The first term in Equation 11.32 comes from the e it /2 part of Cos(t), and the second from e - it . Thus dropping the first term is formally equivalent to writing H' = (V/2)e - it , which is to say, (The latter is required to make the Hamiltonian matrix hermitianor, if you prefer, to pick out the

Chapter 11, Problems #9

The first term in Equation 11.32 comes from the e^iωt/2 part of Cos(ωt), and the second from e^-^iωt. Thus dropping the first term is formally equivalent to writing Ĥ' = (V/2)e^-^iωt, which is to say,

(The latter is required to make the Hamiltonian matrix hermitian—or, if you prefer, to pick out the dominant term in the formula analogous to Equation 11.32 for C_a(t).) Rabi noticed that if you make this so-called rotating wave approximation at the beginning of the calculation, Equation 11.17 can be solved exactly, with no need for perturbation theory, and no assumption about the strength of the field.

(a) Solve Equation 11.17 in the rotating wave approximation (Equation 11.36), for the usual initial conditions: C_a(0) = 1, C_b(0) = 0. Express your results (C_a(t) and C_b(t) in terms of the Rabi flopping frequency,

(b) Determine the transition probability, P_a→b(t), and show that it never exceeds 1. Confirm that |C_a(t) |² + |C_b(t) |² = 1.
(c) Check that P_a→b(t) reduces to the perturbation theory result (Equation 11.35) when the perturbation is “small,” and state precisely what small means in this context, as a constraint on V.
(d) At what time does the system first return to its initial state?