Contribution to the theory of accidental degeneracies

Abstract

An attempt is made to throw light on the problem of degeneracies which cannot be simply attributed to self-evident symmetry properties of a physical system. The present investigation is based on a Hamiltonian which contains two independently continuously variable parameters which enter in a linear way when the operator form of the energy is constructed. The Hamiltonian may therefore contain as many as two independent terms which must commute plus a perturbation ζH'. Emphasis is placed on the question of conditions whereby degeneracies in the unperturbed problem are removed for any perturbation over at least a finite range of ζ. Such points, acnodes in parameter space, are found to occur, in general, only for ζ=0. The fact that such situations arise for uncoupled levels is clearly nontrivial. From a study of the nature of the degeneracies, as they depend on the parameters, between uncoupled states in cases of special dimensionality of the set of coupled states, one can conclude that acnodes are not often encountered and usually, a degeneracy in the unperturbed case will persist in the sense that it will occur for a continuous manifold of points in parameter space. It is pointed out that degeneracies may exist which are independent of the values of the parameters. These so-called intrinsic degeneracies are discussed in an appendix.