time independent schrödinger equation
(H0 + V )|ψi = E|ψi
(A.102)
H0 |ψin i = E|ψin i
(A.103)
with
where |ψin i is the incoming state (E > 0). We assume the potential V to be zero
far away from the target, therefore in asymptotic region, the energy eigenstate
just describe a free-particle motion. The formal solution to this problem can be
expanded as a sum of an incident and a scattered wave according to
|ψi = |ψin i + |ψout i
(A.104)
where the incident state vector |ψin i satisfies
while |ψi satisfies
[G0 ]−1 (z)|ψin i = 0
(A.105)
G−1 (z)|ψi = 0
(A.106)
Here G0 and G are the Green’s function or resolvent for H 0 and H, respectively.
The Green’s function for an arbitrary Hamiltonian H is defined to be
G(z) = (z − H)−1
(A.107)
for any z, real or complex, for which the inverse exists.
The knowledge of the Green’s operator G(z), for all z, of a Hamiltonian H is
equivalent to knowledge of a complete solution of the corresponding eigenvalue
149
problem. The Green’s operator G(z) is analytic except on the spectrum of
H. For each discrete eigenvalue, G(z) has a pole whose position is precisely
the eigenvalue and whose residue determines the corresponding eigenvector (or
subspace of eigenvectors in the case of degeneracy). When H has a continuous
spectrum (as it always does in scattering theory), G(z) has a branch cut running
along the real axis from 0 to ∞; that is, the value of hχ|G(z)|ψi at any E > 0
approached from above (z = E + i0) is different from its value at the same point
approached from below (z = E − i0).
Im(z)
E + i0
x
E4
x
E3
x x
E E
2 1
Re(z)
E − i0
Fig. A.3: The Green’s operator G(z) is analytic except for poles at the bound states
z = E1 , E2 , · · · and a cut from 0 to ∞.
A.8.1
Lippmann-Schwinger equation for G(z)
Knowledge of G(z) for all z is equivalent to a complete solution of the eigenvalue
problem of H. Needless to say, finding G(z) is precisely as hard as solving the
eigenvalue problem, and, generally, one cannot hope to find G(z). For this
reason it is useful to have an equation that relates the unknown G(z) to some
known operator. The usual choice for the latter is the free Green’s operator
G0 (z). The equation relating G and G0 is called the resolvent equation or
Lippmann-Schwinger equation for G(z). It is derived from the simple operator
identity
A−1 = B −1 + B −1 (B − A)A−1
If we set A = z − H and B = z − H 0 this becomes:
G(z) = G0 (z) + G0 (z)V G(z)
(A.108)
or, if we interchange A and B
G(z) = G0 (z) + G(z)V G0 (z)
150
(A.109)
The free Green’s operator G0 is, explicitly known. In the momentum representation, it is diagonal and is given by
hp0 |G0 (z)|pi = hp0 |(z − H)−1 |pi =
1
δ(p − p0 )
z − Ep
(A.110)
obviously, this allows us to calculate the matrix elements of G0 (z) in any other
representation.
To conclude this part on the definition and general properties of Green’s
operator we note that since H = H † it is easily seen that
G(z ∗ ) = [G(z)]†
(A.111)
and similarly for G0 (z). This important identity asserts that the value of the
function G at the point z ∗ is the same as the adjoint of its value at the point z.
A.8.2
The T operator
In scattering theory it is convenient to introduce another operator T (z), which
is defined in terms of G(z) as
T (z) = V + V G(z)V
(A.112)
It is clear that as an analytic function of z the operator T (z) has the same
properties as G(z). That is, T (z) is analytic for all z not in the spectrum of H.
When z approaches the energy of a bound state, T (z) has a pole, and on the
positive real axis T (z) has a branch cut.
Using the Lippmann-Schwinger equation for G (A.108) one can show easily
that
G0 (z)T (z) = G(z)V
(A.113)
and
T (z)G0 (z) = V G(z)
(A.114)
As a first application of these identities we can find an expression for G in
terms of T . By replacing V G by T G0 in the Lippmann-Schwinger equation for
G, we find that
G(z) = G0 (z) + G0 (z)T (z)G0 (z)
(A.115)
This means that knowledge of T implies knowledge of G. Because the converse
is obviously true (by the definition of T ), we see that the information contained
in T is precisely equivalent to that in G.
Using the identity (A.113) we can replace GV in (A.112) by G0 T to obtain
the equation
T (z) = V + V G0 (z)T (z)
(A.116)
which gives T in terms of the known operator G0 . This equation is known as
the Lippmann-Schwinger equation for T (z), and is the starting point of many
methods for calculating T . In particular, when V is sufficiently weak one can
151
hope to obtain a reliable solution by iteration, starting with the so-called Born
approximation T ≈ V . Inserting this into the right-hand side of (A.116) gives
as the second approximation T ≈ V + V G0 V ; and continuing this procedure
produces the infinite series,
T = V + V G0 V + V G0 V G0 V + . . .
(A.117)
This series (which may or may not converge) is known as the Born series.
Using (A.113), the equation (A.106) can easily be solved for the scattered
wave in terms of the unperturbed Green’s function G0 and the T -matrix, yielding
|ψout i = G0 T |ψin i
(A.118)
Equation (A.115) is very useful when we study quantum scattering in lowdimension. Another formula on which we will rely heavily is the Lupu-Sax
formula [41]
TH,V (z) = [1 − TH 0 ,V [GH (z) − GH 0 (z)]]
−1
TH 0 ,V (z)
(A.119)
which relates the T -matrix of the scatter V in the background Hamiltonian H
to the T -matrix for the same scatter but in a different background Hamiltonian
H 0.
Finally we note that the identity (A.111) for G leads to a corresponding
result for T
T (z ∗ ) = [T (z)]†
(A.120)
A.8.3
Relation to the Møller operators
For a collision with the in asymptote labelled by the normalized vector |ψin i =
|φi, the actual state at t = 0 is given by
|ψi = Ω+ |ψin i = Ω+ |φi = lim U (t)† U 0 (t)|φi ≡ |φ+i
t→−∞
(A.121)
Similarly, if the out asymptote were going to be |ψout i = |φi, then the actual
state at t = 0 would have to be
|ψi = Ω− |ψout i = Ω− |φi = lim U (t)† U 0 (t)|φi ≡ |φ−i
t→+∞
(A.122)
Rewriting U † (t)U 0 (t) as the integral of its derivative, led to the result (which
we write for Ω− )
Z ∞
|φ−i = |φi + i lim+
dτ e−²τ U (τ )† V U 0 (τ )|φi
(A.123)
²→0
0
Here the damping factor e−²τ is introduced such that we can replace the proper
vector |φi by improper plane-wave state |pi (an essential step in the discussion of the stationary scattering state in subsection A.8.5) and the integral still
converges.
152
By inserting a complete set of states |pi we obtain
Z
Z ∞
|φ−i ≡ Ω− |φi = |φi+i lim+ d3 p
dτ [e−²τ U (τ )† V U 0 (τ )]|pihp|φi (A.124)
²→0
0
0
Now, the free evolution operator U (τ ) acting on |pi gives just exp(−iEp τ ).
Thus, the operator in brackets in the integrand can be replaced by
[· · ·] = exp[−i(Ep − i² − H)τ ]V
and the integral over τ performed
Z ∞
dτ [· · ·] = −i(Ep − i² − H)−1 V = −iG(Ep − i²)V
0
Substituting back into (A.124) we obtain
Z
|φ−i ≡ Ω− |φi = |φi + lim+ d3 pG(Ep − i²)V |pihp|φi
(A.125)
With the same method one can show
Z
|φ+i ≡ Ω+ |φi = |φi + lim+ d3 pG(Ep + i²)V |pihp|φi
(A.126)
²→0
²→0
A.8.4 Relation to the scattering operator
In this section we establish expression for the scattering operator S in terms of
G(z) and T (z), from which we establish the relevance of the operators G(z) and
T (z) to scattering theory. Our starting point is the equation
hχ|S|φi = hχ|Ω†− Ω+ |φi
³ 0
´³
´
0
0 0
= lim 0 lim hχ| eiH t e−iHt eiHt e−iH t |φi
t→+∞ t →−∞
(A.127)
The order in which we take the limits t → +∞ and t0 → −∞ is immaterial. In
particular, we can take the two limits simultaneously; that is, we can set t0 = −t
and simply let t → +∞, to give
h 0
i
0
hχ|S|φi = lim hχ| eiH t e−2iHt eiH t |φi
(A.128)
t→+∞
writing this expression as the integral of its derivative, in this case
n 0
o
0
0
0
d
[· · ·] = −i eiH t V e−2iHt eiH t + eiH t e−2iHt V eiH t
dt
(A.129)
gives
Z
∞
hχ|S|φi = hχ|φi − i
dthχ|{· · ·}|φi
Z ∞ 0
= hχ|φi − i lim+
dte−²t hχ|{· · ·}|φi
²→0
0
153
(A.130)
If we now replace the proper vectors |χi and |φi by momentum eigenstates |p0 i
and |pi the free evolution operators in the integrand simplify and we get
hp0 |S|pi = δ3 (p0 − p)
Z ∞
n
o
−i lim+
dthp0 | V ei(Ep0 +Ep +i²−2H)t + ei(Ep0 +Ep +i²−2H)t V |pi
²→0
0
= δ3 (p0 − p)
½
µ
¶
µ
¶ ¾
Ep0 + Ep
Ep0 + Ep
1
+ i² + G
+ i² V |pi
+ lim+ hp0 | V G
2 ²→0
2
2
(A.131)
If we next replace V G by T G0 , and GV by G0 T , the free Green’s operators
acting on the momentum eigenstates can be replaced by their eigenvalues to
give
½
¾
1
1
= δ3 (p0 − p) + lim+
+
Ep0 − Ep + i² Ep − Ep0 + i²
²→0
µ
¶
Ep0 + Ep
×hp0 |T
+ i² |pi
2
0
0
0
= δ3 (p − p) − 2πiδ(Ep − Ep ) lim+ hp |T (Ep + i²)|pi
(A.132)
²→0
This is one of the central results of time-independent scattering theory. Comparing it with equation (A.19) we see that
1
f (p0 ← p) = t(p0 ← p) = hp0 |T (Ep + i0)|pi
4π 2 µ
(A.133)
where we have used the notation f (E + i0) to denote the limit of f (E + i²) as
² → 0+ . The matrix t(p0 ← p) is called the on-shell T matrix. It is defined
only for Ep0 = Ep , and is in fact the p0 , p matrix element of the operator T (z)
for the particular values z = Ep + i0 and Ep0 = Ep . For this reason hp0 |T (z)|pi
is known as the off-shell T matrix. The off-shell matrix is more general than
t(p0 ← p), since it is defined for an arbitrary complex numbers and independent
momenta p and p0 .
A.8.5 The stationary scattering states
The stationary scattering states are improper eigenvectors of the Hamiltonian
H = H 0 + V and are denoted by |p+i and |p−i. In particular, the wave
function hx|p+i is just the familiar scattering wave function, often denoted
ψp+ (x), of elementary collision theory,
µ
¶
eipr
−3/2
ip·x
hx|p+i−r−−
(2π)
e
+
f
(A.134)
−
−
→
→∞
r
The scattering amplitude can be expressed in terms of |p+i or |p−i and any
means of computing |p±i therefore provides a method for calculating the amplitude. These method fall into two (closely connected) categories: those using
154
integral equations and those using differential equations. The integral method
depends on the fact that the wave functions hx|p±i satisfy integral equations,
closely related to the Lippmann-Schwinger equation for the operator T (z). The
differential method instead uses the fact that the vectors |p±i are eigenvectors
of H and, hence, that the wave functions hx|p±i satisfy the time-independent
Schrödinger equation. The later is most useful when the potential is spherically symmetric, since in this case the wave function can be decomposed into
angular-momentum eigenfunctions and the Schrödinger equation reduces to a
set of ordinary differential equations.
The stationary scattering states hx|p±i are defined as
|p±i ≡ Ω± |pi
(A.135)
To understand the significance of this definition we consider an orbit with in
asymptote labelled by |φi, which we expand in terms of plane waves as
Z
|φi = d3 pφ(p)|pi
(A.136)
In this case the actual state at t = 0 is
Z
|φ+i = Ω+ |φi = d3 pφ(p)Ω+ |pi
Z
= d3 pφ(p)|p+i
(A.137)
In other words, the actual state |φ+i has the same expansion in terms of |p+i
as does its in asymptote |φi in terms of |pi. Similarly, if an orbit has out
asymptote
Z
|χi =
then the actual state at t = 0 is
|χ−i =
d3 pχ(p)|pi
(A.138)
d3 pχ(p)|p−i
(A.139)
Z
These two results give the primary significance of the improper vectors |p±i.
They are the natural vectors for expanding the actual state at t = 0 to display
explicitly its relation to the momentum expansions of the in and out asymptotes.
Due to (A.13) one can show easily that the vectors |p±i are eigenvectors of the
full Hamiltonian
H|p±i = HΩ± |pi = Ω± H 0 |pi = Ep Ω± |pi = Ep |p±i
(A.140)
The eigenvalue of H acting on |p±i is the same as that of H 0 acting on |pi:
Ep = p2 /2m. This means that the wave function hx|p±i satisfy the timeindependent Schrödinger equation. Besides we have
U (t)|p+i = e−iEp t |p+i
155
(A.141)
U 0 (t)|pi = e−iEp t |pi
(A.142)
0
Note that U (t)|pi is not the asymptote of the actual orbit U (t)|pi. Nonetheless,
R 3 they do satisfy the asymptote condition when “smeared” by the integral
d pφ(p) into proper states.
U (t)|φ+i−−−−−−−→ U 0 (t)|φi
[any proper |φi]
t → −∞
Z
Z
3
0
U (t) d pφ(p)|p+i−−−−−−−→ U (t) d3 pφ(p)|pi
t → −∞
or
Z
(A.143)
Z
d3 pφ(p)[U (t)|p+i]−−−−−−−→
t → −∞
d3 pφ(p)[U 0 (t)|pi]
(A.144)
By expanding |φ±i and |φi in equation (A.125) and (A.126) in terms of the
momentum wave function φ(p) = hp|φi we obtain
Z
Z
d3 pφ(p)|p±i = d3 pφ(p)[|pi + G(Ep ± i0)V |pi]
(A.145)
since this hold for any φ(p) we conclude that
|p±i = |pi + G(Ep ± i0)V |pi
(A.146)
The relation (A.146) can be used to establish an expression for the on-shell T
matrix in terms of |p+i or |p−i. Since we already know that t(p0 ← p) is the
same thing as hp0 |T (Ep + i0)|pi we begin by considering the vector
T (Ep ± i0)|pi = [V + V G(±)V ]|pi
= V [1 + G(±)V ]|pi
(A.147)
Using (A.146) we then obtain the important identities
T (Ep ± i0)|pi = V |p±i
(A.148)
Multiplying the first of these by the bra hp0 | gives
t(p0 ← p) = hp0 |V |p+i
(A.149)
To obtain an equivalent expression in terms of |p+i we use the identity
(A.120), T (z)† = T (z ∗ ) to rewrite the second identity (A.148) (with p replaced
by p0 ) as
hp0 |T (Ep0 + i0) = hp0 − |V
(A.150)
This leads at once to the result
t(p0 ← p) = hp0 − |V |pi
(A.151)
These results allow one to calculate the scattering amplitude from either of the
stationary states |p+i or |p−i.
