Thermodynamics, Statistical Physics, and Quantum Mechanics

5.40 Circle with Field (Colorado, Michigan State)

The perturbation is if weassume the field is in the

The same result is obtained if we assume theperturbation is

in the In order to doperturbationtheory,

we need tofind the matrix element of theperturbationbetweendifferent

eigenstates. For first-order perturbation theory we need

The eigenvalues areunchanged to first-order in the fieldE.
To do second-orderperturbationtheory, weneedoff-diagonalmatrix
elements:

If we recall that then we see that can only
equal for theintegral to benonzero. Indoing second-orderperturbation
theory for thestate theonlypermissibleintermediatestates are

Thissolution isvalid forstates For the ground state,with
the state does not exist, so the answer is

QUANTUM MECHANICS 289