Three-Dimensional Isotropic Harmonic Oscillator and SU3

Abstract

A consideration of the eigenvalue problem for the quantum-mechanical three-dimensional isotropic harmonic oscillator leads to a derivation of a conserved symmetric tensor operator, in addition to the angular momentum vector operator. Upon examining the classical limit, it is found that the symmetric tensor completely specifies the orientation of the elliptical orbit, in a way analogous to the Runge-Lenz vector for the Kepler problem. The tensor operator and the angular momentum vector operator are then related to the infinitesimal operators for the SU3 group.