1st-order WF correction

[cscott]

1. The problem statement, all variables and given/known data

A have a bit of a general question regarding 1st order wave function corrections using perturbation theory.

In a problem like the infinite potential well where you have states numbered like n = 1, 2, 3, ..., how do you compute the sum for the 1st order correction when you have infinite terms?:

\psi_n^{(1)} = \Sigma_{l \ne n} \frac{<\psi_n^{(0)}|H'|\psi_l^{(0)}>}{E_n^{(0)} - E_l^{(0)}} \psi_l^{(0)}

I guess I don't know how to get <n|H'|l> so I can evaluate the sum

[kuruman]

Do you know what your perturbation H' looks like? Sometimes the non-zero terms in the summation result in something that can be summed analytically.

[cscott]

This was the thinking I was missing!

So for H' = constant there is no first-order correction because l \ne n, yes?

[kuruman]

Correct. If you add a constant to your Hamiltonian, you shift the zero of energy but you do not change its eigenstates.