Symmetry of the Two-Dimensional Hydrogen Atom

Abstract

A paradox has arisen from some recent treatments of accidental degeneracy which claim that, for three degrees of freedom, SU(3) should be a universal symmetry group. Such conclusions are in disagreement both with experimentally observed spectra and with the generally accepted solutions of Schrödinger's equation. The discrepancy occurs in the transition between classical and quantum‐mechanical formulations of the problems, and illustrates the care necessary in forming quantum‐mechanical operators from classical expressions. The hydrogen atom in parabolic coordinates in two dimensions, for which the traditional treatment of Fock, extended by Alliluev, requires the symmetry group O(3), is a case for which the newer methods of Fradkin, Mukunda, Dulock, and others require SU(2). Although these groups are only slightly different, SU(2) fails to be the ``universal'' symmetry group on account of the multiple‐valuedness of the parabolic representation. This conclusion extends a result of Han and Stehle: that, for rather similar reasons, SU(2) cannot be the classical symmetry group for the two‐dimensional hydrogen atom.