Most quantum systems cannot be solved exactly. Approximation methods allow us to extract energies, wavefunctions, and transition rates when interactions or potentials are too complicated for closed-form solutions.

If the Hamiltonian is written as $\hat H = \hat H_0 + \lambda \hat V$, and $\lambda$ is small, one can expand the energy and state corrections in powers of $\lambda$.

For a nondegenerate level $n$, the first-order correction is $$E_n^{(1)} = \langle n^{(0)}|\hat V|n^{(0)}\rangle.$$ This is the expected value of the perturbation in the unperturbed state.

When unperturbed levels share the same energy, the perturbation must be diagonalized within the degenerate subspace before corrections are read off.

The variational principle says that any trial wavefunction gives an energy expectation value $\langle H \rangle$ that is at least the true ground-state energy, making it a powerful estimation tool.

These methods are used in atomic physics, spectroscopy, molecular structure, and condensed matter theory.