If $\hat H = \hat H_0 + \lambda \hat V$ and $\lambda$ is small, one expands energies and states in powers of $\lambda$. Rayleigh-Schrödinger perturbation theory is powerful but may be asymptotic rather than convergent.

When unperturbed levels are degenerate, one must diagonalize the perturbation in the degenerate subspace before expanding. This is essential in atoms, solids, and symmetry-broken systems.

The variational principle states that for any normalized trial state $|\phi\rangle$, the expectation value $\langle \phi|\hat H|\phi\rangle$ is an upper bound on the ground-state energy.

In one dimension, WKB approximates solutions in the limit of slowly varying potentials. The exponential tunneling factor is governed by classically forbidden regions where the momentum becomes imaginary.