Canonical transformation and accidental degeneracy. III. A unified approach to the problem

Abstract

We continue the discussion of the groups of canonical transformations responsible for accidental degeneracy in quantum mechanical problems. A general unified treatment is provided for a wide class of two−dimensional physical systems, having an energy spectrum which is a linear combination of two quantum numbers. The general method involves the use of both nonorthonormal and orthonormal sets of states to construct groups of complex or real canonical transformations, mapping the problem under consideration onto the two−dimensional isotropic harmonic oscillator. The group responsible for the accidental degeneracy is then quite obviously SU (2). The problem of an isotropic oscillator in a sector π/q (q integer) was discussed previously using a nonorthonormal basis. In the present paper we carry the analysis in an orthonormal basis to establish the general procedure mentioned above. We also analyze in detail the Calogero problem for three particles which has a spectrum of the type given above, and obtain explicitly the canonical transformation that maps it on the anisotropic oscillator whose ratio of frequencies is 2/3 and subsequently on the isotropic one.