Download Time-Dependent Perturbation Theory: Derivation and Applications and more Exams Quantum Mechanics in PDF only on Docsity! PHY662, Spring 2004, Mar. 18, 2004 18th March 2004 1 Miscellaneous 1. Continue Shankar Ch. 18 for time-dependent perturbation theory. 2. I am out of town next week (exam on Tuesday, we will schedule a makeup class for Thursday). 3. Exam review session on Sunday, 1 PM, Room 204. 4. Today: (a) Continue time-dependent perturbation theory. 2 Time-dependent perturbation theory "You have to wonder how this happened to him," Lundergaard said. "Was he calculating the transition amplitudes between the unperturbed eigen- states due to the presence of the perturbation in order to determine tran- sition probabilities in time-dependent quantum phenomena, and the next day, strapping a TV antenna to his head?" [From the 17 March 2004 issue of The Onion.] We are solving for the time dependence of ψ(t), with ih̄ ∂∂tψ(t) = Hψ(t), where H(t) = H0 +H ′(t). Our approach will be to compute transition amplitudes between the unperturbed eigen- states, where the unperturbed eigenstates |n〉 are the eigenvectors ofH0. By expanding |ψ(t)〉 = ∑ n cn(t)|n〉 and changing to the coefficients dn(t) = cn(t)eiEnt/h̄, we de- rived the exact result ih̄ḋf = ∑ n dn(t)〈f |H ′|n〉eiωfmt . This equation gives the rate of change of the amplitude to be in a final state |f〉 that is the sum over transition amplitudes from n to f that are equal to the matrix elements 1 of H ′. We say that the non-zero matrix elements of H ′ cause transitions between the unperturbed eigenstates. Griffiths carries out this derivation for the more specific case of a 2 × 2 Hamiltonian matrix. 2.1 Zeroth order To lowest order, ḋn = 0. Let’s take the initial condition that ψ(0) is equal to some eigenstate |i〉. Then df = δfi to zeroth order. 2.2 First order The first order result is found by putting in the zeroth order solution for dn on the right hand side of the equation at the end of Sec. 4.1. This gives ih̄ḋf = 〈f |H ′|i〉eiωfit , which has solutions df = δfi − i h̄ ∫ t 0 dt′ 〈f |H ′|i〉eiωfit . This is the important first order result that will get us going on several topics. 2.3 Types of perturbations 2.3.1 Direct computation The first-order formula for df can be directly applied, as Shankar does in Eqns. (18.2.10) through (18.2.13): take a harmonic oscillator with H0 = h̄ω(a†a + 12 ) and a time- dependent perturbation −eEXe−t2/τ2 , X = ( h̄ 2mω )1/2 (a+ a†). Then P0→1 = |d1(∞)|2 = ∣∣∣∣∣ ieEh̄ ( h̄ 2mω )1/2 ∫ ∞ −∞ e−t/τ 2 eiωt ∣∣∣∣∣ 2 = e2E2πτ2 2mωh̄ e−ω 2τ2/2 . What if τ →∞? 2.3.2 Sudden Perturbations Shankar has a section on sudden perturbations that actually has several types of changes in the Hamiltonian: • A brief, finite H ′: in the limit that the duration of the perturbation is short, there will be no change in ψ(t). This results from the fact that for finite total H , the change in ψ over an interval of duration is zero as → 0. 2
