Download Problem Set 3 for Physics 580 - Fall 2007 and more Assignments Quantum Mechanics in PDF only on Docsity! PHYSICS 580 – FALL 2007 PROBLEM SET 3 – DUE: Tuesday, Oct. 2, 2007 1. (LQM 3-1) In the “discrete” Schrödinger equation for a free particle, wi,i−1 = wi,i+1 ≡ w, a constant, and vi = 0. Assume periodic boundary conditions, that is, ψN+1(t) = ψ1(t), and generally ψj+N(t) = ψj(t), where N is a given integer À 1. a) Show that there are N linearly independent solutions of this Schrödinger equation. b) Find the N normalized energy eigenstates, that is, those solutions that vary in time with a fixed frequency. What are the possible energy values for the particle? Show that these go over into the free particle energies, p2/2m, in the continuum limit, i.e., as λ, the size of the intervals goes to 0, with Nλ remaining fixed. c) Define the propagator matrix, K, by ψj(t) = ∑ k Kjk(t, t ′)ψ(k(t′) in analogy with the continuum result. Write down an explicit expression for the matrix elements Kjk(t, t ′). 2. (LQM 3-2) a) Write out δ(r − r′) as the product of three one-dimensional delta functions in spherical coordinates. b) What is the Fourier transform of δ(r−r′)? Write out δ(r−r′) in terms of its Fourier transform. c) Show that 〈p|p′〉 = (2πh̄)3δ(p− p′). 3. (LQM 3-3) What is the representation of the position operator in the momentum basis, i.e., how is 〈p|rop|Ψ〉 related to 〈p|Ψ〉? 4. (LQM 3-4) Suppose that the potential is v(r) = kr2/2. What is the Schrödinger equation written in momentum space; that is, what is the equation of motion of the amplitude 〈p|Ψ(t)〉? 5. (LQM 3-5) What is [yop, Px,op]? Compare these commutation relations for position and momen- tum with the classical Poisson bracket relations satisfied by p and r. 6. (LQM 3-6) What is the expectation value of the kinetic energy of a particle in terms of its wave function? 7. (LQM 3-9) a) Show that the free particle propagator K(rt, r′t′) obeys the free particle Schrödinger equation, ( ih̄ ∂ ∂t + h̄2∇2 2m ) K(rt, r′t′) = 0. b) Suppose that a particle is acted on by a potential v(r, t), and that its wave function at t0 is ψ0(r). Show that the wave function of the particle at a later time t is given as the solution to the integral equation ψ(r, t) = ∫ d3r′K(rt, r′t0)ψ0(r)′+ 1 ih̄ ∫ t t0 dt′ ∫ d3r′ K(rt, r′t′)v(r′, t′)ψ(r′, t′). 8. (LQM 3-10) a) The probability current density j(r,t) is given in terms of the wave function by j(r, t)− 1 2m [ ψ∗(r, t) h̄ i ∇ψ(r, t)− h̄ i ∇ψ∗(r, t) · ψ(r, t) ] . Show directly from Schrödinger’s equation that the probability density, P (r, t) = |Ψ(r, t)2 and the probability current density obey the continuity equation ∂ ∂t P (r, t) +∇ · j(r, t) = 0. Do not assume that the potential vanishes. b) What is the form of j(r, t) when there is a magnetic field present specified by the vector potential A(r, t)? 9. (LQM 3-11) Consider a particle of charge e traveling in the electromagnetic potentials A(r, t) = −∇λ(r, t), φ(r, t) = 1 c ∂λ(r, t) ∂t where λ (r, t) is an arbitrary scalar function. a) What are the electromagnetic fields described by these potentials? b) Show that the wave function of the particle is given by ψ(r, t) = exp [ − ie h̄c λ(r, t) ] ψ(0)(r, t). where ψ(0) solves the Schrödinger equation with λ = 0. c) Let v(r, t) = eφ(t) be a spatially uniform time varying potential. Show that ψ(r, t) = exp [ −ie h̄ ∫ t −∞ φ(t′)dt′ ] ψ(0)(r, t). (Why is the lower limit on the integral −∞?) FIG. 1: 10. (LQM 3-12) Consider doing a “two-slit interference” experiment where the slits are replaced by long conducting tubes. (Fig. 1) The source S emits particles in reasonaby well-defined wave packets, so that one can be sure [if the tubes are long enough] that for a certain time interval, say t0 to t1 seconds after emission, the wave packet of the particle is definitely within the tubes. During this time interval, a constant voltage VA is applied to tube A and a constant voltage VB is applied
