time-independent perturbation theory

Time-independent perturbation theory handles the case where the extra disturbance is static — switched on and left there, not changing with time. We start from a system whose stationary states and energy levels are known exactly, then ask how a fixed small addition to the energy operator shifts those levels and reshapes those states. Because nothing depends on time, we are simply hunting for new, slightly altered standing-wave solutions.

The recipe gives corrections order by order. At lowest order the energy of each state shifts by the average of the disturbance taken over the unperturbed state — a clean, intuitive result. The state itself also bends, mixing in a little of the other unperturbed states, weighted by how strongly the disturbance connects them and divided by how far apart their energies are. Each higher order refines both the energy and the state further.

This is the version most students meet first, and it underlies a great deal of atomic and molecular physics. It explains how an atom's levels split when bathed in a steady electric or magnetic field, and how small relativistic effects nudge spectral lines. Its honesty lies in that dividing by energy gaps: when two unperturbed states sit at the same energy, the gap is zero and this simple version breaks down, demanding the more careful degenerate treatment.