The Schrödinger equation for the two-dimensional simple harmonic oscillator is solved using elliptic coordinates where it is separable. The separability of the problem in such coordinates is independent of the selection of the focal distance. The solutions are labeled by the total number of quanta N and by a set of characteristic values b corresponding to the eigenvalues of an observable B^, which does not commute with L^, the total angular momentum or H${^}_x$, the energy associated with the x degree of freedom. The well-known quantum energies as well as the characteristic values are obtained by imposing physical polynomial solutions.

• Received 28 November 1988

DOI:https://doi.org/10.1103/PhysRevA.40.4215