Quantum Mechanics 1 - Assignment Set 4 | PHYS 5250, Assignments of Quantum Mechanics

Download Quantum Mechanics 1 - Assignment Set 4 | PHYS 5250 and more Assignments Quantum Mechanics in PDF only on Docsity! Leo Radzihovsky, Fall 2007 PHYS 5250: Quantum Mechanics - I Homework Set 4 Issued October 8, 2007 Due October 22, 2007 Reading Assignment: Shankar, Chs. 7, 8, 9, 21.1; Sakurai: 1.6, 2.1-2.6 1. Harmonic Oscillator (a) Verify explicitly in coordinate representation that 2nd and 0th eigenfunctions of a harmonic oscillator are orthogonal. (b) Consider a particle in a potential V (x) = 1 2 mω0x 2, for x > 0 and V (x) = ∞ for x ≤ 0. Find the spectrum and eigenfunctions. Hint: This problem should not require you to do any new computations, just a bit of thinking. (c) Find eigenfunctions and spectrum for a particle in a potential V (x) = 1 2 mω20(x 2− 2cx). Hint: This problem should not require you to do too many new computations, just a bit of thinking. (d) Using the representation of x and p in terms of the creation and annihilation operators a† and a, compute the following expectation values: i. 〈n|x|n〉 ii. 〈n|p|n〉 iii. 〈n|x2|n〉 iv. 〈n|p2|n〉 v. 〈n|∆x∆p|n〉, where ∆x and ∆p are root-mean-squared (rms) deviations of x and p from their average values. (e) Show that 〈n|x4|n〉 = x 4 0 4 (3 + 6n(n + 1)), where x0 = √ h̄/mω0 is the quantum oscillator length. (f) Compute 〈n|x2|n〉 directly in coordinate representation using a generating func- tion for Hermite polynomials, similarly to the way we computed normalization factors in class. Compare to your answer with the above one where you used a and a† representation. (g) At time t = 0 a particle in a harmonic oscillator potential starts out in a state |ψ(0)〉 = 1√ 2 (|0〉+ |1〉). Find: i. |ψ(t)〉, ii. 〈x(0)〉 = 〈ψ(0)|x|ψ(0)〉, 〈p(0)〉, 〈x(t)〉, 〈p(t)〉, iii. 〈ẋ(t)〉 and 〈ṗ(t)〉 using Ehrenfest’s theorem and solve for 〈x(t)〉 and 〈p(t)〉 and compare with part (ii). 2. Coupled Harmonic Oscillators Consider two particles characterized by a familiar Hamiltonian H = p21 2m + p22 2m + 1 2 mω20(x 2 1 + x 2 2 + (x1 − x2)2). (a) Find the spectrum and eigenstates of this Hamiltonian by first going to normal modes of vibration, y1 and y2 that decouple it into two independent harmonic oscillators (with different frequencies) and then solving each by using two types of annihilation and creation operators, b1,2 and b † 1,2 that correspond to y1,2. (b) Compute the expectation value of x21 in the ground state of this coupled harmonic oscillator system. 3. Baker-Campbell-Hausdorff Formula Derive the Baker-Campbell-Hausdorff formula eAeB = eA+B+ 1 2 [A,B] for the simplest case where the two operators A and B have a commutator [A,B] that commutes with A and B, i.e., is a c-number. Do this in two ways: (a) First (only suggestive), by looking at the Taylor expansion in A and B of the two sides of the equation, verifying the equality at least to quadratic order in A and B. (b) Second by considering instead operators eAt, eBt and deriving and solving a simple (first order in t) differential equation for eAteBt. Hint: Consider differentiating this product with respect to t, and then follow your nose. 4. Coherent States Using the representation of a coherent state |z〉 = eza†|0〉 show: (a) a|z〉 = z|z〉, (b) 〈z1|z2〉 = ez ∗ 1z2 , (c) that the evolution operator for a harmonic oscillator in coherents state basis is given by U(z, z′; t) = exp[z∗z′e−iω0t],