Time-Dependent Perturbation Theory - Problem Set 1 | PHYS 581, Assignments of Quantum Mechanics

Download Time-Dependent Perturbation Theory - Problem Set 1 | PHYS 581 and more Assignments Quantum Mechanics in PDF only on Docsity! Physics 581: Quantum Mechanics II Department of Physics, UIUC Spring Semester 2007 Professor Eduardo Fradkin Problem Set No. 1: Time-dependent Perturbation Theory Due Date: 2/5/2007 No late sets will be accepted 1 Harmonic Oscillator Consider a one-dimensional linear harmonic oscillator of mass m. The unper- turbed Hamiltonian is H0 H0 = p2 2m + 1 2 mω2x2 (1) where x is the coordinate of the oscillator and p is the momentum. Note: You may find it useful to write the hamiltonian in terms of raising and lowering operators a† and a, a = √ mω 2~ x + ip√ 2m~ω a† = √ mω 2~ x − ip√ 2m~ω (2) with commutation relations [a, a†] = 1. 1. Write H0 in terms of a and a †. 2. Compute the matrix elements 〈n|a†|m〉, where |n〉 and |m〉 are two eigen- states of H0. 3. Consider a perturbation H1(t) of the form H1(t) = gxe −t2/τ2 (3) and assume that at some initial time t0 ≪ −τ the oscillator is in the eigenstate |n〉. Give a physical interpretation of this perturbation (assume that τ → ∞). 4. Use time-dependent perturbation theory to find the amplitude for finding the system in the state |n+m〉 due to the effects of H1 to the lowest non- vanishing order in perturbation theory. To what order in perturbation theory is this amplitude first different from zero and why? Note: evaluate the expressions only at very long times, t ≫ τ and at very short times t ≪ τ . 1 5. Consider now a perturbation of the form H2(t) = γx 2e−t 2/τ2 (4) and assume that at some initial time t0 ≪ −τ the oscillator is in the eigenstate |n〉. Give a physical interpretation of this perturbation (once again, you may assume that τ → ∞). 6. Use time-dependent perturbation theory to find the amplitude for finding the system in the state |n+m〉 due to the effects of H2 to the lowest non- vanishing order in perturbation theory. To what order in perturbation theory is this amplitude first different from zero and why? Note: evaluate the expressions only at very long times, t ≫ τ and at very short times t ≪ τ . 2 Ionization of the Hydrogen Atom A uniform electric field ~E = E cosωt ~ex (5) is applied to a hydrogen atom. Assume that the proton is heavy enough so that it remains static at all times. For the purposes of this problem you can ignore the electron spin and thus label the hydrogen bound states by |n, l, m〉 where, as usual, the quantum numbers n, l, m are integers satisfying n = 1, . . . ,∞, 0 ≤ l ≤ n − 1 and |m| ≤ l and the energy levels are Enlm = −13.6 eV/n2. We will assume that the perturbation is turned one adiabatically slowly in the remote past (you can accomplish that with a suitable adiabatic prefactor in front of the perturbation, eηt, with η → 0+.) 1. Consider the n = 2 states of hydrogen in the presence of the perturbing electric field, i. e. at time t0 → −∞ the hydrogen atom is in one of the four |2, l, m〉 states. Use time-dependent perturbation theory to find for which value of the principal quantum number n the transition 2 → n is more likely (i. e. it has the largest probability). Calculate the contribution of this process to the lifetime of the n = 2 states of hydrogen. Note: be careful to consider all possible allowed initial and final states. 2. Calculate the ionization rate of hydrogen, initially in its ground state, as a function of ω, i. e. at time t0 → −∞ the hydrogen atom is in its ground state, |1, 0, 0〉. For the sake of simplicity you may assume that the scattering states of hydrogen are well approximated by plane waves. Determine the angular distribution of the emitted electrons. 3 Instantaneous Perturbation Consider a system which at times t < 0 is in some eigenstate |i〉 of a certain Hamiltonian H0. The system is subject to the action of an instantaneous per- turbation Hint(t) = V̂ δ(t). Derive a formula for the transition amplitude, to 2