time-independent perturbation theory
Almost no realistic quantum Hamiltonian can be solved exactly. Time-independent perturbation theory is the workhorse method for the enormous class of problems that are 'close' to one you can solve: take a solvable system and add a small, static disturbance, then compute the correction to the energies and states as a systematic power series in the size of that disturbance. It is how you get from the ideal hydrogen atom to the real one, line by line.
Split the Hamiltonian as H = H_0 + lambda H', where H_0 has known eigenstates |n_0> and energies E_n^0, H' is the perturbation, and lambda is a small bookkeeping parameter. Expand the true energy and state in powers of lambda. For a nondegenerate level, the first-order energy shift is just the expectation value of the perturbation in the unperturbed state: E_n^(1) = <n_0| H' |n_0>. The second-order shift is E_n^(2) = sum over m not equal to n of |<m_0| H' |n_0>|^2 / (E_n^0 - E_m^0), and the first-order correction to the state mixes in other levels with the same denominators. The two facts to carry away: the leading energy correction is simply the average of the perturbation over the old state, and second order always pushes nearby levels apart (a level is repelled downward by states above it and upward by states below it).
The method's reach is vast -- fine structure, the Zeeman and Stark effects, van der Waals forces, and most of atomic and molecular structure are perturbative corrections computed this way. But its honesty must be stated: it is an asymptotic expansion that assumes the perturbation is genuinely small and, critically, that the state you perturb is nondegenerate -- the denominator E_n^0 - E_m^0 blows up if two unperturbed levels coincide. When levels are degenerate you must first use degenerate perturbation theory, and even when they are not, the series can fail to converge and is often only asymptotic, useful term-by-term but not summable to the exact answer.
A charged particle in a harmonic-oscillator potential feels a weak uniform electric field E, adding a perturbation H' = -qE x. The first-order shift <n|H'|n> vanishes because x is odd, so the leading effect is second order; carrying it out gives an exact-looking energy lowering -q^2 E^2/(2 m omega^2), the same for every level -- the oscillator's polarizability, computed with two terms of the series.
When the first-order shift vanishes by symmetry, the second-order term carries the physics -- here the oscillator's polarizability.
The formulas here assume a nondegenerate level; if any E_m^0 equals E_n^0 the second-order sum diverges and you must switch to degenerate perturbation theory first. And perturbation series are typically asymptotic, not convergent -- adding more terms eventually makes the estimate worse, so 'small perturbation' is a real physical prerequisite, not a formality.