Degenerate perturbation theory

When a state being perturbed has a degenerate energy in the original Hamiltonian, non-degenerate perturbation theory no longer applies. This is clear because the first order state correction from that theory $\ket{n\suppar1}$ is undefined.

More generally, non-degenerate perturbation theory fails because the state being perturbed is not necessarily an eigenvector of $\delta H$ . We must choose a set of vectors spanning the degenerate subspace that also diagonalize $\delta H$ in that subspace. The perturbed energy must be smooth from $\lambda=0$ to $\lambda=\epsilon$ , which can only be the case if the state being perturbed is an eigenvector of $\delta H$ .

Call $\delta H$ in the degenerate subspace $[\delta H]$ . We (Zwiebach) then call(s) the states that diagonalize $[\delta H]$ the good states.

To find the good states and their perturbed energy levels we start from the same equations as in the non-degenerate case.

Call the $l$ -th unperturbed state of the $n$ -th energy level $\ket{n\suppar0,l}$ , and let $\ket{\psi_i\suppar0}$ be a good state.

\begin{align*} (H\suppar0 - E_n\suppar0) \ket{\psi_i\suppar0} &= (E_n\suppar1 - \delta H)\ket{\psi\suppar0} \\ 0 &= \braket{n\suppar0,l | E_{ni}\suppar1 - \delta H | \psi_i\suppar0} \\ &= \sum_k (E_{ni}\suppar1 a_{ik} \braket{n\suppar0,l|n\suppar0,k} - a_{ik} \braket{n\suppar0,l | \delta H | n\suppar0,k}) \\ &= \sum_k a_{ik} (E_{ni}\suppar1 \delta_{lk} - [\delta H]_{lk}) \\ &= ([\delta H] - E_{ni}\suppar1 \hat 1) \vec a_i. \end{align*}

This is an eigenvalue equation, with eigenvalues $E_{ni}\suppar1$ and eigenvectors