156
Returning to the equation (A.146) for |p±i we note that, just as with the T
operator in subsection A.8.2, it is convenient to replace the explicit expression
in terms of G(z) by an implicit expression in terms of G0 (z). This can be done
in two simple steps: we replace GV by G0 T , and then we use (A.148) to replace
T |pi by V |p±i. This gives
|p±i = |pi + G0 (Ep ± i0)V |p±i
(A.152)
This equation, which is, of course, an integral equation for the wave function
hx|p±i, is called the Lippmann-Schwinger equation for |p±i.
The equation (A.152) can be also written as
(1 − G0 V )|p±i = |pi
from this we obtain a Born series for |p+i,
|p+i = |pi + G0 V |pi + . . .
(A.153)
If we then use (A.149) to calculate the on-shell T matrix, we obtain
t(p0 ← p) = hp0 |V |pi + hp0 |V G0 V |pi + . . .
(A.154)
which is precisely the Born series resulting from iterating the Lippmann-Schwinger
equation for T (z).
From the Lippmann-Schwinger equation for |p±i it follows that the wave
function hx|p±i satisfy the integral equation
Z
hx|p±i = hx|pi + d3 x0 V (x0 )hx|G0 (E)|x0 ihx0 |p±i
(A.155)
We can use this equation to establish the behavior of the wave function for large
r; where hx|ψi is just a plane wave plus a spherical outgoing wave.
To use the above equation we must calculate the Green’s function hx|G0 (E)|x0 i.
We know that
1
p2
G0 (E)|pi =
|pi, Ep =
(A.156)
E + i0 − Ep
2µ
Thus, by inserting a complete set of states |pi into the required spatial matrix
element, we find that
Z
hx|G0 (E)|x0 i = d3 phx|G0 (E)|pihp|x0 i
=
1
(2π)3
Z
0
d3 p
eip·(x−x )
E + i0 − Ep
(A.157)
If we choose the direction of (x − x0 ) as polar axis, the exponent becomes
ip|x − x0 | cos θ and the angular integral can be performed to give
Z ∞
0
0
eip|x−x | − e−ip|x−x |
−i
pdp
=
4π 2 |x − x0 | 0
E + i0 − Ep
Z ∞
0
peip|x−x |
iµ
dp
(A.158)
=
2π 2 |x − x0 | −∞ p2 − 2µ(E + i0)
157
This integral can be evaluated by contour integration. The final result is
√
i 2µ(E+i0)|x−x0 |
µ
e
hx|G0 (E)|x0 i = −
(A.159)
2π
|x − x0 |
For the incident plane wave |ψin i = |pi we have
E=
p2
2µ
→
[2µ(E + i0)]1/2 = +p.
(A.160)
Substitution of these results into (A.155) gives
hx|p+i = hx|pi −
µ
2π
Z
0
d3 x0
e+ip|x−x |
V (x0 )hx0 |p±i
|x − x0 |
(A.161)
To see what happens when r = |x| is large, we suppose for simplicity that
V (x) ≡ 0 for r greater than some a (although our result actually holds just as
long as V = o(r−3−² ) as r → ∞). In this case the integral is confined to r0 < a,
and for large r we can expand |x − x0 | in powers of (r0 /r),
"
µ 0 ¶2 #
0
1
¡
¢
x
·
x
r
|x − x0 | = x2 − 2x · x0 + x02 2 = r 1 − 2 + o
(A.162)
r
r
The expansion (A.161) then becomes
Z
µe±ipr
d3 x0 exp(∓ipb
x · x0 )V (x0 )hx0 |p±i
hx|p±i = hx|pi −
2πr
·
µ
¶¸
a pa2
× 1+o
+
r
r
·
¸
e±ipr
−3/2
ip·x
2
(2π)
e
−
(2π)
µh±pb
x
|V
|p±i
−−−
−
−
−
−
−
→
r→∞
r
(A.163)
From (A.148) we obtain
−(2π)2 µhpb
x|V |p+i = −(2π)2 µt(pb
x ← p) = f (pb
x ← p)
Thus we can rewrite (A.163) for hx|p+i as
·
¸
eipr
−3/2
ip·x
hx|p+i −−−
(2π)
e
+
f
(pb
x
←
p)
−
−
−
−
−
→
r→∞
r
(A.164)
(A.165)
This establishes that our definition of the amplitude f (p0 ← p) in terms of
hp0 |S|pi is, in fact, the same as the traditional definition in terms of the asymptotic form of the stationary scattering wave function.
158
A.9 Identical Particles
So far we assumed that both particles, involved in the scattering process, are
(at least in principle) distinguishable. This means that we can discriminate
between the 2 scattering diagrams in Fig. A.4 which both give rise to the
same “signal” in the detector, if both particles are identical. The first diagram
corresponds to a scattering amplitude f (k, θ) and the second to f (k, π − θ). As
an example we consider the scattering of two electrons. Oppositely polarized
electrons essentially remain distinguishable. If the detector D counts electrons
of any spin then the probability for both processes just add up
1
2
1
2
θ
2
1
1
π−θ
D
2
D
Fig. A.4: Two scattering processes leading to the same final state for two identical
particles.
dσ
= |f (θ)|2 + |f (π − θ)|2
dΩ
(A.166)
If beams (1) and (2) are both polarized spin up or down, then the detector
cannot distinguish between the incident particles and we have to take the symmetrization into account. The two-particle wavefunction must obey
ψ(x1 , x2 ) = αψ(x2 , x1 )
(A.167)
with α = 1 for bosons and α = −1 for fermions. When we impose this symmetrization condition for the relative wavefunction, we obtain
ψ(x) = αψ(−x)
(A.168)
where x is the relative coordinate x = x1 − x2 . The (anti)symmetrized asymptotic wavefunction can be written as
1
1
eikz + αe−ikz
f (k, θ) + αf (k, π − θ) eikr
√
√
+
ψ(x)−−−
−
−
−
−
−
→
r → ∞ (2πh̄)3/2
r
(2πh̄)3/2
2
2
(A.169)
159
For bosons the amplitudes have to be added
dσ
= |f (θ) + f (π − θ)|2
dΩ b
(A.170)
while for fermions they have to be subtracted
dσ
= |f (θ) − f (π − θ)|2
dΩ f
(A.171)
At an angle θ = π/2 this implies
³π ´
dσ ³ π ´
= 4|f
|2
dΩ b 2
2
dσ ³ π ´
=0
dΩ f 2
(A.172)
so that identical fermions never scatter at an angle of π/2, while the differential
cross section for boson scattering is twice the classical value for this angle.
Using the parity of the Legendre polynomials
Pl (cos(π − θ)) = Pl (− cos θ) = (−1)l Pl (cos θ)
(A.173)
we can deduce that the only partial waves contributing to the cross section for
bosons correspond to even values of l, those for fermions to odd values of l. We
take this into account when we follow the same procedure, that leads to (2.28)
and see that the symmetrization principle doubles the contribution of the even
partial waves for bosons and cancels the contribution of the odd ones and vice
versa for fermions
8π X
σ(k) = 2
(2l + 1) sin2 δl (k) for bosons
(A.174)
k
l even
σ(k) =
8π X
(2l + 1) sin2 δl (k)
k2
l odd
160
for fermions
(A.175)
B. SCATTERING THEORY IN FREE SPACE: MULTIMODE
REGIME
So far we have discussed only processes in which structureless particles undergo
an elastic collision. However, almost all process of experimental interest do
include collisions involving several particles (composite or neutral) and including
inelastic processes such as excitation and disintegration. A typical example from
atomic physics is the set of processes:
e+H →e+H
e + H → e + H∗
e+H →e+e+p
(elastic scattering)
(excitation)
(ionization or breakup)
(where H ∗ denotes any of the excited states of hydrogen); or in nuclear physics:
p +12 C → p +12 C
p +12 C → n +12 N
12
p + C → p +3 He +9 Be, etc
(charge exchange)
or in particle physics:
Π− + p → Π− + p
Π− + p → K 0 + Λ0
−
Π + p → Π− + K 0 + Π0 + p, etc
Each of the different sets of final particles in each of these examples is called a
channel, and processes of this kind are called multichannel processes.
In this chapter we set up a description of the collision process in terms of
asymptotic free states and define a unitary S operator that maps each in state
onto the corresponding out state.
B.1 Channels
To illustrate the essential features of multichannel scattering we assume a system
of N spinless particles, which interact via short-range two-body potentials. A
channel α is specified by the grouping of the N particles into nα freely moving
fragments with 2 ≤ nα ≤ N , each fragment being either one of the original N
particles or a definite bound state of some subset (we do not include among the
channels the bound states of all N particles for which nα = 1. These remain
bound at all times and do not communicate with the scattering states, in which
we are interested.)
Particles
a
b
c
Bound States
(bc)
(bc)∗
(ac)
and (possible bound states of all three particles)
Tab. B.1: The Model Three Particle System.
Channel number
0
1
2
3
Channel
a+b+c
a + (bc)
a + (bc)∗
b + (ac)
Tab. B.2: Possible Channels.
To illustrate the essential features of multichannel scattering lets discuss a
system of three spinless particles, a, b and c, which interact via short-range
two-body potentials. We shall suppose that the particles b and c have two
bound states, a ground state (bc) and an excited state (bc)∗ ; that a and c have
one bound state (ac); and that, apart from possible bound states of all three
particles, there are no other bound states. For example, we can think of a, as a
proton, b as a neutron and c as some stable nuclear “core” such as 16 O; in this
case the (bc) bound state is 17 O, the (ac) state is 17 F , and we have a simple
model for the scattering of protons off 17 O.
For definiteness we suppose that we are interested in the disintegration process
a + (bc) → a + b + c
(B.1)
However, the given initial state a + (bc) will in general lead to several different
final states in addition to the particular one of interest. The possible grouping
of the particles a, b and c into stable subsystems or channels are shown in Table
B.2.
The process of interest (B.1) leads from an in state in channel 1 to an out
state in channel 0. Clearly, however, an in state in channel 1 can generally lead
to out state, in any of the four channels 0, 1, 2, or 3,
a + (bc) → a + b + c
a + (bc) → a + (bc)
a + (bc) → a + (bc)∗
a + (bc) → b + (ac)
(break up)
(elastic scattering)
(excitation)
(rearrangement)
(B.2)
In exactly the same way the final state of interest can arise from several different
initial states. In fact, a final state in any one of the four channels 0, 1, 2, or
162
3 can arise from an initial state in any of the same four channels. Therefore,
there are 16 qualitatively distinct processes to be considered.
A schematic view of multichannel scattering, which illustrates the name
channel, is shown in Fig. B.1. The various possible in channels are shown as
tube, or channels, through which a fluid of probability can flow into a junction.
This junction represents the actual collision, and from it lead the various possible
out channels. In practice the in states always lies in a definite channel, which
means that all of the fluid enters by one channel. The corresponding out state is
usually a superposition of the various possible channels and the fluid therefore
leaves through several channels in some definite proportions.
0
0
1
1
In
Out
2
2
3
3
Fig. B.1: A Schematic view of multichannel scattering as a quantum-mechanical irrigation system.
In practice, experiments are performed with initial states whose energy and
momentum are rather well defined. Since energy and momentum are conserved,
this means that at certain energies some of the channels may not be accessible.
For example, the original process of interest (B.1), cannot occur when the incident kinetic energy is insufficient to overcome the binding energy of (bc). In
fact, in our model there are four threshold energies, at each of which one of the
four channels “opens up”. Thus, if we suppose that the energies of the three
bound states (bc), (bc)∗ , and (ac) occur in the order
E(bc) < E(bc)∗ < E(ac) < 0
then for energies below E(bc) there may be bound states of all three particles
but there are no scattering states. In the energy range between E(bc) and E(bc)∗
we can scatter a off (bc) but no inelastic processes can occur, thus, only elastic
scattering of a + (bc) is possible and, as we shall see, it is given by an amplitude
with all the general properties (invariance properties, partial-wave decomposition, etc.) of the elastic amplitude of App. A. For energies between E(bc)∗
and E(ac) excitation is possible, and there are two open channels allowing four
possible processes. At E(ac) the channel b + (ac) opens up and there are then
three open channels and nine possible processes. And finally at E = 0, the
disintegration channel a + b + c opens up, therefore in the case of E ≥ 0 there
are four open channels, and all 16 processes are possible.
163
Our simple model should make clear all of the essential descriptive ideas related to the concept of a channel. In general a channel is simply a set of particles
(elementary or composite) that can enter or leave a collision. In nonrelativistic theory one deals with systems of a fixed number of elementary particles,
i = 1, ..., N . We shall always take as channel 0 that in which all N particles
move freely. The remaining channels α = 1, 2, ... are groupings of the particles
into nα stable fragments (2 ≤ nα < N ) each of which is either one of the original
particles or some definite bound state of some of them. The reason that the
fragment must be stable is that the channels specify the grouping of particles
in the asymptotic free states, which are defined as t → ±∞. Since only a stable
particle can live an infinitely long time, it is clear that -in principle at least- it
makes no sense to speak of a channel containing unstable fragments.
Nonetheless, scattering experiments are done with unstable particles. The
atomic physicist measures cross sections for excitation of atoms, even though
the excited atom will eventually decay back to its ground state. One of the
nuclear physicist’s most important tools is the unstable neutron; and almost all
elementary particles are unstable -many with exceedingly short lifetime. The
point is that whenever one speaks of a scattering experiment involving unstable
incident or outgoing fragment, the lifetime of these fragments, however short,
is nonetheless much longer than the characteristic time of the actual collision.
Thus, one can wait for what is (as regards the collision) an “infinitely” long
time and still observe the unstable fragments well before they decay.
In this work we shall always assume that for any process under consideration
all initial and final fragments of interest are stable. It should be emphasized that
to identify a channel it is necessary to specify both the grouping of the particles
and the internal state of each group. Thus, in our three-particle example, the
channels
a+b+c
a + (bc)
b + (ac)
are distinguishable by the arrangement of the particles into groups, while the
channels
a + (bc)
a + (bc)∗
have the same grouping and are distinguished by different internal states of (bc)
and (bc)∗ .
The number of channels for a given system can be either finite or infinite. In
our model, which can be regarded as typical of nuclear physics in this respect,
the number is finite. But when there are attractive Coulomb forces the number
is generally infinite. (For example, the e − H system has an infinite number
of channels, since the hydrogen atom has infinitely many bound states.) For
simplicity of discussion we shall suppose that the number of channels, α =
0, 1, ..., n, is finite, although very little is changed when n becomes infinite.
