How Do You Compute the 1st Order Wave Function Correction in Quantum Mechanics?

[cscott]

Homework Statement

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.