PHYS 5250: Quantum Mechanics Homework Set 5 - Fall 2007 by Leo Radzihovsky - Prof. Leo Rad, Assignments of Quantum Mechanics

Download PHYS 5250: Quantum Mechanics Homework Set 5 - Fall 2007 by Leo Radzihovsky - Prof. Leo Rad and more Assignments Quantum Mechanics in PDF only on Docsity! Leo Radzihovsky, Fall 2007 PHYS 5250: Quantum Mechanics - I Homework Set 5 Issued October 22, 2007 Due November 5, 2007 Reading Assignment: Shankar: Chs. 10, 11; Sakurai: Secs. 1.6, 1.7, Chs. 4, 6 1. Using path integral formulation, derive a free particle time evolution operator in co- ordinate representation, U0(x, x ′; t, 0). Rather than following shortcuts we discussed in class, please do this by explicitly evaluating N − 1 gaussian integrals that arise in discretizing the path integral for U(x, x′; t, 0). Hint: The most direct route is to first demonstrate and then utilize the ”closure” property of a propagator, namely to show that U0(x, x ′′; t, t′′) = ∫∞ −∞ dx ′U0(x, x ′; t, t′)U0(x ′, x′′; t′, t′′). To show this you will simply need to do a single gaussian integral, taking advantage of the calculus that we developed in class. There are other, less direct but technically simpler ways of doing it. Can you find one or two other ways of doing it? 2. Consider a Hamiltonian of two interacting particles H = p21/2m1 +p 2 2/2m2 +V (r1−r2) moving in 3d. (a) Show that by performing a unitary transformation to the center of mass Rcm,Pcm and relative r,p coordinates, above Hamiltonian decouples into H = Hcm +Hrel, where Hcm = P 2 cm/2M and Hrel = p 2/2µ + V (r) are the center of mass and relative coordinate Hamiltonians, whereM and µ are the total and reduced masses respectively whose expressions in terms of m1 and m2 you should derive. (b) Verify that this transformation is indeed unitary, i.e., that the commutation re- lations between Rcm and Pcm, and between r and p remains canonical. Do this by finding the relation between old and new variables and then computing the commutation relations between all new variables ([Ricm, p j], [Ricm, P j cm], etc.) using commutation relations for the old variables. (c) Take advantage of the above separability of H to find the Rcm dependence of the eigenstates of H, and derive the effective Schrodinger equation satisfied by the part of the wavefunction that only depends on the relative coordinate r. (d) Considering a physical case in which V (r1−r2) = V (|r1−r2|) (i.e., that the inter- action is rotationally invariant, depending only on the distance between the two particles), show that the Schrodinger equation for ψrel(r) can be further separated in a spherical coordinate system by reducing to three independent eigenvalue dif- ferential equations for the radial, polar and azimuthal part respectively dependent on r, θ, and φ. 3. When an energy measurement is made on a system of three spinless bosons in a box, the n values obtained were 3, 3, and 4. Write down a symmetrized, normalized state vector. 4. Imagine a situation in which there are three particles and only three states a, b, and c available to them. What is the total number of allowed, distinct configurations for this system if the particles are: (a) labeled, i.e., distinguishable (b) indistinguishable bosons (c) indistinguishable fermions 5. Bloch theorem and discrete translational invariance Consider a particle in a periodic potential with period a. With the particle an electron and the periodic potential due to a positive ions arranged in a periodic arrangement in a crystalline solid, this problem is at the very heart of much of modern solid-state physics. (a) Show explicitly that the corresponding Hamiltonian commutes with a translation operator T only for translations that are a multiple of a, i.e.,  = na, n ∈ Z. (b) Use above result together with the observation that Ta is a unitary operator to clearly argue that the eigenstates of H are (the so-called) Bloch states, given by ψk,n(x) = e ikxuk,n(x), where k is a continuous (for an infinite system) quantum number limited to a finite range (called 1st Brillouin zone) that can be taken to be 0 < k < 2π/a and uk,n(x) is periodic part of the wavefunction with period a, labeled by k and another discrete quantum number n (∈ Z). (c) Derive an effective Schrodinger equation satisfied by uk,n(x). (d) What can you say about the property of the energy eigenvalues En(k)? Note that althought the eigenfunctions are not periodic with period a, as expected on physical grounds the probability density Pk,n(x) = |ψk,n(x)|2 of finding a particle at positive x is. This is no different than in a special case of this problem of a constant V (x) (e.g., 0), where the eigenfunctions are plane waves eikx that are less symmetric (i.e., not translationally invariant) than the Hamiltonian.