164
It should also be emphasized that there are various different possible choices
for the zero of energy. In the preceding discussion we have taken the energy when
all three particles are well separated and stationary as our zero. Theoretically,
this is the most natural choice (in nonrelativistic problems) since the total energy
is then just the sum of all kinetic energies plus all potentials (each of which goes
to zero for large separation). However it should be borne in mind that for
an experimental starting, for example, in the channel a + (bc), a very natural
choice for zero-point would be the energy of a and (bc) when well separated and
stationary. This choice differs from the previous one by the amount E(bc) .
B.2 Channel Hamiltonian and Asymptotic States
In this section we discuss the quantum-mechanical evolution of a collision experiment. The discussion will proceed as a natural generalization of the one-channel
discussion of App. A. In particular, the three essential results that lead up to
the introduction of the S operator -the asymptotic condition, the orthogonality
theorem and asymptotic completeness- are close analogues of the corresponding
results of the one channel case.
The time development of any state is determined by the Hamiltonian which
we take to have the form
N
N
−1 X
N
X
X
p2i
H=
+
Vij (xij ) ≡ H 0 + V
2m
i
i=1
i=1 j=i+1
(B.3)
where xij = xi − xj and H 0 is the sum of the N kinetic energies. In terms of
this Hamiltonian the general orbit of the system has the usual form (h̄ = 1)
U (t)|ψi ≡ e−iHt |ψi
(B.4)
where now, of course, |ψi is any vector in the N -particle Hilbert space H =
L2 (R3N ) defined by wave functions of the N coordinates,
ψ(x1 , ..., xN ) ≡ ψ(e
x)
Here we have introduced a tilde to denote the set of all particle coordinates
x
e ≡ (x1 , ..., xN ).
Returning to our three-particle model, the Hamiltonian, takes the form
H = H0 + V =
p2
p2
p2a
+ b + c + Vab (xab ) + Vac (xac ) + Vbc (xbc ) (B.5)
2ma
2mb
2mc
Let us now consider a scattering orbit originating in channel 0; that is,
which originated as three freely moving particles, a + b + c. For such an orbit
we naturally expect that
e−iHt |ψi
−−−−−−−−−→
t → −∞
165
0
e−iH t |ψin i
(B.6)
for some in state |ψin i. We shall see that this result, and a corresponding result
for out asymptotes are indeed true for the appropriate states |ψi, and that they
are true for exactly the same reason as before - as the three particles move apart
all of their interactions cease to have any effect.
Suppose however we consider an orbit U (t)|ψi which originated in channel
1, a + (bc). When the two particles a and (bc) move apart (as we follow the
orbit back in time) the interactions Vab and Vac between the particle a and the
two particles b and c become ineffective. On the other hand the interaction Vbc
between b and c can never lose its importance. Indeed, it is only because of V(bc)
that the bound state (bc) remains bound. Thus, for an orbit that originated in
channel 1, the part of H that is effective long before the collision is not H 0 but,
rather, the “channel 1 Hamiltonian”
H1 =
p2a
p2
p2
+ b + c + Vbc
2ma
2mb
2mc
(B.7)
and for such an orbit we must expect that
e−iHt |ψi
−−−−−−−−−→
t → −∞
1
e−iH t |ψin i
(B.8)
for some |ψin i.
It is important to note that if the orbit U (t)|ψi does originate in channel
1, then the wave function of |ψin i must describe a state in which the position
xa of particle a and xbc of the center of mass of (bc) move arbitrarily, but the
relative motion of b and c is fixed as that appropriate to the bound state (bc);
that is,
he
x|ψin i = χ(xa , xbc )φ(bc) (xbc )
(B.9)
Here χ(xa , x) describes the motion of the incident particles a and (bc) and is an
arbitrary normalizable function of xa and the center of mass xbc of b and c. On
the other hand φ(bc) (xbc ) is uniquely determined as the wave function for the
internal motion of the bound state (bc). Thus, not every vector in H can label
an in asymptote of channel 1; only those vectors in the subspace L1 ⊂ H made
up from wave functions of the form (B.9) can label an in asymptote of channel
1. This subspace L1 is called the channel 1 subspace and consist of those vectors
that can label in or out asymptote in channel 1.
The action of the channel Hamiltonian H 1 on the vectors of L1 is especially
simple. By going into the relative and center of mass coordinates of the particles
b and c we can write
H1 =
p2a
p2
p2
+ bc + ( bc + Vbc )
2ma
2Mbc
2mbc
(B.10)
(B.11)
where pbc and pbc are the total and relative momentum operators for b and
c, while Mbc and mbc are their total and reduced masses. It is clear that the
166
asymptotic behavior (B.8) can be rewritten as
· µ 2
¶ ¸
pa
p2bc
−iHt
−iH 1 t
e
|ψi −−−−−−→ e
|ψin i = exp −i
+
+ E(bc) t |ψin i
t → −∞
2ma
2Mbc
(B.12)
That is, the asymptotic behavior is just that of two freely moving particles of masses ma and Mbc = (mb + mc ) - except for the additional phase factor
exp(−iE(bc) t). Here E(bc) is the eigenenergy of the relative Hamiltonian corresponding to the bound state φ(bc) (xbc ); that is
µ 2
¶
pbc
+ Vbc φ(bc) (xbc ) = E(bc) φ(bc) (xbc )
(B.13)
2mbc
Exactly parallel consideration apply to all other channels. In the general N particle case, for a channel α, the corresponding channel Hamiltonian H α is
obtained by deleting from H those potentials that link different fragments,
X0
Hα = H −
Vij
(B.14)
P0
Where
denotes a sum over all pairs ij for which particles i and j belong to
different fragments of channel α - when the two particles i and j move apart
(as we follow the orbit back in time) the interactions Vij become ineffective -.
The in and out asymptotes in channel α are identified by wave functions in
the subspace Lα comprising those functions with the form
χ(y1 , ..., ynα )φ1 (z1 )...φnα (znα )
where χ is an arbitrary function of the centers of mass y1 , ..., ynα of the nα
fragments. The term φν (zν ) is the bound-state wave function of the ν th fragment
with internal coordinates zν . (If the ν th fragment happens to be a single particle,
then φν ≡ 1.)
B.2.1
Asymptotic condition
If the particle interactions Vij (xij ) all satisfy our usual assumptions, then for
every vector |ψin i in any channel subspace Lα there is a vector |ψi that satisfies
e−iHt |ψi
−−−−−−−−−→
t → −∞
α
e−iH t |ψin i
(B.15)
and which is given in terms of a channel Møller operator Ωα
+ as
α
iHt −iH t
|ψi = Ωα
e
|ψin i
+ |ψin i = lim e
t→−∞
(B.16)
and similarly for every |ψout i in Lα as t → +∞, with |ψi = Ωα
− |ψout i. It should
be emphasized that there are separate Møller operators Ωα
± for each channel α
and that they are defined by the limit (B.16) only for those vectors in Lα . In
fact, we shall find that Lα is the largest space on which we need to define Ωα
±.
167
The asymptotic condition guarantees that every vector in the channel subspace Lα labels a possible in or out asymptotic in channel α. If |ψin i is in Lα
then the vector
|ψi = Ωα
(B.17)
+ |ψin i
is that actual state of the system which has developed from the in state labelled
by |ψin i in channel α. Similarly, if |ψout i is in Lα , then
|ψi = Ωα
− |ψout i
(B.18)
is the state that will develop into the out state |ψout i in channel α.
With this result we can calculate the probability that a system that enters
a collision in channel α with in asymptote |φi (in Lα ) be observed to leave in
0
channel α0 with out asymptote |φ0 i (in Lα ). If the in state was |φi in channel
α, then the actual state at t = 0 would be Ωα
+ |φi. If the out state was going to
0
0
0
0
be |φ i in channel α , then the actual state at t = 0 would have to be Ωα
− |φ i.
The required probability amplitude is just the overlap of these two states
0
† α
2
w(φ0 , α0 ← φ, α) = |hφ0 |Ωα
− Ω+ |φi|
B.2.2
(B.19)
Orthogonality and asymptotic completeness
Just as in single-channel scattering any state that has developed from some
in asymptote (or will develop into some out asymptote) should be orthogonal
to any bound state of all N particles. In addition, we would expect that any
two states that have developed from in asymptotic in (or will develop into out
asymptotes of) different channels should be mutually orthogonal. These results
are the content of the orthogonality theorem.
Orthogonality Theorem. Let |φi be any bound state of all N particles
and let
|ψi = Ωα
+ |ψin i
0
(B.20)
0
|ψ 0 i = Ωα
+ |ψin i
0
0
with |ψin i in Lα and |ψin
i in Lα and α 6= α0 . Then
hφ|ψi = hφ|ψ 0 i = hψ|ψ 0 i = 0
α
and likewise with Ωα
+ replaced by Ω− (and “in” by “out”) in (B.20).
By defining B to be the subspace spanned by the bound states of all N
α
α
particles and Rα
± to be the ranges of Ω± , i.e., R+ is the subspace of all states
α
that originated in the in channel α, while R− is the subspace of all states that
will terminate in the out channel α, the theorem can be stated alternatively as:
α
B⊥Rα
+ ⊥R+
0
[all α, α0 ;
α 6= α0 ]
0
[all α, α0 ;
α 6= α0 ].
and similarly,
α
B⊥Rα
− ⊥R−
168
So far we have discussed only those scattering orbits that have developed
from a definite in channel, or will develop into some definite out channel. These
are certainly not the most general kind of scattering orbits. For example, suppose that |ψ 1 i and |ψ 2 i are states that have developed from in asymptotes in
two different channels, α = 1 and α = 2 (say),
1
|ψ 1 i = Ω1+ |ψin
i
and
2
i
|ψ 2 i = Ω2+ |ψin
Then the superposition principle asserts that the vector |ψi = |ψ 1 i+|ψ 2 i defines
an allowed physical state, and we can ask the question: what is the asymptotic form of the orbit defined by |ψi? Now, we know that as t → −∞
1
e−iHt |ψ 1 i
−→
1
i
e−iH t |ψin
e−iHt |ψ 2 i
−→
2
i
e−iH t |ψin
and
2
Adding these two results we immediately see that the orbit defined by |ψ 1 i+|ψ 2 i
has the asymptotic form
e−iHt (|ψ 1 i + |ψ 2 i)
−→
1
2
1
2
e−iH t |ψin
i + e−iH t |ψin
i
That is, this orbit originates as a superposition of states in the two in channels
1 and 2.
Having once recognized the existence of orbits that originate as a superposition of two in channels, we must obviously expect that the most general
scattering orbit would be one which originated as a superposition of all possible
in channels.
e−iHt |ψi
−−−−−−−−−→
t → −∞
0
n
0
n
e−iH t |ψin
i + · · · + e−iH t |ψin
i
(B.21)
α
where each ψin
lies in the appropriate channel subspace Lα and
0
n
|ψi = Ω0+ |ψin
i + · · · + Ωn+ |ψin
i
(B.22)
α
In the asymptotic form (B.21) it is natural to regard each term exp(−iH α t)|ψin
i
as the component of the in asymptote in the channel α. The general in asymptote can be identified by giving the sequence
© 0
ª
n
|ψin i = |ψin
i, . . . , |ψin
i
(B.23)
in the space
Has = L0 ⊕ · · · ⊕ Ln
169
(B.24)
α
where each |ψin
i identifies the component of the incoming asymptotic behavior
in the corresponding channel α. The new space Has , which we call the space
of asymptotic states, is (in an obvious sense) bigger than H, since L0 alone is
equal to H.
In the same way we should expect the general scattering orbit to evolve as
t → +∞ into a superposition of all possible out channels
e−iHt |ψi −−−−−−−−−→
t → +∞
0
n
0
n
i + · · · + e−iH t |ψout
i
e−iH t |ψout
(B.25)
with
0
n
|ψi = Ω0− |ψout
i + · · · + Ωn− |ψout
i
This asymptotic form is, of course, identified by the sequence
© 0
ª
n
|ψout i = |ψout
i, . . . , |ψout
i
(B.26)
(B.27)
in space Has .
It should be noted that in practice the in state always lies in one definite
channel α and, hence, is given by a sequence
α
{0, . . . , 0, |ψin
i, 0, . . . , 0}
(B.28)
On the other hand the out state is usually a superposition of several channels and
has the general form (B.27) with several nonzero components (except of course
in the special case where all inelastic processes are energetically impossible).
According to (B.22), we can define a linear operator Ω+ on our new space Has
as
© 0
ª
n
Ω+ |ψin i = Ω+ |ψin
i, . . . , |ψin
i
0
n
= Ω0+ |ψin
i + · · · + Ωn+ |ψin
i
(B.29)
which maps Has into H and for each in asymptote labelled by |ψin i, the corresponding actual state at t = 0 is just |ψi = Ω+ |ψin i. Similarly, we can define
Ω− so that the actual state at t = 0 corresponding to any out asymptotic |ψout i
is just |ψi = Ω− |ψout i.
Asymptotic Completeness. A multichannel scattering theory is asymptotically complete if the scattering states |ψi [satisfying (B.21) and (B.25)] together with the bound states should span the space H of all states. In this case
H can be written as the direct sum
H=B⊕R
Where B is the space spanned by the bound states of all N particles, while R
is the space spanned by the scattering states, which has to be the direct sum:
R = R0+ ⊕ · · · ⊕ Rn+ = R0− ⊕ · · · ⊕ Rn−
170
(B.30)
and Rα
± is the subspace of all states that originated as (or will develop into) an
asymptotic state in channel α.
Asymptotic completeness for a three-body system with suitable potentials
was proved by Faddeev in 1965 [96] and his proof was extended to the N -body
case by Hepp in 1969 [97].
α
It should be emphasized that the subspace Rα
+ and R− are not in general
α
α
the same. Indeed, if it were true that R+ = R− , then every orbit that came
from the in channel α must necessarily evolve into the same out channel α; that
is, no inelastic processes could occur. The observed fact that inelasticity does
0
n
occur implies that each Rα
+ overlaps several of the spaces R− , . . . , R− and vice
versa.
The two decompositions of R on the direct sums (B.30) express the fact that
every scattering state can be written as a superposition of states each of which
originated in a definite in channel α, and also as a superposition of states each
of which will develop into a definite out channel α. This implies that every
scattering orbit has in and out asymptotes of the forms (B.23) and (B.27).
Provided the theory is asymptotically complete, Ω± are actually isometric
from Has onto R, the space of scattering states, that is, its domain is Has , its
range R, and it preserves the norm. The situation is as follows:
isometric
isometric
Has −−−−−−−−−→ R ←−−−−−−−−− Has
Ω+
Ω−
(B.31)
or, in terms of vectors,
|ψin i
−−−−−−−→
Ω+
|ψi ←−−−−−−−
Ω−
|ψout i
(B.32)
The analogy with the single-channel problem is now complete. In particular,
we can invert
|ψi = Ω− |ψout i
(B.33)
to give
|ψout i = Ω†− |ψi = Ω†− Ω+ |ψin i
(B.34)
Thus, if we define the scattering operator
S = Ω†− Ω+
(B.35)
then S maps each in asymptote onto the corresponding out asymptote
|ψout i = S|ψin i
(B.36)
Since S is isometric from Has onto Has , it is actually unitary.
