Quantum Mechanics II: Homework Set 3 - Probability of Transitions in Atoms and Oscillators, Assignments of Quantum Mechanics

Download Quantum Mechanics II: Homework Set 3 - Probability of Transitions in Atoms and Oscillators and more Assignments Quantum Mechanics in PDF only on Docsity! Leo Radzihovsky, Spring 2008 PHYS 5260: Quantum Mechanics - II Homework Set 3 Issued February 11, 2008 Due February 25, 2008 Reading Assignment: Shankar, Ch.18, 21 1. Consider a hydrogen atom in its ground state at t → −∞, subjected to a uniform electric field E(t) = ẑEe−t2/τ2 . Show that the probability that at time t→∞ the atom ends up in any of the n = 2 states is, to first order, P (n = 2) = ( eE h̄ )2 (215a20 310 ) πτ 2e−ω 2τ2/2, (1) where ω = (E2`m − E100)/h̄. Does the answer depend on whether or not the spin is incorporated in the picture? 2. Shifted harmonic oscillator (revisited) (a) Consider a charged particle (charge q, mass m) confined to move in 1D in a pres- ence of a harmonic potential V = 1 2 mω2x2 and subjected to a uniform, constant electric field E . If at time t = 0− the particle is its ground state but at t = 0+ the electric field is suddenly shut off, compute the average energy change of the oscillator at subsequent time. (b) What is the final energy change after the electric field is instead shut off adiabat- ically and how does it compare to that in part (a)? (c) How does your answer compare to the classical result? Hint: It is convenient to represent the state of the original displaced oscillator in terms of the displacement operator (by distance α) Dα = e −α∂x acting on the state of the undisplaced oscillator. Expressing this displacement operator in terms of creation and annihilation operators and using Baker-Campbell-Hausdorff formula eA+B = eAeBe− 1 2 [A,B] (valid when [A,B] is a c number) allows a very efficient computation of the probability of finding the system in the n-th quantum state. 3. Perturbed harmonic oscillator transitions (a) Consider, once again, a charged particle in a ground state of a 1D periodic po- tential V (x) = 1 2 mω2x2, this time perturbed at time t = 0 by a weak oscillating electric field E cosωet. Calculate a transition rate at time t out of the ground state. What is the asymptotic, long-time rate? (b) Repeat above analysis if instead of an electric field, the perturbation is a peri- odically modulated oscillator frequency, i.e., ω → ω(t) = ω + ω0 cosωet, where amplitude of frequency modulation is weak, ω0  ω. 4. Consider a single spin 1/2 that has been “prepared” (using e.g., a strong polarizing magnetic field pointing along ẑ, that now has been shut off) to be in the spin-up (“along” ẑ) eigenstate. (a) What are the amplitudes d↑, d↓ of finding the spin in the spin-up and spin- down eigenstate, respectively, after an application of a weak (so that 1st order perturbation theory applies) magnetic field pulse (τ is the duration of time over which pulse is applied, taken to be much smaller than any other scale in the problem) i. B = ẑB0τδ(t), i.e., along z, parallel to the initial spin configuration? ii. B = ŷB0τδ(t), i.e., along y, transverse to the initial spin configuration? Comment: Take for concreteness the interaction of the spin with a magnetic field to be described by a Hamiltonian H = −µBσ ·B (b) Repeat analysis of all of the above questions for a magnetic field of magnitude B0 constant during time 0 < t < τ and zero otherwise, i.e., turned on at t = 0 and shut off at t = τ . Notice that this case should reduce to the previous case for a vanishing “on” duration τ . (c) For the last case with B0 along ŷ, deduce the duration τ∗ beyond which the per- turbation theory breaks down, and interpret this time physically, e.g., by thinking about the full spin dynamics (at least classically first) for this case. (d) Solve the problem (b) exactly by explicitly finding the expression for the spinor dσ(t) at time t, and compare the lowest order Taylor expansion with the perturba- tive solutions above. Also for case (ii) calculate the magnetic field pulse duration τπ for which the spin is in the spin-down eigenstate with 100% certainty, and compare it to τ∗ 5. Berry’s phase As we discussed in class, for an extremely slowly (adiabatically) changing Hamiltonian, H(α(t)) (where α(t) is some parameter that changes slowly in time) we expect that a system that starts out in an eigenstate |n, α(0)〉 separated from other states by a large gap will remain in an instantaneous eigenstate of the time-dependent Hamiltonian, i.e.,