Assume that we solve the unperturbed energy eigenvalue problem exactly:
Now we add a perturbation that depends on time,
Our problem is now inherently time dependent so we go back to the
time dependent Schrödinger equation.
Now dot into this equation to get the time dependence of one coefficient.
Now we want to calculate transition rates. To first order, all the are small compared to , so the sum can be neglected.
Note that, if there is a large energy difference between the initial and final states, a slowly varying perturbation can average to zero. We will find that the perturbation will need frequency components compatible with to cause transitions.
If the first order term is zero or higher accuracy is required, the second order term can be computed. In second order, a transition can be made to an intermediate state , then a transition to . We just put the first order into the sum.
Transitions of a 1D harmonic oscillator in a transient E field.*
Jim Branson 2013-04-22