The probability amplitude for a system that enters a collision with in asymptote given by |φi in Has to be observed leaving with out asymptote |φ0 i is just
the matrix element hφ0 |S|φi,
w(φ0 ← φ) = |hφ0 |S|φi|2
171
(B.37)
or
w(φ0 ← φ) = |hφ0 |Ω†− Ω+ |φi|2
(B.38)
In practice, the experimental initial state is prepared in a definite channel and
the in asymptote |φi has the form
|φi = {0, . . . , 0, |φi, . . . , 0}
with |φi in a definite Lα . The corresponding out state S|φi is naturally not
(in general) in any definite channel. However, in practice one always monitors
for a final state that does lie in a definite channel, that is, in practice we are
interested in a |φ0 i of the form
|φ0 i = {0, . . . , 0, |φ0 i, . . . , 0}
0
with |φ0 i in a definite Lα . For these states we have
and
Ω+ |φi = Ωα
+ |φi
(B.39)
0
Ω− |φ0 i = Ωα
− |φ i
(B.40)
Thus, for such states the probability (B.37) can be written as
0
† α
2
w(φ0 , α0 ← φ, α) = |hφ0 |Ωα
− Ω+ |φi|
(B.41)
This is precisely the result quoted in (B.19) and serves to emphasize that if
one is interested solely in the computation of transition probabilities between
two different channels, it is actually unnecessary to introduce Has . It is only if
we wish to express the required probability amplitudes as matrix elements of a
single unitary operator S that this space is needed.
B.3 The Momentum-Space Basis Vectors
The momentum-space basis of Has is a generalization of the basis of planewave states |pi used in the one-particle case, and of the corresponding states
|p1 , p2 i ≡ |p, pi in the two-particle problem. For simplicity we discuss first our
simple model of three particle a, b, c introduced in the previous sections.
Since Has is the direct sum of the channel spaces Lα we can begin by constructing a basis for each Lα . We start with the channel 0, in which a, b, and
c all move freely, and for which the general asymptote is labelled by an arbitrary wave function χ(xa , xb , xc ) of all these positions. The appropriate basis
is clearly given by the three particle momentum eigenstates:
|pa , pb , pc ; 0i ≡ |e
p, 0i
(B.42)
with corresponding wave functions
he
x|e
p, 0i = (2π)−9/2 exp [i(pa · xa + pb · xb + pc · xc )]
172
(B.43)
and normalization
he
p0 , 0|e
p, 0i = δ3 (p0a − pa )δ3 (p0b − pb )δ3 (p0c − pc ) ≡ δ9 (e
p0 − pe)
(B.44)
[e
p stands for the set (pa , pb , pc ) and 0 is the channel number α = 0.] These
vectors are, of course, eigenvectors of the free Hamiltonian,
µ 2
¶
p2b
p2c
pa
0
p, 0i
(B.45)
H |e
p, 0i =
+
+
|e
p, 0i ≡ Ep0 |e
2ma
2mb
2mc
In channel 1, a + (bc), the asymptotes are identified by wave functions
χ(xa , xbc )φ(bc) (xbc )
(B.46)
where χ is an arbitrary function of the position xa and the center of mass xbc
of b and c, while φ(bc) is the bound-state wave function of (bc). Since φ(bc) is
fixed these functions can be spanned by products of plane waves in xa and xbc
with the fixed function φ(bc) (xbc ). The appropriate basis vectors are:
|pa , pbc ; 1i ≡ |e
p, 1i
(B.47)
he
x|e
p, 1i = (2π)−3 exp [i(pa · xa + pbc · xbc )] φ(bc) (xbc )
(B.48)
with wave functions
and normalization
he
p0 , 1|e
p, 1i = δ3 (p0a − pa )δ3 (p0bc − pbc ) ≡ δ6 (e
p0 − pe)
(B.49)
These vectors represent states in which particle a moves with momentum pa ,
while b and c are bound together into the bound state (bc), whose CM moves
with momentum pbc .
The energy operator for asymptotic states in channel 1 is the channel Hamiltonian H 1 , and the vectors |e
p, 1i are, of course, eigenvectors of this operator,
µ 2
¶
pa
p2
1
|e
p, 1i
(B.50)
H 1 |e
p, 1i =
+ bc + E(bc) |e
p, 1i ≡ Ee
p
2ma
2Mbc
where E(bc) is the internal energy of the bound state (bc). Thus, the improper
vectors |e
p, 1i will represent in and out states of definite energy E 1 in channel 1.
e
p
The momentum basis vectors for the other channels are constructed in exactly the same way.
The construction of the corresponding bases of Lα in the general N -particle
problem is entirely straightforward and need not be spelled out here. We mention only that, since the general channel α has nα freely moving fragments, the
corresponding basis vector has the form |e
p, αi where pe labels the nα momenta
of these fragments.
Since Has is the direct sum
Has = L0 ⊕ · · · ⊕ Ln
173
(B.51)
we can obtain an orthonormal basis of Has by combining any orthonormal bases
of L0 , . . . , Ln . In particular, the vectors |e
p, αi (α fixed, all pe) are an (improper)
orthonormal basis of Lα and, hence, the set of vectors
{0, . . . , 0, |e
p, αi, 0, . . . , 0} ≡ |e
p, αi [α = 0, . . . , n; all pe]
(B.52)
is the desired momentum basis of Has . It should be noted that to span Has we
must include all momenta and all channels.
The basis vector (B.52) represents an asymptote with momenta pe lying completely in the channel α. We use the same symbol |e
p, αi both for the basis vectors
of Lα in H, and for the sequence in Has . In practice it will always be clear which
kind of vector is being used, and this imprecise usage will cause no confusion.
With this convention the orthonormality of our basis of Has is expressed as
he
p0 , α0 |e
p, αi = δα0 α δ(e
p0 − pe)
(B.53)
where the factor δα0 α reflects the orthogonality (as vectors of Has ) of any two
vectors belonging to different channels. With respect to this basis the S operator
becomes a matrix with elements he
p0 , α0 |S|e
p, αi and knowledge of all of these
elements is equivalent to knowledge of S. Similarly, an operator equation like
the unitarity equation S† S = 1 becomes the matrix equation
XZ
de
p00 he
p0 , α0 |S† |e
p00 , α00 ihe
p00 , α00 |S|e
p, αi = δα0 α δ(e
p0 − pe)
(B.54)
α00
To conclude this section we remark that so far we have assumed that all
particles are spinless and that all bound states have zero orbital angular momentum. The inclusion of either kind of angular momentum into our formalism
is completely straightforward. Suppose, for instance, that in our three-particle
model the particle a has spin s and that the state (bc) has orbital angular momentum l. Let us then consider an asymptotic state (either in or out) in which
a and (bc) move freely. There are various possible orientations of the spin of
particle a, and similarly, of the orbital momentum of (bc); and it is natural to
regard these as being different “spin” orientations all within the same channel.
Thus, we denote as channel 1 all free states of a and (bc), and the corresponding
momentum eigenvectors we label
|pa , pbc , ma , mbc ; 1i
(B.55)
where pa and pbc are the momenta of the two fragments, ma is the z component
of the spin of particle a, and mbc is the z component of the orbital angular
momentum of (bc). The corresponding wave function has the form
(2π)−3 exp [i(pa · xa + pbc · xbc )] χma φ(rbc )Ylmbc (b
xbc )
(B.56)
where χma is the appropriate spinor for particle a, and the last two factors are
the radial and angular wave functions of the (bc) bound state.
It should be clear that, at this stage, fragments with spin (either intrinsic
or “orbital”) are no more than a slight notational complication, and we shall,
for the most part, proceed under the simplifying assumption that none of the
fragments in any channel have angular momentum.
174
B.4 Conservation of Energy and the On-Shell T Matrix
We are now in a position to establish conservation of energy, which follows in almost exactly the same way as in the one-channel case. By using the intertwining
relations
α
α
HΩα
[on Lα ]
(B.57)
± = Ω± H
which follow at once from the definition
iHt −iH
Ωα
e
± = lim e
α
t
t→∓∞
[on Lα ]
(B.58)
exactly as in App. A, one can show easily that, the S-matrix element he
p0 , α0 |S|e
p, αi
α
α0
between initial and final states of energies E and E 0 -which we shall abbrevie
p
e
p
ate as E and E 0 when there is no danger of confusion - is zero unless the initial
and final energies are equal; that is, S conserves energy and its matrix elements
contain the expected factor δ(E 0 − E).
In addition to conserving energy, S also conserves total momentum. This
is because all potentials involve only the relative positions and the system is
therefore translationally invariant.
This means that S commutes with the total
P
momentum operator p =
pi and, hence, that its matrix elements contain a
factor δ3 (p0 − p). In fact, we can go further: By using as independent variables
the overall CM position X and any suitable choice of N − 1 relative coordinates
we can factor the space H (and similarly Has ) as
H = HCM ⊗ Hrel
(B.59)
Here, just as in the two-particle case, HCM describes the motion of the overall
CM , and Hrel the relative motion of the particles. The Hamiltonian can then
be written as
H = HCM + Hrel
(B.60)
where HCM = p2 /2M and M is the total mass of the system; and just as in
two-particle scattering, the S operator factors as
S = 1CM ⊗ S
(B.61)
The operator S describes the relative motion and is the scattering operator one
would obtain directly from the Hamiltonian Hrel . Just as in the one-channel
case all of the physically interesting information is contained in S and much of
our subsequent analysis will be in terms of S.
Returning to the matrix elements of S, by combining the consequences of
energy and momentum conservation, one obtains
he
p0 , α0 |S|e
p, αi = δα0 α δ(e
p0 − pe) − 2πiδ(E 0 − E)δ3 (p0 − p)t(p0 , α0 ← p, α) (B.62)
Here the on-shell T matrix t(p0 , α0 ← p, α) depends only the sets of initial and
final relative momenta, which we denote with a single underscore as p and p0 .
(That is, p denotes the set of nα − 1 suitably chosen relative momenta for the
175
fragments of channel α.) It is defined only on the energy shell E 0 = E, i.e.,
for those p and p0 consistent with conservation of energy. It can, of course, be
computed directly from the operator S of the relative motion, whose matrix
elements hp0 , α0 |S|p, αi have a decomposition similar to (B.62), but without the
factor δ3 (p0 − p).
The interpretation of the decomposition (B.62) is much the same as in the
one-channel case. The first term is the S matrix for no scattering and leaves all
momenta and channels unchanged. The second represents the actual scattering;
it conserves total energy and total momentum but can in general change the
relative momenta and the channel.
The fact that S can only connect states of the same energy leads to important
restrictions on what we can call the “channel structure” of the S matrix. To
illustrate this, let us return to our three-particle model with its four channels α =
0, ..., 3. We can regard the discrete channel indices α0 and α in hp0 , α0 |S|p, αi or
t(p0 , α0 ← p, α) as labelling the rows and columns of a matrix in “channel space”,
each element of which is a function of the momenta p0 and p. In our model we
have a matrix with up to four rows and columns. However, at certain energies
the dimension is actually less than four as we now discuss. For definiteness we
shall confine our attention to the CM frame (that is, we discuss the S matrix of
the relative motion), and we now imagine the energy to be increased from some
large negative value.
All states in a given channel have energy at least as great as the channel’s
threshold energy. Thus for energies less than the lowest threshold E(bc) there
are no states in any channel and hence, no S matrix. When the energy increases
into the range E(bc) ≤ E < E(bc)∗ there are states in channel 1 but none in any
other. Thus, in this energy range, S is a 1 × 1 “channel-space” matrix
hp0 , 1|S|p, 1i
(B.63)
which has many of the properties of the one-channel S matrix. For example, if
we choose an angular momentum basis of the channel 1 subspace and assume
rotational invariance, then exactly the arguments of section A.6 show that
hE 0 , l0 , m0 ; 1|S|E, l, m; 1i = δ(E 0 − E)δl0 l δm0 m e2iδl (E)
(E(bc) ≤ E < E(bc)∗ )
(B.64)
where the phase shift δl (E) is real.
If we now increase the energy into the range E(bc)∗ ≤ E < E(ac) there are
states in channel 1 and 2 but in no others. Thus, S becomes a 2 × 2 “channelspace” matrix:
µ
¶
hp0 , 1|S|p, 1i hp0 , 1|S|p, 2i
(B.65)
hp0 , 2|S|p, 1i hp0 , 2|S|p, 2i
clearly, as we increase the energy past each threshold the corresponding channel
opens up and the matrix gains one dimension. Finally, when E ≥ 0 all channels
are open and the S matrix has its full complement of four rows and four columns.
In conclusion, note the relative motion in any two-body channel is specified
by a single momentum: the relative momentum of the two fragments. Thus, as
176
long as only two-body channels are open, each element hp0 , α0 |S|p, αi is in fact
labelled by just two momenta p0 and p, as indicated for example in (B.65). In
this energy range the multichannel S matrix is more complicated than that of the
one-channel problem only inasmuch as it has more than one element. Once the
channels with three or more bodies open up, the corresponding matrix elements
depend on several variables and the situation is markedly more complicated.
B.5 Cross Section
We must now set about computing the observable cross sections in terms of the
on-shell T matrix. Since it is an experimental fact that almost all processes of
current interest have two-body initial states, we shall confine our discussion to
such processes, and consider a process leading from a two-body in channel α
to an arbitrary out channel α0 (with n0 bodies). In our three-particle model
we could consider any of the processes (B.2). Finally, we work in the CM
frame, considering just the relative motion, and take an initial state |φi in Lα ,
prescribed by its momentum-space wave function φ(p) in the relative momentum
p of the two initial particles which we assume to be well peaked about the mean
incident momentum p0 . The corresponding in asymptote must, of course, be
written as
|ψin i = |φi = {0, . . . , 0, |φi, 0, . . . , 0}
where |φi occupies the αth position.
The probability that, with the given in state |ψin i = |φi in channel α, the
final particles be observed in channel α0 with their momenta in some prescribed
volume ∆0 about some p0 in the (3n0 − 3)-dimensional space of the relative
momenta of this channel is given by
Z
w(∆0 , α0 ← φ, α) =
dp0 |hp0 , α0 |S|φi|2
0
Z ∆
≈
dp0 |hp0 , α0 |S|p, αi|2
(B.66)
∆0
0
In principle, ∆ can be an arbitrary volume in the space of the final momenta.
In the case that the final channel is a two-body channel, then p0 reduces to
a single momentum p0 (the relative momentum of the two final particles) the
natural choice for ∆0 is the familiar cone defined by an element of solid angle
about any fixed direction.
