Notes 21 : Bound - State Perturbation Theory

Bound state perturbation theory applies to the bound states of perturbed systems, for which the energy levels are discrete and separated from one another. The system may also have a continuous spectrum, but the interest is attached to the discrete states. The effects of perturbations on states of the continuous spectrum, or on discrete states imbedded in the continuum, require different techniques, such as time-dependent perturbation theory. The version of bound-state perturbation theory that you are probably familiar with is called Rayleigh-Schrödinger perturbation theory. In these notes, we will begin with a variation on RayleighSchrödinger perturbation theory, called Brillouin-Wigner perturbation theory. The Brillouin-Wigner theory is simpler and less messy to derive than the Rayleigh-Schrödinger theory, and it also gives more accurate answers in many circumstances. Its disadvantage is that the unknown energy levels are given only implicitly, in terms of the solutions of nonlinear equations. But if explicit formulas for the energy levels are desired, it is easy to expand the solutions in powers of the perturbation parameter, whereupon the results of the Rayleigh-Schrödinger theory are recovered. In my opinion, this is the easiest and most elegant way to derive Rayleigh-Schrödinger perturbation theory.