Quantum Mechanics: Time-Dependent Perturbation Theory Solutions, Physics 512, Winter 2003, Assignments of Physics

Download Quantum Mechanics: Time-Dependent Perturbation Theory Solutions, Physics 512, Winter 2003 and more Assignments Physics in PDF only on Docsity! Physics 512 Winter 2003 Homework Set #9 – Due Friday, March 21 1. We consider first order time dependent perturbation theory for a Hamiltonian H(t) = H0 +V (t). Suppose that a system is initially in an eigenstate |s〉 of H0 at time t0. Let O be some observable of the system. Show that the expectation value of O at time t is given to first order in V (t) by 〈ψ(t)|O|ψ(t)〉 = 〈s|O|s〉 − i h̄ ∫ t t0 〈s|[Õ(t), Ṽ (t′)]|s〉 dt′ where Õ(t) and Ṽ (t) are in the interaction picture. 2. Apply first-order time dependent perturbation theory to a forced harmonic oscillator H = (a†a+ 12 )h̄ω + f(t)a+ f ∗(t)a† which is initially in the ground state, and compare the transition probability with the exact result Pn←0 = 1 n! ∣∣∣∣g(ω)h̄ ∣∣∣∣ 2n e−|g(ω)/h̄| 2 where g(ω) = ∫ ∞ −∞ e−iωt ′ f(t′) dt′ is the Fourier transform of the generalized force f(t). Calculate the energy transfer to the oscillator exactly and also in perturbation theory. Show that the energies agree, even though the probabilities do not (see Exercise 19.2). 3. This is similar to Sakurai, Chapter 5, Problem 25. The unperturbed Hamiltonian of a two-state system is represented by H0 = ( E01 0 0 E02 ) There is, in addition, a time-dependent perturbation V (t) = λ ( 0 eiωt e−iωt 0 ) (λ real ) a) At t = 0 the system is known to be in the first state, represented by ( 1 0 ) . Using time-dependent perturbation theory and assuming that E01−E02 is not close to ±h̄ω, derive an expression for the probability for the system to be found in the second state represented by ( 0 1 ) as a function of t (for t > 0). b) Why is this procedure not valid when E01 −E02 is close to ±h̄ω? 1