Principles of Quantum Mechanics · Problem 18.1

Time-Dependent Perturbation Theory — R. Shankar

Physics — Quantum Mechanics · time-dependent-perturbation-theory

Problem

Consider a two-state system with states

|1\rangle

and

|2\rangle

, with

E_1 < E_2

. Suppose that at

t = 0

we have

c_1(0) = 1

and

c_2(0) = 0

. Now add a time-dependent perturbation

H'

such that

H'_{11} = H'_{22} = 0

and

H'_{12} = (H'_{21})^* = \gamma e^{i\omega t}

where

\gamma

is a real constant. Solve the coupled equations for

c_1(t)

and

c_2(t)

exactly (that is, without making the approximation that

c_1 \approx 1

throughout). Express your answer in terms of

\Omega_r \equiv \sqrt{(\omega_0 - \omega)^2/4 + (\gamma/\hbar)^2}

(the generalized Rabi flopping frequency), where

\omega_0 \equiv (E_2 - E_1)/\hbar

. What is

|c_2(t)|^2

? Check that

|c_1(t)|^2 + |c_2(t)|^2 = 1