We can replace the S-matrix element by the appropriate multiple of the onshell T matrix. (For elastic scattering, α0 = α, this requires that we avoid the
forward direction; for inelastic scattering this restriction is unnecessary.) One
can obtain for the effective cross section of the target for scattering the packet
φ into the volume ∆0 in channel α0 [1]
Z
Z
1
0
0
4
0
d3 p δ(E 0 − E)|t(p0 , α0 ← p, α)|2 |φ(p)|2
dσ(∆ , α ← φ, α) = (2π) µ
dp
p
k
∆0
(B.67)
177
where µ denotes the reduced mass of the initial two particles in channel α and
pk is the component of p along p0 . Provided the in wave packet is sufficiently
well peaked about its mean momentum (which we now call p) the integral over
p disappears as the normalization integral for φ and we obtain an answer that
is independent of the shape of φ and so can be written as
Z
0
0
4µ
dp0 δ(E 0 − E)|t(p0 , α0 ← p, α)|2
dσ(∆ , α ← p, α) = (2π)
(B.68)
p ∆0
Here we just discuss the simplest case of a two-body final channel [ e.g. any of the
first three processes in (B.2)]. In this case the final relative momenta p0 reduce
to a single momentum p0 and the volume ∆0 is a volume in the corresponding
three-dimensional space. Because of energy conservation there is no interest in
measurement of the magnitude of p0 , and the smallest interesting volume ∆0
is the familiar cone defined by the infinitesimal solid angle dΩ. For this case
(B.68) becomes
Z ∞
µ
p02
p2
dσ(∆0 , α0 ← p, α) = (2π)4 dΩ
p02 dp0 δ( 0 + Wα0 −
− Wα )
p
2µ
2µ
0
⊗|t(p0 , α0 ← p, α)|2 (B.69)
2
p
+ Wα , where Wα denotes the
Since the energy E in a two-body channel α is 2µ
channel threshold. Rewriting the left-hand side in the familiar form (dσ/dΩ)dΩ
this given
dσ 0 0
p0
(p , α ← p, α) = (2π)4 µµ0 |t(p0 , α0 ← p, α)|2
dΩ
p
(α and α0 two − body channels)
(B.70)
In the case of elastic scattering, α0 = α, this result reduces to
dσ 0 0
(p , α ← p, α) = (2π)4 µ2 |t(p0 , α0 ← p, α)|2
dΩ
(B.71)
which has exactly the form of the one-channel result and includes the later as
a special case. For inelastic processes, α0 6= α, the result (B.70) differs from
(B.71) in two respects. The reduced masses µ and µ0 may be different [as, for
example, in the rearrangement collision a + (bc) → b + (ac)]. And the initial
and final momenta may be different, since p0 is fixed by energy conservation to
satisfy
p02
p2
+ Wα0 =
− Wα
0
2µ
2µ
For example, in the excitation process a + (bc) → a + (bc)∗ ,
p0 = [p2 − 2µ(E(bc)∗ − E(bc) )]1/2
178
(B.72)
It is convenient, as in the one-channel case, to introduce a scattering amplitude,
defined as
f (p0 , α0 ← p, α) = −(2π)2 (µ0 µ)1/2 t(p0 , α0 ← p, α)
(B.73)
in terms of which the cross section for any process with two-body initial and
final states is
p0
dσ 0 0
(p , α ← p, α) = |f (p0 , α0 ← p, α)|2
(B.74)
dΩ
p
To conclude this section we remark that if we take the whole of the momentum space of channel α0 for the volume ∆0 , then we obtain the total cross section
for scattering into channel α0 , which we denote σ(α0 ← p, α). (In particular,
for α0 = α this is the total elastic cross section.) If we then sum this over all
final channels α we obtain the total cross section,
Z
X
µX
σ(α0 ← p, α) = (2π)4
σ(p, α) =
dp0 δ(E 0 − E)|t(p0 , α0 ← p, α)|2
p 0
0
α
α
(B.75)
while if we restrict this sum to α0 6= α, we obtain the total inelastic cross section.
B.6 Rotational Invariance
If our multichannel system is rotationally invariant, then the rotational operator
R commutes with the Hamiltonian H and H α . It follows that it commutes with
Ωα
± and, hence, with S,
S = R† SR
(B.76)
If all of the particles and their bound states are spinless, the effect of the rotation
R on the channel basis vectors |p, αi is just
R|p, αi = |pR , αi
(B.77)
where pR denotes the effect of rotating all of the relative momenta p. Thus,
taking matrix elements of (B.76), we obtain the very natural result
hp0 , α0 |S|p, αi = hp0R , α0 |S|pR , αi
(B.78)
and corresponding equalities for the T matrix t(p0 , α0 ← p, α). If any of the
fragments of channels α and α0 have spins (due either to the intrinsic spins of
the original particles, or to the orbital momentum of the bound states), then
this result is complicated by the transformation properties of the spin indices.
As, in the one-channel case, the simplest way to exploit rotational invariance
is to work in an angular-momentum basis. This is especially simple if we consider
an energy at which only two-body channels are open and suppose further that
none of the bodies have spin. In this case the angular-momentum basis (of the
relative motion) in each channel is a simple orbital basis with vectors |E, l, m; αi.
Rotational invariance implies that the corresponding S matrix has the form
hE 0 , l0 , m0 ; α|S|E, l, m; αi = δ(E 0 − E)δl0 l δm0 m slα0 α (E)
179
(B.79)
That is, the scattering for given E and l is determined by an n × n matrix sl (E)
in “channel-space”, n being the number of open channels at the energy E.
The passage between the momentum and angular-momentum bases proceeds
exactly as in the one-channel case. Corresponding to the definition
fl (E) =
sl (E) − 1
2ip
(B.80)
of the one-channel amplitude, we define the multichannel partial-wave amplitude
as
sl 0 (E) − δα0 α
fαl 0 α (E) = α α
(B.81)
2i(pα0 pα )1/2
Here pα denotes the momentum in channel α when the total energy is E - that
is,
pα = [2mα (E − Wα )]1/2
(B.82)
where Wα is the the threshold of channel α - and the Kronecker delta δα0 α is
to ensure that the amplitude is related to the matrix elements of the operator
S − 1 in the normal way. With our definition, the multichannel partial-wave
series has the simple form (for α and α0 both two-body channels)
X
(2l + 1)fαl 0 α (E)Pl (cos θ)
(B.83)
f (p0 , α0 ← p, α) =
l
The unitarity of S implies that the n × n matrix sl (E) of (B.79) is a unitary
matrix. Since the dimension of this matrix depends on the energy, it is of
some interest to discuss the unitary equation as a function of energy. We start
with the energy E just above the threshold of the lowest channel (which we
label α = 1), and imagine E to be increased steadily past the various threshold
Wα (which we label in order of increasing energy). When W1 ≤ E ≤ W2 , the
matrix sl (E) is a one-dimensional unitary matrix and hence has a single element
of modulus one |sl11 | = 1, or
sl11 (E) = e2iδl (E)
[W1 ≤ E < W2 ]
(B.84)
with δl (E) real. If we consider the partial-wave amplitude defined in (B.81) this
l
lies on the unitary circle (see Fig. B.2).
means that the quantity p1 f11
When E moves above the first inelastic threshold, sl (E) becomes a 2 × 2
unitary matrix; and more generally, when E moves above Wn (still assume to
be a two-body threshold) sl (E) becomes an n × n matrix satisfying
sl (E)† sl (E) = 1
(B.85)
If we consider the (1, 1) matrix element of this equation, we see that
|sl11 |2 + |sl21 |2 + · · · + |sln1 |2 = 1
(B.86)
Clearly, as soon as there is any inelsticity, the original elastic element sl11 must
have modulus less than one, sl11 < 1 or
sl11 (E) = ²l (E)e2iδl (E)
180
(B.87)
The Unitary Circle
.
i
2
First Inelastic
Threshold W2
Elastic Threshold W1
Fig. B.2: Typical behavior of the elastic partial-wave amplitude as a function of energy.
where δl is still real but ²l , which is called the inelasticity factor, satisfies
0 ≤ ²l (E) < 1
Alternatively we can write,
sl11 (E) = e2iηl (E)
(B.88)
with ηl complex. In this form we can say that inelasticity forces the “phase
shift” ηl to have a positive imaginary part. The result means that the elastic
partial-wave amplitude moves inside the unitary circle as shown in Fig. B.2 .
The result (B.87) is easily understood. Much as in the one-channel case,
|sl11 |2 is the ratio of outgoing to ingoing flux in channel 1, when the incident
beam is pure l wave in channel 1. As long as only channel 1 is open, this ratio
must be one; but as soon as other competing channels open up it must be less
than one.
It is clear from (B.86), and its equivalent with 1 replaced by α, that every
element slα0 α must have modulus less than one. In particular, for any elastic
l
element the number pα fαα
must lies inside the unitary circle. However only the
l
first amplitude f11 has a purely elastic interval.
Once a three-body channel opens up the situation becomes more complicated. The basis vectors for the relative motion in a three-body channel are
labelled by E, l, m plus an internal energy (a continuous variable) and two internal angular momenta. The discussion of such channel is obviously much more
difficult. Thus, if we return to (B.86) and suppose that the channel α has three
bodies, then the simple term |slα1 |2 is replaced by an integral and sum over the
additional labels. However, this new term is still positive and our conclusion
that |sl11 | ≤ 1 remains valid.
181
B.7 Time-Reversal Invariance
The discussion of parity invariance for multichannel scattering is little different
from that for the one-channel case, and need not be spelled out here. However,
in the case of time-reversal invariance, the multichannel problem offers some
striking new possibilities.
If the dynamics are invariant under time reversal then we can show just as
before that S = T † S † T and, hence, that
hχ|S|φi = hφT |S|χT i
(B.89)
For example, if all particles are spinless, T invariance simply implies that
hp0 , α0 |S|p, αi = h−p, α|S| − p0 , α0 i
(B.90)
The interesting new feature of this result is that it relates the amplitudes for
two qualitatively different processes, α0 ← α and α ← α0 .
B.8 Fundamentals of Time-Independent Multichannel Scattering
In the previous sections, we have set up the time-dependent theory of multichannel scattering. In this section we discuss the corresponding time-independent
theory. Its main purpose is to furnish a means for the actual computation of the
T matrix for given interaction; and for this reason it is the time independent
theory that is the day to day concern of the practising physicist.
In one-channel scattering we saw that the time-independent formalism centers around the Green’s operators G(z) and G0 (z), the T operator T (z), and
the stationary scattering states |p±i. Both T (z) and |p±i were given in terms
of G0 (z) by Lipmann-Schwinger equation, which provided essentially equivalent
approaches to finding the quantity of real interest, the on-shell T matrix.
In this chapter we shall see how the one-channel results extend to the multichannel case. Superficially at least, the most important complication is that
all quantities acquire additional channel labels. As we have already seen the
single free Hamiltonian H 0 of the one-channel case is replaced by a family of
“free” channel Hamiltonians H α , describing the “free” evolution in the various
different channels. In place of the single free Green’s operator G0 (z), there is
a family of channel Green’s operators Gα (z) = (z − H α )−1 . In place of the
stationary states |p±i, there is a family of stationary states |p, α±i. In place
of the single T operator there is a double family of T operators T βα (z), one for
each pair of channels α and β.
B.8.1
The stationary scattering states
In single-channel the stationary states |p±i were eigenvectors of the full Hamiltonian H defined as Ω± |pi. We define multichannel stationary scattering states
as
(B.91)
|p, α±i = Ωα
± |p, αi
182
where as usual p denotes a set of (nα − 1) relative momenta of the nα bodies
in channel α and |p, αi is the corresponding “free” plane-wave state. Using
the intertwining relations (B.57) we can immediately see that these states are
eigenstates of the full Hamiltonian H with energy E = Epα . (This means that
their wave functions hx|p, α±i satisfy the time-independent Schrödinger equation, which provides one route to the actual computation of the states.) In
the same loose sense as applied to the single-channel case, |p, α+i (|p, α−i) can
be regarded as the actual state at t = 0 coming from an in asymptote (would
evolve into an out asymptote) in channel α with momenta p.
Just as in the one-channel case we can derive an expression for |p, α±i in
terms of |p, αi and the Green’s operator. We first note, in the now familiar way,
that for a proper normalizable vector |φi in the channel α subspace,
α
iHt −iH t
Ωα
e
|φi
± |φi = lim e
t→∓∞
Z ∓∞
α
= |φi + i
dteiHt V α e−iH t |φi
(B.92)
0
where we have introduced the channel α scattering potential
V α = H − Hα
(B.93)
This consists of all potentials that link different freely moving fragments in
channel α; that is, V α is precisely that part of the potential which becomes
ineffective as the particles move apart in channel α. If V α were zero, an in state
in channel α would not be scattered.
The known convergence of the integral (B.92) allows as to insert the familiar
damping factor e±²t with the limit ² → 0+ ; and in this form we can apply (B.92)
to the improper states |p, αi to give
Z
|p, α±i = |p, αi + lim+ i
²→0
∓∞
α
dte±²t eiHt V α e−iH t |p, αi
(B.94)
0
Since |p, αi is an eigenvector of the channel Hamiltonian H α with energy E = Epα
this gives
Z ∓∞
|p, α±i = |p, αi + lim+ i
dte−i(E±i²−H)t V α |p, αi
(B.95)
²→0
or
0
|p, α±i = |p, αi + G(E ± i0)V α |p, αi
(B.96)
where we have introduced the full Green’s operator G(z) = (z−H)−1 . The result
(B.96) is the analogue of the one channel result |p±i = |pi + G(E ± i0)V |pi,
and with its help we can obtain expressions for the on shell T matrix. The
momentum-space S-matrix elements can be written as
†
0
hp0 , β|S|p, αi = hp0 , β|Ωβ− Ωα
+ |p, αi = hp , β − |p, α+i
183
(B.97)
Now, using (B.96) we can write |p, α+i as
|p, α+i = |p, α−i + [G(E + i0) − G(E − i0)]V α |p, αi
(B.98)
which can be substituted into (B.97) to give
hp0 , β|S|p, αi = δβα δ(p0 − p) + hp0 , β − |[G(E + i0) − G(E − i0)]V α |p, αi (B.99)
Finally, since the bra hp0 , β − | is an eigenvector of H with eigenvalue E 0 = Epβ0 ,
the factor in brackets comes outside as
[
1
1
−
] = −2πiδ(E 0 − E)
0
E − E + i0 E − E 0 − i0
to give
hp0 , β|S|p, αi = δβα δ(p0 − p) − 2πiδ(E 0 − E)hp0 , β − |V α |p, αi
(B.100)
That is, the on-shell T matrix is given by
t(p0 , β ← p, α) = hp0 , β − |V α |p, αi
(B.101)
Similarly, if we were to rewrite the bra hp0 , β − | of (B.97) in terms of hp0 , β + |
we would find the alternative result
t(p0 , β ← p, α) = hp0 , β|V β |p, α+i
(B.102)
These two expressions are the multichannel versions of the one-channel results
(A.149) and (A.151). The version in (B.101) contains the scattering potential
V α of the initial channel α and is therefore referred to as the “prior” version.
That in (B.102) has the potential V β appropriate to the final channel β and is
called the “post” version.
B.8.2 The Lippmann-Schwinger equations
It is clear from the result (B.101 ) and (B.102) that knowledge of the states
|p, α+i or |p, α−i implies knowledge of the on-shell T matrix. Just as in the
one-channel case the explicit expressions |p, α±i = |p, αi + G(±)V α |p, αi are of
little direct use since they require knowledge of the full Green’s operator G(z).
