# The Handbook of Mathematics, Physics and Astronomy Data is provided

The Handbook of Mathematics, Physics and Astronomy Data is provided KEELE UNIVERSITY EXAMINATIONS, 2012/13 Level II Thursday 17th January 2013, 16:00 – 18:00 PHYSICS/ASTROPHYSICS PHY-20006 QUANTUM MECHANICS Candidates should attempt ALL of PART A and TWO questions from PART B. PART A yields 40% of the marks, PART B yields 60%. A sheet of useful formulae can be found on page 9. NOT TO BE REMOVED FROM THE EXAMINATION HALL PHY-20006 Page 1 of 9 PART A A1 Answer all TEN questions Give an expression for the probability of observing a particle in a thin shell of radius r and infinitesimal width dr for a particle with a spherically symmetric wave function Ψ(r, t). A2 [4] Sketch the following function and give two reasons why it cannot be part of a realistic wave function (a > 0 is a real constant). eax x < 0 ax x ≥ 0 f (x) = [4] A3 Calculate the value of A for the following wave function and explain your method. Ψ(x, t) = A sech(x) e−iωt You may use the following integral without proof in your answer. Z sech2 (x) dx = tanh(x) + C N.B. tanh(x) → ±1 for x → ±∞. [4] /Cont’d PHY-20006 Page 2 of 9 A4 State three differences between the predictions of classical physics and the predictions of quantum mechanics for the properties of a particle in the semi-infinite square well potential, ∞ x<0 V (x) = −V0 0 ≤ x ≤ a 0 x>a [4] A5 Sketch the energy eigenfunctions for the ground state and first excited state of a particle trapped in the region 0 < x < a by the 8 8 potential shown below assuming that the energy of the particle is E < VB in both cases. V(x) VB 0 x a 0 [4] A6 Calculate the expectation value hEi for the energy of a particle with the wavefunction 1 1 Ψ(x, t) = √ ψ1 (x)e−iE1 t/h̄ − √ ψ2 (x)e−iE2 t/h̄ , 2 2 √ where En = n eV is the energy for the state ψn . [4] /Cont’d PHY-20006 Page 3 of 9 A7 Show that the wave function Ψ(x, t) = e−i(kx+wt) is an eigenfunction of the momentum operator, p̂, and state the eigenvalue. [4] A8 The wave function for 2 identical particles in a 1-dimensional potential is Ψ(x1 , x2 ). State and explain the possible values of Ψ(x2 , x1 ). [4] A9 List all possible values for the magnitude of the total angular momentum, J, for an electron in the 2p state of the hydrogen atom. Give your answers in units of h̄. [4] A10 A particle has the wave function 3 a2 Ψ(x, t) = √ x e−ax/2 e−iωt , 2 x > 0. Calculate an approximate correction to the energy of the particle if the potential is perturbed by a field V 0 (x) = ²x, where ² = 0.02 eV/nm and a = 1 nm−1 . You may use the following integral without proof in your answer. Z ∞ n! rn e−ar dr = n+1 0 a [4] /Cont’d PHY-20006 Page 4 of 9 PART B A particle of mass m is trapped in the following potential: 1 kx2 2 V (x) = 0 ∞ x<0 0≤x≤a x>a V(x) 8 B1 Answer TWO out of FOUR questions 0 a x For some values of k it is possible to write the solution of the time independent Schrödinger equation as ψ(x) = 2 ψA = Ae−αx ψB = B cos(βx) + C sin(βx) 0 x<0 0≤x≤a x>a (a) Apply the appropriate boundary condition at x = 0, and hence show that B = A and C = 0. [6] (b) Apply the appropriate boundary condition at x = a and hence derive the possible values of β. [5] (c) Show that ψB is a solution of the time independent Schrödinger equation, and hence derive an expression for the energy, E, in terms of a. [8] (d) Discuss the behaviour of ∂ψ ∂x at x = a [5] (e) Describe and explain what the solutions of the time independent Schrödinger equation would look like in the alternative case k → ∞. [6] /Cont’d PHY-20006 Page 5 of 9 B2 The energy of a particle with mass m in a 2-dimensional harmonic oscillator potential 1 1 V (x, y) = mω 2 (x2 + y 2 ) = kr2 2 2 is given by Enx ,ny = (nx + ny + 1)h̄ω nx = 0, 1, 2, . . . ny = 0, 1, 2, . . . The angular momentum operator for a 2-dimensional system is L̂2 = x̂pˆy − ŷ pˆx , ∂ , and similarly for pˆy . where pˆx = −ih̄ ∂x (a) Write down the energy in units of h̄ω for the first three energy levels. State the degeneracy and the values of nx and ny for each energy level. [6] (b) Show that Y (x, y) = y − ix is an eigenfunction of the operator L̂2 . State the values of the expectation value hL2 i and the uncertainty ∆L2 for the particle in this state. [8] (c) Show that the commutator for L̂2 and x̂ has the value [L̂2 , x̂] = ih̄y. State briefly what your result implies for the observed values of L2 and x. [10] (d) Give a physical reason why eigenfunctions of the L2 operator can be used to find a solution to the time independent Schrödinger equation for this potential. [6] /Cont’d PHY-20006 Page 6 of 9 B3 The energy of a particle with mass m in the harmonic oscillator potential V (x) = 12 kx2 is given by Ã ! 1 En = n + h̄ω, 2 where ω = n = 0, 1, 2, . . . , q k/m. The energy eigenfunction for the first excited state is Ã ψ1 (x) = A1 ! x −x2 /2a2 e , a where a is a constant. (a) Show that ψ1 has definite parity and state its value. [4] (b) Sketch the position probability distribution function for the state ψ1 . [4] (c) The spectrum of H35 Cl shows spectral features due to changes in vibration state spaced equally in frequency by 8.66×1013 Hz. Calculate the bond strength, k, for this molecule. State any assumptions you have made in your calculation. [8] (d) Describe how the energy eigenfunctions ψj , j = 0, 1, 2 . . ., for this potential can be used to represent a time-dependent wave function Ψ(x, t) given an initial state Ψ(x, 0). [8] (e) Discuss briefly whether the following two statements are consistent with each other. • The momentum, p, of a particle with kinetic energy E1 is given by p2 = 2mE1 . • The expectation value of the momentum for the particle described by ψ1 (x) is hpi = 0. [6] /Cont’d PHY-20006 Page 7 of 9 B4 The wave functions for an electron in a simple model of the hydrogen atom have the form Ψn,`,m` (r, θ, φ, t) = u(r) Y`,m` (θ, φ)e−iEt/h̄ . r (a) State the physical quantity most closely associated with each of the quantum numbers n, ` and m` , and state the possible values for each quantum number for an electron in a 3p state. [6] (b) The function u(r) is a solution of the radial Schrödinger equation for the effective potential `(` + 1)h̄2 e2 Ve (r) = . − 2me r2 4π²0 r Explain the origin of the two terms in this equation for the effective potential. [4] (c) With the aid of a labelled diagram, describe the main features and results of the Stern-Gerlach experiment. Explain how this experiment shows that the eigenfunctions Ψn,`,m` do not give a complete description for the properties of an electron in a hydrogen atom. [12] (d) The 3p state of hydrogen is split into two energy levels split by about 0.05 meV. Explain the origin of this fine structure in the energy spectrum of hydrogen. [8] /Cont’d PHY-20006 Page 8 of 9 Quantum Mechanics formulae Time independent Schrödinger equation h̄2 d2 ψ Ĥψ = − + V (x) ψ = Eψ 2m dx2 PHY-20006 Page 9 of 9

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

### Related manuals

Download PDF

advertisement