 Degenerate perturbation theory

Degenerate perturbation theory treats a perturbation of an energy eigenvalue shared by several independent quantum states. Ordinary nondegenerate formulas contain vanishing energy denominators in this situation. Instead, for the orthogonal projection $P$ onto that eigenspace, diagonalize $PVP$ within the eigenspace before treating coupling to other levels.

Its eigenvalues are the first-order energy shifts, and its eigenvectors choose combinations with a well-defined leading response. For a two-state block $V=\left(\begin{array}{cc}0& v\\ v^{\ast }& 0\end{array}\right)$, the shifts are $\pm |v|$: zero diagonal entries in the original basis do not imply zero shifts. Coupling to states outside the block enters at higher order when their energy separations remain large compared with the perturbation. This method explains how a weak field or broken symmetry splits a degenerate energy level.