Exactly solvable quantum Hamiltonians are rare. Perturbation theory approximates the spectrum of $$H=H_0+\lambda V,$$ when $H_0$ is solvable and the perturbation $\lambda V$ is sufficiently weak.

For a nondegenerate unperturbed eigenstate $$H_0|n^{(0)}\rangle=E_n^{(0)}|n^{(0)}\rangle,$$ the first-order energy correction is $$E_n^{(1)}=\langle n^{(0)}|V|n^{(0)}\rangle.$$ Thus $$E_n\approx E_n^{(0)}+\lambda E_n^{(1)}.$$ The first-order correction to the state contains admixtures of other unperturbed eigenstates: $$|n^{(1)}\rangle= \sum_{m\ne n} \frac{\langle m^{(0)}|V|n^{(0)}\rangle} {E_n^{(0)}-E_m^{(0)}}|m^{(0)}\rangle.$$

As a simple example, if a perturbation adds a constant $V_0$ throughout the region where a normalized state lives, then $$E_n^{(1)}=\langle n|V_0|n\rangle=V_0,$$ so every level shifts by the same amount while transition energy differences remain unchanged.

Near degeneracies require degenerate perturbation theory: one must first diagonalize the perturbation within the degenerate subspace. Small denominators also warn that the naive expansion may fail.

Perturbation theory turns exact idealized models into useful approximations for real systems. External fields, weak couplings, relativistic corrections and interactions often appear as perturbations to simpler Hamiltonians.