degenerate perturbation theory
Degenerate perturbation theory is the careful version we must use when several unperturbed states share exactly the same energy. The ordinary formulas of perturbation theory divide by the gap between energy levels, and when that gap is zero they explode into nonsense. The root of the trouble is subtle: when states are degenerate, the system has no unique 'starting' state to perturb, because any blend of the degenerate states is equally valid.
The fix is to let the disturbance itself choose the right starting states. Within the cramped space of degenerate states, we write the disturbance as a small matrix and find its special directions — the particular combinations that the disturbance does not mix together. These privileged combinations are the correct zeroth-order states, and the disturbance's effect on each of them gives the first-order energy shifts directly, no division required.
Physically, this is the machinery behind level splitting. A degenerate level that looked like a single line often fans out into several distinct levels once a disturbance is applied, and degenerate perturbation theory predicts both how it splits and which combinations of states emerge. The splitting of hydrogen levels in an electric field and the lifting of degeneracies by spin-orbit coupling are classic examples where this more careful approach is essential.
Inside the shared-energy subspace, diagonalizing the disturbance picks the right states and gives the level splittings.
A disturbance need not lift a degeneracy completely. Symmetry can protect some states from splitting, so a level may break into fewer pieces than the count of degenerate states suggests.