Explicit Solution for the Wavefunction and Energy in Degenerate Rayleigh–Schrödinger Perturbation Theory

Abstract

The Rayleigh–Schrödinger perturbation series for the energy and wavefunction are derived for the case that the zeroth-order state is degenerate. The solution, embodied in four “rules,” is a generalization of the nondegenerate formulas of Huby and Brueckner. The derivation consists of redefining the unperturbed Hamiltonian and the perturbation (in a particular way) so as to remove the degeneracy, then rearranging the terms in the nondegenerate series in the new perturbation according to the order in the old perturbation. The “choice” of “correct” zeroth-order functions falls out in a natural way, as do the components of the wavefunction in the degenerate unperturbed subspace. The solution given here is not directly related to the degenerate problems solved previously by Kato and by Bloch, in that their solutions do not lead directly to the Rayleigh–Schrödinger series for the energy and wavefunction. The present solution is more related to the one given by Hirschfelder (both being solutions of the same problem), but it differs in not involving the recursively defined operators Qi(n), in being expressed in terms of quantities directly related to the unperturbed Hamiltonian and the perturbation, and in the method of derivation. (An explicit representation of Hirschfelder's Qi(n) is given in Appendix C.)