In this problem we develop time-dependent perturbation theory for a multi-level system, starting with the generalization of Equations 11.5 and 11.6: At time t = 0 we turn on a perturbation H' (t), so that the total Hamiltonian is (a) Generalize Equation 11.10 to read and show that where (b) If the system starts out in the state N

Chapter 11, Problems #24

In this problem we develop time-dependent perturbation theory for a multi-level system, starting with the generalization of Equations 11.5 and 11.6:

At time t = 0 we turn on a perturbation H' (t), so that the total Hamiltonian is

(a) Generalize Equation 11.10 to read

and show that

where

(b) If the system starts out in the state ΨN, show that (in first-order perturbation theory)

and

(c) For example, suppose Ĥ' is constant (except that it was turned on at t = 0, and switched off again at some later time . Find the probability of transition from state N to state M (M ≠ N), as a function of T.

(d) Now suppose Ĥ' is a sinusoidal function of time: Ĥ' = V cos(ωt). Making the usual assumptions, show that transitions occur only to states with energy EM = EN ± ћω, and the transition probability is

(e) Suppose a multi-level system is immersed in incoherent electromagnetic radiation. Using Section 11.2.3 as a guide, show that the transition rate for stimulated emission is given by the same formula (Equation 11.54) as for a two-level system.