On a Generalized Oscillator System - Interbasis Expansions

This article deals with a nonrelativistic quantum mechanical study of a dynamical system which generalizes the isotropic harmonic oscillator system in three dimensions. The Schrodinger equation for this generalized oscillator system is separable in spherical, cylindrical, and spheroidal (prolate and oblate) coordinates. The quantum mechanical spectrum of this system is worked out in some details. The problem of interbasis expansions of the wavefunctions is completely solved. The coefficients for the expansion of the cylindrical basis in terms of the spherical basis, and vice-versa, are found to be analytic continuations (to real values of their arguments) of Clebsch-Gordan coefficients for the group SU(2). The interbasis expansion coefficients for the prolate and oblate spheroidal bases in terms of the spherical or the cylindrical bases are shown to satisfy three-term recursion relations. Finally, a connection between the generalized oscillator system (projected on the z-line) and the Morse system (in one dimension) is discussed.