03 Rayleigh-Schroedinger Perturbation theory Part 3

Degenerate perturbation theory

[himanshu]

I think we can safely assume the higher order corrections to be orthogonal to unperturbed state. Suppose higher order correction have some overlap with unperturbed state. Then we can combine all the overlaps of corrections with unperturbed state at one place with the zero order correction. Now the higher order correction are orthogonal to unperturbed state. So, in perturbation expansion, the coefficient of zero order correction would be (1+f(lambda)), where f(lambda) is not known and lambda is the extent factor of perturbation. Now we can divide the whole wavefunction by (1+f(lambda)) (it just scale each coefficient). And now the coefficient of unperturbed state in the expansion would be 1. And the coefficient which is in denominator of the higher order corrections can be absorbed into the higher order corrections because both are unknown. I think it won't change anything because anyways in the end we will normalize the perturbed state. But in the whole process we divide the wavefunction by a term in which the contribution of overlap of higher order correction with the unperturbed state is there.
I still don't know how to argue that that series converges to finite value because only then that division make sense. If it blows up, then all the corrections will vanish.