perturbation theory
Perturbation theory is a strategy for problems that are almost, but not quite, ones we can already solve exactly. The idea is to split the problem into two pieces: a clean, solvable part whose energies and states we know by heart, plus a small extra disturbance that nudges everything a little. We then build the answer as the known solution corrected by a series of progressively smaller terms.
The crucial word is small. We introduce a bookkeeping parameter, often written as a tiny number, that measures how strong the disturbance is, and we expand the energies and wavefunctions as a power series in it. The first correction captures the leading effect, the next correction refines it, and in good cases each successive term matters less than the one before, so a handful of terms gives an excellent approximation.
This method is one of the workhorses of quantum mechanics, because exactly solvable problems are rare while almost-solvable ones are everywhere. The fine splitting of atomic spectral lines, the response of an atom to a weak electric or magnetic field, and the slow shifts caused by tiny interactions are all handled this way. It is honest about its limits, though: it only works when the disturbance really is gentle compared with the gaps between the unperturbed energy levels.
Split the Hamiltonian into a solvable part plus a small disturbance, then expand in powers of its strength.
A power series in the disturbance need not converge; sometimes adding more terms eventually makes things worse. The series is often only asymptotic, useful for a few terms even when its full sum does not exist.