degenerate perturbation theory
Ordinary perturbation theory has a fatal blind spot: it divides by the energy gap between levels, so when two or more unperturbed states share exactly the same energy -- a degeneracy -- the formulas blow up. Degenerate perturbation theory is the fix. The insight is that when several states are tied, the perturbation itself gets to choose which combinations of them are the 'right' starting states, and once you let it choose, the divergence disappears and the correct splitting emerges.
Suppose a level is g-fold degenerate, with unperturbed states |1>, ..., |g> all at energy E^0. The trouble is that any linear combination of them is equally an eigenstate of H_0, so there is no unique 'zeroth-order' state to perturb. The resolution: build the g-by-g matrix of the perturbation within the degenerate subspace, W_ij = <i| H' |j>, and diagonalize it. Its eigenvalues are the first-order energy shifts -- the perturbation splits the single degenerate level into (up to) g distinct levels -- and its eigenvectors are the 'good' zeroth-order states, the specific combinations that the perturbation does not mix. Symmetry is your friend here: choosing basis states that are already eigenstates of an operator commuting with both H_0 and H' makes W block-diagonal (or diagonal), and you can often read off the splitting with no matrix work at all.
This is the machinery behind nearly every 'splitting' in physics. The Zeeman effect splits a magnetic level, the Stark effect splits it in an electric field, and spin-orbit coupling splits the accidentally degenerate hydrogen levels into fine structure -- each is a degenerate perturbation calculation that identifies the good states and lifts the degeneracy. The core lesson is conceptual: a symmetry produces degeneracy, and a perturbation that breaks the symmetry resolves it, with the pattern of splitting revealing exactly which symmetry was broken.
Hydrogen's n=2 level is fourfold degenerate (2s and three 2p states, ignoring spin). A weak electric field (the linear Stark effect) mixes the 2s and 2p_z states; diagonalizing the 2-by-2 perturbation matrix splits them by an amount proportional to the field, while the other two 2p states are unshifted -- the good states are the sum and difference of 2s and 2p_z, a textbook degenerate-perturbation result.
The linear Stark effect: the field's good states are combinations of 2s and 2p_z, split symmetrically about the original level.
You cannot pick the zeroth-order states arbitrarily when there is degeneracy -- you must let the perturbation matrix select them by diagonalization, or the first-order theory is simply wrong. A perturbation that happens to be already diagonal in your chosen degenerate basis is the lucky case; in general it is not, and off-diagonal elements are exactly what the method handles.