Degenerate RS perturbation theory

A concise, systematic procedure is given for determining the Rayleigh-Schrödinger energies and wavefunctions of degenerate states to arbitrarily high orders even when the degeneracies of the various states are resolved in arbitrary orders. The procedure is expressed in terms of an iterative cycle in which the energy through the (2n + 1)th order is expressed in terms of the partially determined wavefunction through the n th order. Both a direct and an operator derivation are given. The two approaches are equivalent and can be transcribed into each other. The direct approach deals with the wavefunctions (without the use of formal operators) and has the advantage that it resembles the usual treatment of nondegenerate perturbations and maintains close contact with the basic physics. In the operator approach, the wavefunctions are expressed in terms of infinite order operators which are determined by the successive resolution of the space of the zeroth-order functions. The operator treatment has some similarity to that of Choi (1969), but it is more closely related to that of either Hirschfelder (1969) or Silverstone-Holloway (1971). The operator expressions are useful for: double perturbations, expectation values of physical properties, and problems involving finite dimensional Hilbert space (for example, wavefunctions approximated by linear basis sets). The use of variational principles for degenerate perturbation problems is discussed.