And, again, just as in the one-channel case, they can be converted into implicit
Lippmann-Schwinger equation in terms of certain “free” Green’s operators, as
we now show.
The first step in establishing the Lippmann-Schwinger equation is to use the
familiar identity
A−1 = B −1 + B −1 (B − A)A−1
with A = (z − H) and B = (z − H α ). This gives the resolvent equations
G(z) = Gα (z) + Gα (z)V α G(z)
184
(B.103)
where, we have defined the Channel Green’s operators Gα (z) corresponding to
the “free” channel Hamiltonian H α , as
Gα = (z − H α )−1
(B.104)
Returning to (B.96) we multiply through by Gα V α to give
Gα V α |p, α±i = Gα V α |p, αi + Gα V α GV α |p, αi
(B.105)
Since by (B.103) Gα V α G = (G − Gα ), this reduces to the important result
Gα (E ± i0)V α |p, α±i = G(E ± i0)V α |p, αi
(B.106)
Finally substitution into (B.96) gives the desired Lippmann-Schwinger equation
|p, α±i = |p, αi + Gα (E ± i0)V α |p, α±i
(B.107)
This equation, which is an integral equation for the corresponding wave function, and the Schrödinger equation provide the two principle approaches to the
computation of the stationary scattering states.
It should be emphasized that the multichannel Lippmann-Schwinger equation is considerably harder to handle than its single-channel equivalent. Some
of the difficulties are readily apparent. For example, the Green’s operator on
the right is the channel Green’s operator Gα (z) = (z − H α )−1 . Except for
the case α = 0 (all particles free), H α contains some potentials and the corresponding Green’s operator Gα cannot, in general, be exactly calculated (in
contra-distinction to the one-channel situation where G0 is known exactly). A
less obvious though more profound, difficulty is that where the one-channel
equation is (or can be easily converted to) a non-singular integral equation,
the multichannel equations are in general highly singular. This makes both the
theoretical study and practical use of the equations much more difficult.
B.8.3
The T operators
Just as in the one-channel case, the time-independent multichannel theory has
two essentially equivalent formulations, one in terms of the stationary scattering
states, the other in terms of the T operators. Having established the former, we
can now easily set up the latter.
We start with the expression
t(p0 , β ← p, α) = hp0 , β − |V α |p, αi
(B.108)
into which we substitute
|p, β−i = |p, βi + G(E − i0)V β |p, βi
(B.109)
This gives
t(p0 , β ← p, α) = hp0 , β|[V α + V β G(E + i0)V α ]|p, αi
185
(B.110)
or
t(p0 , β ← p, α) = hp0 , β|T βα (E + i0)|p, αi
(B.111)
where we have defined the T operator,
T βα (z) = V α + V β G(z)V α
(B.112)
As anticipated here is a double family of T operators T βα (z). The on-shell T
matrix for a transition from channel α to channel β is given by the on-shell
matrix elements of the appropriate T βα (z) between the free states |p0 , βi and
|p, αi.
One feature of the definition (B.112) of T βα (z) that deserves comment is
the apparent asymmetry between the indices α and β. There seem no obvious
reason why it is the potential V α and not V β that appears in the first term.
Indeed, if we had started with the “post” form hp0 , β|V β |p, α+i for the on-shell
T matrix, we would have been led to a result similar to (B.111) but with the T
operator
Teβα (z) = V β + V β G(z)V α
(B.113)
If the channel α and β have different grouping of the particles, then the channel
Hamiltonians H α and H β are different and hence, V α 6= V β . Thus, the two T
operators T βα (z) and Teβα (z) are generally different. Nevertheless both leads
to the same on-shell T matrix, since
hp0 , β|(T βα − Teβα )|p, αi = hp0 , β|(V α − V β )|p, αi
= hp0 , β|(H β − H α )|p, αi
= (E 0 − E)hp0 , β|p, αi
(B.114)
which is zero on the energy shell.
As one would expect, the explicit definition (B.112) of T βα (z) can be replaced by an implicit Lippmann-Schwinger equation. To do this we multiply
(B.112) on the left by Gβ (z) to give
Gβ (z)T βα (z) = (Gβ + Gβ V β G)V α = G(z)V α
(B.115)
substitution back into (B.112) gives the desired equation
T βα (z) = V α + V β Gβ (z)T βα (z)
B.8.4
(B.116)
Asymptotic form of the wave function; Collision without rearrangement
In this section we discuss the multi-channel analogue of the one-channel asymptotic form (A.165). This is quite complicated in general, and we shall therefore
consider just a few simple examples. We begin by considering our three-particle
model with light particles a and b and a heavy fixed particle c. We allow for several bound states (bc)1 , . . . , (bc)n and (ac)1 , . . . , (ac)n0 , and perhaps also some
bound states of a and b. There are then n channels of the form
a + (bc)1 , . . . , a + (bc)n
186
(B.117)
and n0 of the form
b + (ac)1 , . . . , b + (ac)n0
(B.118)
In addition, there is the channel 0 (a + b + c) and perhaps further channels of
the form (ab) + c.
We shall consider a process in which particle a is incident with momentum
p on the ground state (bc)1 . We label this channel as channel 1, and the corresponding stationary state is then |p, 1+i. This has wave function hxa , xb |p, 1+i
depending on the two variables xa and xb , and the question immediately arises:
Do we want the asymptotic behavior as xa → ∞, or xb → ∞, or perhaps both? In fact, all of these cases are of interest, as can readily be seen. If
xa → ∞ with xb fixed, then the wave function should show the effects of those
channels in which particles a moves far away from b and c (and which are open
at the energy considered). However, we would not expect to see those channels
in which a is captured [a + (bc)1 → b + (ac)i ], nor those in which a and b move
off together [a + (bc)1 → (ab)i + c]. Thus, as xa → ∞ with xb fixed, the wave
function hxa , xb |p, 1+i should consist of an incident plane wave in channel 1
plus outgoing waves in the channels a + (bc)i and a + b + c. Instead, if we let
xb → ∞ with xa fixed, then we should see the outgoing waves in the channels
b + (ac)i and a + b + c, but not the others.
We begin with the case that xa → ∞ with xb fixed. We proceed in close analogy with the one-channel analysis of Section A.8, starting from the LippmannSchwinger equation
|p, 1+i = |p, 1i + G1 (E + i0)V 1 |p, 1+i
(B.119)
To get at the wave function hxa , xb |p, 1+i we need to know the spatial matrix
elements hxa , xb |G1 (z)|x0a , x0b i of the channel 1 Green’s operator. In the onechannel case we evaluated the matrix element hx|G0 (z)|x0 i of the free Green’s
operator by inserting a complete set of eigenvectors |pi of the free Hamiltonian H 0 . In the present case we seek analogous eigenvectors of the channel 1
Hamiltonian, which we write as
H1 =
p2a
p2
p2
+ b + Vbc ≡ a + H(bc)
2ma
2mb
2ma
(B.120)
Here the second term is the Hamiltonian of particle b in the field of the fixed
particle c. From this it is clear that we can take as eigenfunctions of H 1 the
products,
(2π)−3/2 eip·xa φα (xb )
(B.121)
where φα (xb ) denotes any of the eigenfunctions of H(bc) with energy Eα . These
are of two types; there are the n bound states (bc)1 , . . . , (bc)n with wave functions φ1 (xb ), . . . , φn (xb ). Also, there are the continuum states, which for example we could choose to be the outgoing wave scattering states of b in the field
of c. In either case the function (B.121) are eigenfunctions of H 1 with energy
(p2 /2ma ) + Eα .
187
When φα (xb ) is one of the bound-state wave functions φ1 , . . . , φn , the function (B.121) is precisely the free wave function of the channel a + (bc)α . The
same is not true of the continuum functions; in this case the function (B.121)
is not the free wave function of any channel. Therefore, the basis of eigenfunctions (B.121) of the channel Hamiltonian H 1 is, a curious hybrid. The
discrete part consists of free channel wave functions hxa , xb |p, αi where α runs
over all channels with the arrangement a + (bc)α . The continuous part consists
of functions that have no simple relation to any channels. In fact, one shall
be mainly interested in the discrete part of the basis, and it is for this reason
that we identify the functions φα (xb ) with the same label α as used to identify
channels. However, it must be remembered that this α runs over some of the
channels (specifically those with the same arrangement as channel 1), and it
also runs over a continuous range corresponding to the continuum states of b in
the potentialR VbcP
. To emphasize this point we shall write “sums” over α with
the symbol ( + ).
Now we can insert a complete set of the states (B.121) into the required
Green’s function, to give
Z
Z
X eip·(xa −x0a ) φα (xb )φα (x0 )∗
b
)
hxa , xb |G1 (z)|x0a , x0b i = (π)−3 d3 p( +
2 /2m ) − E
z
−
(p
a
α
α
(B.122)
The integral over p can be performed, exactly as in the one-channel case. Each
term in the sum over α has a pole at p = [2ma (z − Eα )]1/2 ; and for the case
z = E + i0 the result is
Z
X eipα |xa −x0a | φα (xb )φα (x0 )∗
ma
b
hxa , xb |G1 (E +i0)|x0a , x0b i = −
( +
)
(B.123)
0|
2π
|x
−
x
a
a
α
where pα ≡ [2ma (E − Eα )]1/2 is the momentum of particle a if the target (bc)
is left with energy Eα . In particular, we shall want the Green’s function for
ra À ra0 , in which case,
Z
X eipα ra
0
ma
xa ·xa
hxa , xb |G1 (E + i0)|x0a , x0b i−−−
−
(
+
e−ipαb
)
−
−
−
−
−
−
→
xa → ∞
2π
ra
α
×φα (xb )φα (x0b )∗
(B.124)
We are now ready to establish the asymptotic form of the stationary wave
functions. From the Lippmann-Schwinger equation (B.119) it follows that
Z
Z
hxa , xb |p, 1+i = hxa , xb |p, 1i + d3 x0a d3 x0b hxa , xb |G1 (E + i0)|x0a , x0b i
×V 1 (x0a , x0b )hx0a , x0b |p, 1+i
(B.125)
Substituting (B.124) for the Green’s function we obtain
Z
X eipα ra
−3/2 ip·xa
2
)
(2π)
[e
φ
(x
)
−
(2π)
m
(
+
−−−
−
−
−
−
−
−
→
1 b
a
xa → ∞
ra
α
b, α|V 1 |p, 1+i]
×φα (xb )hpα x
188
(B.126)
or
hxa , xb |p, 1+i −−
−−−−→ (2π)−3/2 [eip·xa φ1 (xb )
x−
a →∞
Z
X
eipα ra
ba , α ← p, 1)
+( +
)f (pα x
φα (xb )]
ra
α
(B.127)
This result is a natural generalization of the well-known one-channel result
(A.165). The first is the expected incident plane wave in channel 1; the second
term has a sum over α. This sum includes n discrete values, corresponding to
the bound states (bc)1 , . . . , (bc)n , plus continuous values corresponding to the
continuum states of the target (bc). For the moment we focus attention on the
n discrete terms. Each of these is the product of three factors: an amplitude, a
spherically spreading wave for particle a, and the target function φα (xb ). Since
the momentum of the spherical wave is:
pα = [2ma (E − Eα )]1/2
(B.128)
we see that each term in the sum represents the particle a travelling outwards
with energy E − Eα having excited the target to the state (bc)α with energy
Eα .
If the total energy E is less than Eα , then the momentum (B.128) in channel
α is pure imaginary. In this case the contribution of channel α to the asymptotic
form (B.127) vanishes exponentially as ra → ∞. Thus, the sum in (B.127) can
be taken to include only those channels that are open at energy E. In particular,
if the energy is below that necessary to disintegrate the target, the “sum” in
(B.127) is a genuine sum with no contribution from the continuum.
If the incident energy is sufficient to break up the target, then the “sum” in
the asymptotic form (B.127) includes an integral over the continuum states. It
can be shown that these terms represent outgoing waves in the breakup channel
a + b + c. It is also not hard to extend these results to more general systems,
in which both projectile and target are composites made up of several particles
(atom-atom collisions, for example). However, we shall not go into these cases.
The cross sections for elastic scattering and for excitation can easily be read
off from the asymptotic form (B.127). The first term represents a steady incident
p
flux per unit area = (2π)−3 m
, in channel 1. The αth term in the sum represents
a scattered flux per unit solid angle = (2π)−3 |f (pα , α ← p, 1)|2 pmα in channel α.
Thus we obtain
dσ
scattered flux/solid angle
pα
(pα , α ← p, 1) =
=
|f (pα , α ← p, 1)|2
dΩ
incient flux/area
p
(B.129)
as expected.
B.8.5 Asymptotic form of the wave function; Rearrangement collisions
So far we have considered the behavior of the stationary wave function hx|p, 1+i
as the coordinate of the original projectile approaches infinity, with all target coordinates fixed. As expected this displayed the effects of these channels with the
189
same arrangement as that of the incident channel 1. If we wish to see the effects
of rearrangement collision, then we must let some of the target coordinates go
to infinity. Here we shall consider the same three-particle model as above with
a incident on the ground state (bc)1 , and examine the wave function as xb goes
to infinity with xa fixed. This should display the effects of the rearrangement
collisions
a + (bc)1 → b + (ac)β
The analysis of this case is quite similar to the previous one. The first step is
to obtain some analogue of the Lippmann-Schwinger equation (B.119) in terms
of the final Channel Green’s operator Gβ (z) = (z − H β )−1 , where H β is the
Hamiltonian of the final channel under consideration,
Hβ =
p2b
p2
+ a + Vac
2mb
2ma
(B.130)
We start with the original Lippmann-Schwinger equation
|p, 1+i = |p, 1i + G1 (E + i0)V 1 |p, 1+i
(B.131)
and rewrite G1 in terms of Gβ as
G1 = Gβ + Gβ (V β − V 1 )G1
(B.132)
This gives
|p, 1+i = |p, 1i + Gβ V 1 |p, 1+i + Gβ (V β − V 1 )G1 V 1 |p, 1+i
(B.133)
Since G1 V 1 |p, 1+i in the last term is the same as |p, 1+i − |p, 1i, this gives
where
|p, 1+i = Gβ V β |p, 1+i + |oi
(B.134)
|oi = |p, 1i − Gβ (V β − V 1 )|p, 1i
(B.135)
It can be shown that the vector |oi is orthogonal to all vectors in the channels
of interest, i.e., the channels b + (ac)β . Therefore it need not concern us here
and we shall omit it from the next few equations. The first term in (B.134) can
be treated by the techniques of the previous section.
