Perturbation theory studies a Hamiltonian of the form $\hat{H}=\hat{H}_0+\lambda\hat{V}$, where $\lambda$ is small. One expands energies and states as series in $\lambda$. First-order energy corrections are often enough to capture small shifts due to weak interactions.

If the unperturbed system has degenerate states, a perturbation can mix them strongly. The correct approach is to diagonalize the perturbation within the degenerate subspace. This is common in atomic, molecular, and condensed-matter systems.

The variational principle states that for a normalized trial state $|\psi\rangle$, the expectation value $\langle\psi|\hat{H}|\psi\rangle$ is greater than or equal to the ground-state energy. By choosing a trial wavefunction with adjustable parameters, one can approximate the true ground state and optimize those parameters.