Quantum Mechanics Exam Fall 2011 SAMPLENAME (Please Print):

Solve 4 out of the following 5 problems

Problem 1. Continuity Equation From electrodynamics you are familiar with the continuity equation for the flow of electrical ~ ~ ∂ ~ charge: ∇ · j + ∂t ρ = 0, where ρ and j are the charge and current densities, respectively. ~ ~ ∂ ~ The same equation ∇ · j + ∂t ρ = 0 holds in quantum mechanics, where now ρ and j represent the probability density and probability current density for a quantum system, respectively.

(a) Show that in non-relativistic quantum mechanics the probability current density of the wave function for a particle of mass m is given by i ~j = ~ (ψ∇~ ψ∗ − ψ∗∇~ ψ) 2m if ρ is taken as the usual definition for the probability density of a wave function ψ, i.e. ρ = ψ∗ψ = |ψ|2. Hint: Start from the time-dependent Schroedinger equation and its complex conju- gate.

(b) Show that for a free quantum-mechanical particle with normalized wave function i (~p·~x−Et) ~p ψ = N e ~ that is moving with velocity ~v = m , the usual relationship

~j = ρ ~v

between a current density and velocity (see e.g. electrodynamics) also holds in quantum mechanics for probabilities.

Problem 2. Compatible observables Using the canonical commutation relations for position and momentum:

(a) Determine whether the z component of angular momentum vector (Lz = xpy − ypx) is compatible with each component of position vector, i.e. x, y and z.

2 2 (b) Show that [Lz, r ] = 0. Next, given also [Lz, p ] = 0, show that the observables 2 total energy E and Lz are compatible when potential energy V = ar , where a is a positive real constant.

1 Problem 3. One-Dimensional Infinite Square Well Perturbation The energy eigenstates and wave functions for a particle with mass m in an infinite one- SAMPLEdimensional square well  0 if 0 ≤ x ≤ a V (x) = ∞ otherwise are n2π2 2 r2 x E = ~ and ψ (x) = sin nπ )(n = 1, 2,...), respectively. n 2ma2 n a a

Suppose we add a delta function spike in the center of the infinite square well:

H0 = α δ(x − a/2) , where α is some constant.

(1) (a) Find the first-order perturbative correction En to the energy levels En due to the delta spike.

(b) Explain why the energies En are actually not perturbed for even n.

(1) (c) Find the first three non-zero terms for the first-order correction ψ1 to the ground state wave function ψ1.

Problem 4. Two-particle system Two non-interacting particles, each of mass m, are in the one-dimensional harmonic os- 1 2 2 cillator potential V = 2 mω x . (a) Construct the composite wave function for the ground state of the two-particle system in terms of one-particle stationary states ψn(x) for the three cases of distinguishable particles, identical bosons, and identical fermions. (b) For each case, normalize the wave function and give its energy eigenvalue Problem 5. Time Dependent Infinite Square Well A particle in an infinite square well has the initial wave function:

ψ(x, 0) = Ax(a − x), (0 ≤ x ≤ a),

for some positive constant A (see figure). Outside the well ψ = 0.

(a) Find ψ(x, t). (b) Calculate the constant A.

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