To evaluate the spatial matrix elements of Gβ (z) we need a set of eigenfunctions for the channel Hamiltonian H β of (B.130). These have the form
(2π)−3/2 eip·xb χβ (xa )
where χβ (xa ) stands for any of the (ac) bound-state wave functions or the
corresponding continuum functions. This gives an expression for the matrix
190
elements of Gβ analogous to (B.124) for G1 (but with xa , ma , and φα replaced
by xb , mb , and χβ ). Substitution into the expression (B.134) for |p, 1+i gives
Z
X eipβ rb
−3/2
2
hxa , xb |p, 1+i−−−
(2π)
[−(2π)
m
(
+
)
χβ (xa )
−
−
−
−
−
−
→
b
xb → ∞
rb
β
bb , β|V β |p, 1+i]
×hpβ x
(B.136)
Bearing in mind that for the rearrangement
a + (bc)1 → b + (ac)β
the amplitude is
f (pβ , β ← p, 1) = −(2π)2 (ma mb )1/2 t(pβ , β ← p, 1)
(B.137)
we can rewrite (B.136) in the final form
µ
hxa , xb |p, 1+i−−−
−−∞
−−→ (2π)−3/2
x−−→
b
mb
ma
¶1/2 Z
X
bb , β ← p, 1)
( +
)f (pβ x
β
×
eipβ rb
χβ (xa )
rb
(B.138)
where pβ = [2m(E − Eβ )]1/2 .
This result has exactly the expected form; there is no incident wave. The
discrete part of the sum runs over those open channels with the arrangement
b + (ac) (the closed-channel terms go to zero exponentially and can be dropped)
and each term represents the particle b moving out in a spherical wave leaving particle a in the bound state χβ (xa ). Finally, it can be shown that the
continuum terms represent the channel a + b + c as before.
The relevant cross sections can be read off from (B.138). The scattered flux
per unit solid angle in any of the discrete channels β is
(2π)−3
mb
pβ
|f (pβ , β ← p, 1)|2
ma
mb
while the incident flux in channel 1 is (2π)−3 p/ma as before. Thus,
dσ
pβ
(pβ , β ← p, 1) =
|f (pβ , β ← p, 1)|2
dΩ
p
(B.139)
as expected.
B.9 Multichannel Scattering with Identical Particles
In this section we proceed to the general multichannel problem and shall find
that the scattering theory for systems of identical particles can be obtained from
191
that for distinct particles by the use of symmetrizing projectors, much as in the
single-channel case.
We consider an arbitrary system of N particles, at least some of which are
identical, and which has Hamiltonian H. We can, of course, imagine H to be
the Hamiltonian of N distinct particles (the fact that H commutes with the
relevant permutation operators does not prevent this) in which case we already
understand the corresponding scattering theory as follows.
The general state, which is labelled by an arbitrary function of the N variables xi , can be written as a superposition of the bound states (of all N particles)
and scattering states. The latter are the states of interest and can themselves
be classified according to the channels from which they have evolved as t → −∞
(or to which they will evolve as t → +∞).
The general scattering state can be expressed as a superposition of states
each coming from some definite in channel, and similarly of states each going
into some definite out channel.
For an arbitrary wave function ψ the orbit e−iHt ψ does not represent an
allowed orbit of the actual system (with identical particles) since ψ does not have
the required symmetry. However, we have only to multiply by the appropriate
projector Λ to obtain an orbit that does. Under the action of Λ the situation
described in the previous paragraph changes. In particular, the number of
channels is reduced in two ways.
The first change can be easily understood by considering the example of
electron-Hydrogen scattering. Here, for every channel with the grouping
e1 + (e2 p)
there is a second channel of the form
e2 + (e1 p)
obtained by permuting the two electrons. If the electrons are distinguishable
these two channels are physically distinct. In reality, however, the electrons
are indistinguishable and the distinction between the two channels is without
physical meaning. Nonetheless, in the formalism that we choose to adapt, the
two channels are mathematically distinct, wave functions of the former having
the form
χ(x1 )φ(x2 )
(where χ is an arbitrary wave packet, and φ is the hydrogen bound-state wave
function) while those of the latter have the form
χ(x2 )φ(x1 )
In general, whenever a channel has two or more identical particles distributed
among its freely moving fragments there is a whole family of channels obtained
simply by permuting identical particles among the different fragments. These
192
channels, while mathematically distinct, are physically indistinguishable, any
one of them can be used to identify the same physical situation.
To illustrate the second reduction in the number of channels we consider the
slightly more complicated example of electron-helium scattering. The point is
that if the three electrons are distinguishable then there are channels in which
the electrons of the helium atom have a symmetric bound-state wave function.
As soon as we act with the anti-symmetrizing projector these wave functions
are annihilated, corresponding to the obvious fact that the real helium atom has
only anti-symmetric states. In enumerating the channels of a system containing
identical particles we shall count only those channels in which the particles of
each fragment have the correct allowed symmetry. By the same token we shall
always use channel wave functions which are properly symmetrized with respect
to the internal coordinates of each fragment. Thus, the wave function describing
an electron incident on a helium atom will have the form
χ(x1 )φ(x2 , x3 )
where χ(x1 ) is the incident wave packet and φ(x2 , x3 ) is the properly antisymmetric helium wave function.
B.9.1
Transition probabilities and cross section
Having classified the channels of our N −particle system we can now set up
a time-dependent description of the collision process and compute the various
scattering probabilities. If we multiply the appropriate results for distinct particles by the symmetrizer Λ we immediately obtain the asymptotic condition for
identical particles. An orbit that originates in channel α has the form
α
Λe−iHt |ψi−−−−−−−−−→ Λe−iH t |ψin i
t → −∞
[|ψi = Ωα
+ |ψin i]
(B.140)
while an orbit which is going to terminate in channel α has the form
α
Λe−iHt |ψi−−−−−−−−−→ Λe−iH t |ψout i [|ψi = Ωα
− |ψout i]
t → +∞
(B.141)
Suppose now that we wish for the probability that an in state |φi in channel
α lead to the out state |φ0 i in channel α0 . If the in state was given by |ψin i = |φi
in Lα , then according to (B.140) the actual state at t = 0 is
aΛΩα
+ |φi
(B.142)
where a is some normalization factor. If the out asymptote were going to be
0
|φ0 i in Lα , then according to (B.141) the actual state at t = 0 would have to
be
0
0
a0 ΛΩα
− |φ i
(B.143)
with a0 the appropriate normalization factor. The required transition amplitude
is just the overlap of these two functions, and hence,
0
0
α
2
w(φ0 , α0 ← φ, α) = (a0 a)2 |hΛΩα
− φ |ΛΩ+ φi|
193
0
0
α
2
= (a0 a)2 |hΛΩα
− φ |Ω+ φi|
X
0
0
α
2
∝|
ηΠ hΠΩα
− φ |Ω+ φi|
(B.144)
Π
The interpretation of this result is completely natural. The first term inside
the sum (that coming from Π = 1) is the amplitude one would obtain for the
process if the particles were all distinct; the other terms are the amplitudes for
the various “exchange” processes differing from the original one by permutations
of the final (identical) particles. Since the particles are really indistinguishable,
the actual amplitude is obtained by summing all of these amplitudes, multiplied
by the appropriate factors ±1 if the particles are fermions.
To determine the constant of proportionality in (B.144) we determine the
normalization factors a and a0 in the states (B.142) and (B.143). The calculation
of these factors in general (i.e., with arbitrary numbers of particles of arbitrarily
many different types and with arbitrary channels α and α0 ) is straightforward
but tedious. For important special case of a collision between a single particle
and a target containing N particles of the same type as the projectile (this includes the important examples of collisions between an electron and N -electron
atom, and between a nucleon and an N -body nucleus with proton and neutrons
treated as identical using the isospin formalism), for processes in which the in
and out channels have the same arrangements (i.e., only elastic scattering and
excitation), the normalization factors a and a0 are (N + 1)1/2 . Therefore we
have
0
0
α
2
w(φ0 , α0 ← φ, α) = (N + 1)2 |hΛΩα
− φ |Ω+ φi|
¯
¯2
¯
1 2 ¯¯X
¯
α0 0 α
=( ) ¯
ηΠ hΠΩ− φ |Ω+ φi¯
¯
N! ¯
(B.145)
Π
To simplify the sum in (B.145) it is convenient to rewrite each of the permutations Π of the N + 1 particles 0, 1, . . . , N in the form
Π = Π0 Π00
where Π0 exchanges 0 with one of the variables 0, . . . , N leaving all others alone,
while Π00 is a permutation of 1, . . . , N only. The sum over the (N + 1)! permutations Π can then be replaced by a sum over the (N +1) permutations Π0 and the
N ! permutations Π00 . Now, the wave function on which Π operates is already
symmetrized with respect to particles 1, . . . , N and so each of the N ! different
permutations Π00 produces the same result. Summing over these exactly cancels
the factor 1/N ! outside to give
¯
¯2
¯X 0
¯
¯
¯
0
0
α0 0
α
w(φ , α ← φ, α) = ¯
ηΠ hΠΩ− φ |Ω+ φi¯
(B.146)
¯
¯
Π
P0
where now
denotes a sum over the N + 1 permutations that exchange the
original projectile 0 with one of the particles 0, . . . , N leaving all other particles
undisturbed.
194
P0
Finally we can divide the sum
into one term corresponding to Π = 1
and the remaining N terms in which particle 0 changes place with one of the
target particles 1, . . . , N . Since the wave function on which Π acts is already
symmetrized with respect to particles 1, . . . , N then last N terms are all the
same and we arrive at the final result.
¯
¯2
0
¯
¯
0 α
α0 0
α
(B.147)
φ
|Ω
φi
+
ηN
hΠ
Ω
φ
|Ω
φi
w(φ0 , α0 ← φ, α) = ¯hΩα
01
−
+
−
+ ¯
where as usual η = ±1 depending on whether the particles are bosons or fermions
and Π01 denotes the permutation which just exchanges particles 0 and 1.
From this result we can immediately proceed to the calculation of the observed cross section. We assume spinless particles and target for simplicity. We
obtain
dσ 0 0
p0
(p , α ← p, α) = |fb(p0 , α0 ← p, α)|2
(B.148)
dΩ
p
where
fb(p0 , α0 ← p, α) = fdi (p0 , α0 ← p, α) + ηN fex (p0 , α0 ← p, α)
(B.149)
Here fdi is the amplitude one would calculate for the “direct” process
direct :
0 + (12 · · · N )α → 0 + (12 · · · N )α0
(B.150)
on the assumption that the particles are distinct; that is
fdi (p0 , α0 ← p, α) = f (p0 , α0 ← p, α)
(B.151)
if, as usual, we use f to denote the amplitude calculated for distinct particles.
The amplitude fex is the corresponding “exchange” amplitude for the ejection
of particle 1 with momentum p0 and the simultaneous capture of particle 0
exchange :
0 + (12 · · · N )α → 1 + (02 · · · N )α0
fex (p0 , α0 ← p, α) = f (p0 , α
e0 ← p, α)
(B.152)
(B.153)
where α
e0 denotes the exchange channel of (B.152). Since the original particle
0 can exchange with any one of the N equivalent particles in the target, the
amplitude fex appears multiplied by N in the expression (B.149) for fb.
195
C. ALKALI ATOMS IN A MAGNETIC GUIDE:MATRIX
ELEMENTS
The matrix elements of the Hamiltonian (8.10) can be calculated analytically
by exploiting the recurrence relation of the associated Laguerre polynomials
[94, 95]. The matrix reads
1
[hi | p2 | ji + hi | r sin θ cos ϕ(Sx + αIx ) | ji + hi | r sin θ sin ϕ
2
×(Sy + αIy ) | ji − 2hi | r cos θ(Sz + αIz ) | ji + βhi | I · S | ji] (C.1)
hi | H | ji =
Here i = (n, l, mi , ms ), j = (n0 , l0 , m0i , m0s ),
hi | p2 | ji =
[−h̄2 hRnζ | (
+ hRnζ |
0
∂2
2 ∂
i
+
) | Rnζ 0 ihYlm | Ylm
0
∂r2
r ∂r
0
1
i]hmi , ms | m0i , m0s i (C.2)
| Rnζ 0 ihYlm | L2 | Ylm
0
r2
with m = M − mi − ms and m0 = M − m0i − m0s ,
hi | r sin θ cos ϕ(Sx + αIx ) | ji
0
i
= hRnζ | r | Rnζ 0 ihYlm | sin θ cos ϕ | Ylm
0
× hmi , ms | (Sx + αIx ) | m0i , m0s i (C.3)
hi | r sin θ sin ϕ(Sy + αIy ) | ji =
hi | r cos θ(Sz + αIz ) | ji =
0
i
hRnζ | r | Rnζ 0 ihYlm | sin θ sin ϕ | Ylm
0
× hmi , ms | (Sy + αIy ) | m0i , m0s i (C.4)
0
i
hRnζ | r | Rnζ 0 ihYlm | cos θ | Ylm
0
× hmi , ms | (Sz + αIz ) | m0i , m0s i
(C.5)
and
0
hi | I · S | ji = hRnζ | Rnζ 0 ihYlm | Ylm
ihmi , ms | I · S | m0i , m0s i
0
(C.6)
C.1 Radial Matrix Elements
The Laguerre polynomials satisfy the following recursion relation
rLln = −(n + l)Lln−1 + (2n + l + 1)Lln − (n + 1)Lln+1
(C.7)
(ζ)
This yields the following relation for the basis functions Rn
(ζ)
(ζ)
rRn(ζ) = −(n + 1)Rn+1 + (2n + 1)Rn(ζ) − nRn−1
(C.8)
From which we obtain
1
(ζ)
| Rn0 i = δn,n0
r2
ζhRn(ζ) |
4ζhRn(ζ) |
∂2
2 ∂
(ζ)
| Rn0 i =
+
∂r2
r ∂r
ζ 2 hRn(ζ) |
1
(ζ)
| Rn0 i =
r
(C.9)
n0 (n0 − 1)δn,n0 −2 − 2(n02 + n0 + 1)δn,n0
+ (n0 + 1)(n0 + 2)δn,n0 +2
(C.10)
−n0 δn,n0 −1 + (1 + 2n0 )δn,n0 − (1 + n0 )δn,n0 +1
(C.11)
(ζ)
ζ 3 hRn(ζ) | Rn0 i
= (n0 − 1)n0 δn,n0 −2 − 4n02 δn,n0 −1 + (2 + 6n0 (1 + n0 ))δn,n0
− 4(1 + n0 )2 δn,n0 +1 + (1 + n0 )(2 + n0 )δn,n0 +2
(C.12)
(ζ)
ζ 4 hRn(ζ) | r | Rn0 i = −(n0 − 2)(n0 − 1)n0 δn,n0 −3 + 3(n0 − 1)n0 (2n0 − 1)δn,n0 −2
−3n0 (1 + 5n02 )δn,n0 −1 + 2(1 + 2n0 )(3 + 5n0 (1 + n0 ))δn,n0
−3(1 + n0 )(6 + 5n0 (2 + n0 ))δn,n0 +1 + 3(1 + n0 )(2 + n0 )(3 + 2n0 )δn,n0 +2
−(1 + n0 )(2 + n0 )(3 + n0 )δn,n0 +3
(C.13)
C.2 Angular Matrix Elements
For the angular matrix elements the following relations are needed
s
s
(l
−
m)(l
+
m)
(l − m + 1)(l + m + 1)
m
m
cos θ | Ylm i =
| Yl−1
i+
| Yl+1
i
(2l − 1)(2l + 1)
(2l + 1)(2l + 3)
(C.14)
s
sin θe
±iϕ
|
Ylm i
=±
s
∓
(l ∓ m)(l ∓ m − 1)
m±1
| Yl−1
i
(2l − 1)(2l + 1)
(l ± m + 2)(l ± m + 1)
m±1
| Yl+1
i
(2l + 3)(2l + 1)
197
(C.15)
From which we obtain
hYlm
| cos θ |
s
0
Ylm
i
0
=
s
(l0 − m0 )(l0 + m0 )
δl,l0 −1 δm,m0
(2l0 − 1)(2l0 + 1)
(l0 − m0 + 1)(l0 + m0 + 1)
δl,l0 +1 δm,m0
(2l0 + 1)(2l0 + 3)
+
(C.16)
s
0
hYlm | sin θe±iϕ | Ylm
i=±
0
s
∓
(l0 ∓ m0 )(l0 ∓ m0 − 1)
δl,l0 −1 δm,m0 +1
(2l0 − 1)(2l0 + 1)
(l0 ± m0 + 2)(l0 ± m0 + 1)
δl,l0 +1 δm,m0 −1 (C.17)
(2l0 + 1)(2l0 + 3)
0
0
0
hYlm | sin θ cos ϕ | Ylm
i = 12 hYlm | sin θeiϕ | Ylm
i + 21 hYlm | sin θe−iϕ | Ylm
i
0
0
0
(C.18)
0
0
1 m
i
hY | sin θeiϕ | Ylm
0
2i l
0
1
i
− hYlm | sin θe−iϕ | Ylm
0
2i
i=
hYlm | sin θ sin ϕ | Ylm
0
(C.19)
C.3 Spin Matrix Elements
For spin matrix elements the following relations are needed
Ix | m i i
Iy | m i i
=
=
h̄ p
(i + mi + 1)(i − mi ) | mi + 1i
2
h̄ p
+
(i − mi + 1)(i + mi ) | mi − 1i
2
h̄ p
(i + mi + 1)(i − mi ) | mi + 1i
2i
h̄ p
−
(i − mi + 1)(i + mi ) | mi − 1i
2i
Iz | mi i = h̄mi | mi i
(C.20)
(C.21)
(C.22)
i is integer or half-integer and −i ≤ mi ≤ i. From these equations we obtain
hmi | Ix | m0i i =
h̄ p
(i + mi + 1)(i − mi )δmi ,m0i +1
2
h̄ p
+
(i − mi + 1)(i + mi )δmi ,m0i −1
2
198
(C.23)
hmi | Iy | m0i i =
h̄ p
(i + mi + 1)(i − mi )δmi ,m0i +1
2i
h̄ p
−
(i − mi + 1)(i + mi )δmi ,m0i −1
2i
hmi | Iz | m0i i = h̄mi δmi ,m0i
(C.24)
(C.25)
For S-Matrix elements we should just put s = 1/2 instead of i in above equations.
