On the Degeneracy of the Two-Dimensional Harmonic Oscillator

Abstract

A further study of the degeneracy of the two dimensional harmonic oscillator is made, both in the isotropic and anisotropic cases. By regarding the Hamiltonian as a linear operator acting through the Poisson bracket on functions of the coordinates and momenta, a method applicable generally to bilinear Hamiltonians, it is shown how all possible rational constants of the motion may be generated. They are formed from eigenfunctions of the Hamiltonian which are linear combinations of the coordinates and momenta, and which belong to negative pairs of eigenvalues. Canonical coordinates, which may be visualized geometrically for the isotropic oscillator in terms of the Hopf mapping, place the symmetry group responsible for the accidental degeneracy clearly in evidence. Surprisingly, one finds that the unitary unimodular group SU2, is the symmetry group in all cases, even including that of an anisotropic oscillator with incommensurable frequencies. The lack of a quantum-mechanical analogy in the latter case is due to a lack of the necessary transcendental roots of the operators involved in attempting to use the correspondence principle, rather than to the lack of a symmetry group for the classical problem.