Download Time-Dependent Perturbation Theory: Periodic Perturbations and Transition Amplitudes and more Study notes Quantum Mechanics in PDF only on Docsity! PHY662, Spring 2004, Mar. 30, 2004 30th March 2004 1 Miscellaneous 1. Reading: Continue Shankar Ch. 18 for time-dependent perturbation theory, Grif- fiths Ch. 9. 2. Time-dependent perturbation theory, especially periodic perturbations. 2 Periodic perturbations REMINDER: We are solving for the time dependence of ψ(t), with ih̄ ∂∂tψ(t) = Hψ(t), where H(t) = H0 +H ′(t). Our approach generally will be to compute transition amplitudes between the unper- turbed eigenstates, where the unperturbed eigenstates |n〉 are the eigenvectors of H0. By expanding |ψ(t)〉 = ∑ n cn(t)|n〉 and changing to the coefficients dn(t) = cn(t)eiEnt/h̄, we derived the first order result df = δfi − i h̄ ∫ t 0 dt′ 〈f |H ′|i〉eiωfit . This formula approximates the amplitude for ψ(t) to be in the state |f〉, which is an eigentstate of H0. Let us consider the specific case H ′(t) = { 0 t < 0 V (~r) cos(ωt) t ≥ 0 . This is a useful arrangement that allows us to cleanly consider the initial eigenstates for t < 0 and the effect of adding an oscillating term to the Hamiltonian at later times. Usuallly, the perturbation is present at all times, but the state is considered in an eigen- state of H0 at time t = 0. (An alternative derivation to obtain essentially the same results has the perturbation turned on slowly.) 1 In any case, for this H ′(t), we can compute the first-order transition amplitudes to an arbitrary final state |f〉 6= |i〉: df = − i h̄ ∫ t 0 dt′ 〈f |V (~r)|i〉 [ ei(ω+ωfi)t + ei(−ω+ωfi)t 2 ] = − i〈f |V (~r)|i〉 2h̄ [ ei(ω+ωfi)t − 1 ω + ωfi + ei(−ω+ωfi)t ωfi − ω ] = h̄−1〈f |V (~r)|i〉 [ ei(ω+ωfi)t/2 ei(ω+ωfi)t/2 − e−i(ω+ωfi)t/2 2i(ω + ωfi) + ei(ω−ωfi)t/2 ei(ω−ωfi)t/2 − e−i(ω−ωfi)t/2 2i(ω − ωfi) ] = h̄−1〈f |V (~r)|i〉 { ei(ω+ωfi)t/2 sin[(ω + ωfi)t/2] ω + ωfi + ei(ω−ωfi)t/2 sin[(ω − ωfi)t/2] ω − ωfi } . This amplitude can then be used to compute the probability of being in |f〉 at time t: |df |2 = |〈f |V (~r)|i〉|2 h̄2 { sin2[(ω + ωfi)t/2] (ω + ωfi)2 + sin2[(ω − ωfi)t/2] (ω − ωfi)2 + (crossterm) } . The simplest thing to do at this point is to assume that we are looking at cases where ωfi is near ω. Then, the second term dominates and we get |df |2 ≈ |〈f |V (~r)|i〉|2 h̄2 { sin2[(ω − ωfi)t/2] (ω − ωfi)2 } . Note that the transition probability is oscillatory in time! This can be viewed as an artifact of turning on the potential suddenly. We can also “smooth” out this oscillation by making standard assumptions, which we will do soon. This form is somewhat familiar, though, as it should remind us of the oscillations seen in the probabilities in the problem of magnetic resonance. 3 Averaging to smooth: Fermi’s golden rule The expression for |df |2 is put into a standard transition rate expression taking t to be large and by either: 1. Taking a smooth distribution of final states. Example: an electron in an atom being excited by an oscillating field from a bound (discrete) state to an unbound state - the unbound states have a continuum distribution as the wave vectors ~k in free space are smoothly distributed. 2. Taking a smooth incoherent mixture of perturbing frequencies. Example: an electron in an atom making a transition from one bound state to another, stimu- lated by a range of frequencies, simultaneously. Here, the states are discrete, but we average over ω. 2
