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Journal of Physics A Mathematical and General

On the hidden symmetry of a one-dimensional hydrogen atom

Levon S. Davtyan, GEORGE S. POGOSYAN, A. N. Sissakian, V. M. Ter-Antonyan

Published Jul 11, 1987• 30 citations
Publisher

Abstract

The Fock method is applied to the problem of a one-dimensional hydrogen atom. Integral Fock equations are obtained in discrete and continuous spectra; the case of zero energy is studied and wavefunctions and normalisation constants are calculated in the momentum representation.
DOI
10.1088/0305-4470/20/10/020

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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.
We construct the systems of generalised coherent states for the discrete and continuous spectra of the hydrogen atom. These systems are expressed in terms of elementary functions and are invariant under the SO(3,2)SO(3, 2)SO(3,2) (discrete spectrum) and SO(4,1)SO(4, 1)SO(4,1) (continuous spectrum) subgroups of the dynamical symmetry group SO(4,2)SO(4, 2)SO(4,2) of the hydrogen atom. Both systems of coherent states are particular cases of the kernel of integral operator which interwines irreducible representations of the SO(4,2)SO(4, 2)SO(4,2) group.
Fock (1935) explained the degeneracy of the energy levels of the Kepler problem (or hydrogen atom) in terms of the dynamical symmetry group SO(4). He showed that the problem is equivalent to the geodesic flow on the sphere S 3 . The 'hidden' symmetry SO(4) is made manifest. The classical n-dimensional Kepler problem can be better understood by enlarging the phase space of the geodesical motion on S n and including time and energy as canonical variables: a following symplectomorphism transforms the motion on S n in the Kepler problem. The authors prove that the Fock procedure is the implementation at 'quantum' level of the above-mentioned symplectomorphism.