hmi , ms | I · S | m0i , m0s i =
hmi | Ix | m0i ihms | Sx | m0s i
+ hmi | Iy | m0i ihms | Sy | m0s i
+ hmi | Iz | m0i ihms | Sz | m0s i
199
(C.26)
D. BOSE-FERMI MAPPING THEOREM
In 1940 Nagamiya noted [102] that in the “fundamental sector” z1 ≤ z2 ≤
. . . ≤ zN the ground state wave function of a spatially uniform, 1D hard-core
Bose gas can be written as an ideal Fermi gas determinant, continuation into
other permutation sectors being effected by imposing overall Bose symmetry
under all permutations zi ↔ zj in spite of the fermionic anti symmetry under
permutations of orbitals (not coordinates) in the fundamental sector. After
that in 1960 the Fermi-Bose mapping method which is much more general was
introduced independently by Girardeau [52] and Stachowiak [103]. The mapping
theorem holding also in the presence of external potentials and/or finite twoparticle or many-particle interactions in addition to the hard core interaction
[52]. It also applies to the 1D time-dependent many-body Schrödinger equation
and has been used to treat some time-dependent interference properties of the
1D hard core Bose gas [104, 105, 106, 107, 108]
We now briefly review the mapping theorem. In general the N -boson Hamiltonian is assumed to have the structure
2
H1D = −(h̄ /2m)
N
X
∂z2i + V (z1 , . . . , zN )
(D.1)
j=1
where the real, symmetric function V contains all external potentials (e.g., a
longitudinal trap potential) as well as any finite interaction potentials not including the Hard-sphere repulsion, which is instead treated as a constraint on
allowed wave functions ψB (z1 , . . . , zN )
ψB = 0
if |zj − zk | < a,
(1 ≤ j < k ≤ N )
(D.2)
Let ψF (z1 , . . . , zN ) be a fermionic solution of H1D ψ = Eψ which is antisymmetric under all pair exchanges zj ↔ zk , hence all permutations. One can
consider ψF to be either the wave function of a fictitious system of “spinless
fermions”, or else that of a system of real, spin-aligned fermions. Define a “unit
antisymmetric function” [52]
Y
sgn(zk − zj ),
(D.3)
A(z1 , . . . , zN ) =
1≤j<k≤N
where sgn(z) is the algebraic sign of the coordinate difference z = zk − zj , i.e.,
it is +1 (−1) if z > 0 (z < 0). For given antisymmetric ψF , define a bosonic
wave function ψB by
ψB (z1 , . . . , zN ) = A(z1 , . . . , zN )ψF (z1 , . . . , zN )
(D.4)
which defines the Fermi-Bose mapping. ψB satisfies the hard core constraint
(D.2) if ψF does, is totally symmetric (bosonic) under permutations, obeys the
same boundary conditions as ψF , and H1D ψB = EψB follows from H1D ψF =
EψF [52, 109]. In the case of periodic boundary conditions (no trap potential, spatially uniform system) one must add the proviso that the boundary
conditions are only preserved under the mapping if N is odd, but the case of
even N is accommodated by imposing periodic boundary conditions on ψF but
anti periodic boundary conditions on ψB .
The mapping theorem leads to explicit expressions for all many-body energy
eigenstates and eigenvalues under the assumption that the only two-particle
interaction is a zero-range hard repulsion, represented by the a → 0 limit of the
hard-core constraint, the “TG gas”. Such solutions were obtained in Sec. 3 of
[52] for periodic boundary conditions and no external potential.
201
E. BOUNDARY CONDITIONS
In the limit r tends to zero, u in (5.17) goes to zero. This boundary condition
can be satisfied easily by putting u(r = 0) = 0 and u−j = uj in the vicinity of
d2
r = 0. The latter is needed for approximating the derivatives dx
2 u(r(x)) and
d
u(r(x))
near
the
point
r
=
0.
dx
The boundary condition for u approximating the asymptotic form (5.10) at
large r can be written in the form
(1)
(2)
(3)
(4)
uj + αj uj−1 + αj uj−2 + αj uj−3 + αj uj−4 = gj
j = N − 2, N − 1, N
(E.1)
where αj ’s are diagonal Nθ × Nθ matrices and gj is a Nθ -dimensional vector.
The above boundary conditions are constructed by eliminating the unknown
amplitudes fnn0 from the asymptotic equations (5.10) written for a few rj s
neighboring to the point rN = R. This gives the values for the coefficients αj ’s
and gj . In general for up to four open channels we have
h
i
(l)
αj
mm0
³
´ r φn (ρm )
m
m
e
j
j
= (−1)i Tjl−1,m + Tjl,m
eikne (|zj |−|zj−l |) δmm0
m
rj−l φne (ρj−l )
(l = 1, 2, 3, 4)
(E.2)
and
[gj ]m = 2
p
ikne |zjm |
λm rj φne (ρm
j )e
ne X
4
X
m
m
(−1)i e−ikne |zj−l | eikn zj−l
n=0 l=0
×(Tjl−1,m + Tjl,m )
φn (ρm
j−l )
φne (ρm
j−l )
(E.3)
For the case of bosonic (fermionic) collisions we must consider just the even
m
(odd) part of the equation (E.3), i.e. we just need to replace the term eikn zj−l
m
m
). Here φn (ρ) = φn,0 (ρ, ϕ) is the eigenfunction of the
(i sin kn zj−l
by cos kn zj−l
m
transverse trapping Hamiltonian [see (5.6)], ρm
j = rj sin θm , zj = rj cos θm , n
is the channel number of the initial state, and ne is the number of transversely
excited open channels. The complex numbers Tji,m ’s are given through
m
m
m m
m m
m m
Tjl,m = δl,0 + δl,1 (am
j + bj + cj ) + δl,2 (aj bj−1 + aj cj−1 + bj cj−1 )
m
m
+δl,3 am
j bj−1 cj−2
(E.4)
If ne < 1 (single-mode regime) am
j = 0, otherwise
m
m
m
m
m
m
m m
m
am
j = {Ξ0,j − (1 + bj + cj )Ξ0,j−1 + (bj + cj + bj cj−1 )Ξ0,j−2
m
m
m
m
m
m
−bm
j cj−1 Ξ0,j−3 } × {Ξ0,j−1 − (1 + bj + cj )Ξ0,j−2
m
m m
m
m m
m
−1
+(bm
j + cj + bj cj−1 )Ξ0,j−3 − bj cj−1 Ξ0,j−4 }
(E.5)
if ne < 2, bm
j = 0, otherwise
© m
ª
m
m
m m
bm
j = Ξ1,j − (1 + cj )Ξ1,j−1 + cj Ξ1,j−2
©
ª−1
m
m
m m
× Ξm
1,j−1 − (1 + cj )Ξ1,j−2 + cj Ξ1,j−3
(E.6)
if ne < 3, cm
j = 0, otherwise
© m
ª © m
ª−1
m
m
cm
j = Ξ2,j − Ξ2,j−1 × Ξ2,j−1 − Ξ2,j−2
m
i(kn −kne )|zj
where Ξm
n,j = e
m
| φn (ρj )
φne (ρm
).
j
kn and kne are given by (5.9).
203
(E.7)
F. FAST IMPLICIT MATRIX ALGORITHM
Following the idea of the LU decomposition [61] and the sweep method [62] (or
the Thomas algorithm [63]), we search the solution of the system of N vector
equations with any coefficient being a Nθ × Nθ matrix (5.24) in the form
(1)
(2)
(3)
uj = Cj uj+1 + Cj uj+2 + Cj uj+3 ,
j = 1, ..., N − 3
(F.1)
Here we define u(r(xj )) as uj for simplicity. The Cj s are unknown Nθ × Nθ matrices. To find the solution, first we should calculate the unknown Cj matrices.
The plan is the following: Due to (F.1) we have
(1)
(2)
(3)
uj−p = Cj−p uj−p+1 + Cj−p uj−p+2 + Cj−p uj−p+3 ,
p = 1, 2, 3 ,
(F.2)
then one obtains
(1)
(2)
(3)
uj−1 = Cj−1 uj + Cj−1 uj+1 + Cj−1 uj+2
(F.3)
h
i
h
i
(1)
(1)
(2)
(1)
(2)
(3)
(1)
(3)
uj−2 = Cj−2 Cj−1 + Cj−2 uj + Cj−2 Cj−1 + Cj−2 uj+1 + Cj−2 Cj−1 uj+2
(F.4)
and
h
i
(1)
(1)
(1)
(1)
(2)
(2)
(1)
(3)
uj−3 = Cj−3 Cj−2 Cj−1 + Cj−3 Cj−2 + Cj−2 Cj−1 + Cj−2 uj
h
i
(1)
(1)
(2)
(1)
(3)
(2)
(2)
+ Cj−3 Cj−2 Cj−1 + Cj−3 Cj−2 + Cj−2 Cj−1 uj+1
h
i
(1)
(1)
(3)
(2)
(3)
+ Cj−3 Cj−2 Cj−1 + Cj−2 Cj−1 uj+2
(F.5)
By substituting uj defined by (F.3-F.5) into (5.24) one can calculate uj in terms
of uj+1 , uj+2 , and uj+3 . Then, by comparing with (F.1) we find a recurrence
formula for calculating the unknown matrices Cj :
h
i
(1)
(1)
(1)
(2)
(1)
(3)
(2)
(2)
−DCj = Ajj−3 Cj−3 Cj−2 Cj−1 + Cj−3 Cj−2 + Cj−3 Cj−1
h
i
(1)
(2)
(3)
(2)
+ Ajj−2 Cj−2 Cj−1 + Cj−2 + Ajj−1 Cj−1 + Ajj+1
(F.6)
(2)
−DCj
h
i
(1)
(1)
(3)
(2)
(3)
= Ajj−3 Cj−3 Cj−2 Cj−1 + Cj−3 Cj−1
(1)
(3)
(3)
+ Ajj−2 Cj−2 Cj−1 + Ajj−1 Cj−1 + Ajj+2
(F.7)
and
(3)
−DCj
= Ajj+3 .
(F.8)
Here
D
h
i
(1)
(1)
(1)
(1)
(2)
(2)
(1)
(3)
= Ajj−3 Cj−3 Cj−2 Cj−1 + Cj−3 Cj−2 + Cj−3 Cj−1 + Cj−3
i
h
(1)
(2)
(1)
(1)
+ Ajj−2 Cj−2 Cj−1 + Cj−2 + Ajj−1 Cj−1 + Ajj + 2(²I − Vj ) (F.9)
By using the left-side boundary conditions and (F.1) one can calculate the Cj
matrices for j = 1, 2, and 3. Then by using (F.9-F.6) we calculate all the
matrices Cj . Subsequently by using the right-side boundary conditions (E.1)
and recurrence formula (F.1) we first calculate uj for j = N − 2, N − 1, and N
and then uj for j = 1, ..., N − 3.
205
G. ACKNOWLEDGMENTS
The financial support for doing the work which appears in this thesis has been
provided by the Ministry of Science, Research and Technology of Iran,
in the form of a scholarship. This provided me with the wonderful opportunity
to study without being concerned about my finances.
I would like to express my deepest appreciation to my advisor Prof. Peter
Schmelcher, I began working with him about four years ago and during this
time I have learned an immense amount from him.
I am grateful to prof. Vladimir Melezhik for helpful discussion and for
teaching me numerical methods.
I would like to thank Dr. Igor Lesanovsky who was a PhD student of
Peter’s group when I began working there. Discussion with him was really
worthful for me.
I like to thank Prof. Jochen Schirmer for accepting to be the second
examiner of this work.
Special thanks to all the members of groups of Prof. Schmelcher, Prof.
Schmiedmayer, and Prof. Pan particularly Bernd Hezel, Florian Lenz,
Torsten Straßel, Dennis Heine, Budhaditya Chatterjee, Sascha Zöllner,
Christoph Petri, and Florian Koch.
I would like to thank my parents and my brother. They have been a great
source of support and encouragement throughout of my life.
I am very grateful to my wife Azam and my children Pourya and Yekta
for everything. I would like to dedicate this dissertation to my family.
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