Perturbation theory

If the eigenvalues and eigenfunctions of H⁰ are known, and if H^’ is a perturbing Hamiltonian, then corrections to the energies and wave functions produced by the perturbation are obtained as follows:

Perturbation on non-degenerate states	Perturbation on degenerate states
First order energy First order wave function Second order energy	Zeroth order wave functions and first order energies are obtained by diagonalising the perturbation matrix, which is order n, where n is the degeneracy. If an operator A commutes with H^0 and H’, then the correct zeroth order wave function will be a linear combination of those unperturbed functions that have the same eigenvalue of A.