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Journal of Physics A Mathematical and General

On the Fock quantisation of the hydrogen atom

Bruno Cordani

Published Jul 21, 1989• 4 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/22/14/020

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Journal of Mathematical Physics•Sep 1, 1984•5 citations
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.
Quantization of the conformal Kepler problem is defined and studied in order that the quantized system, which will be referred to as a conformal hydrogen atom, may associate the harmonic oscillator with the hydrogen atom. The conformal hydrogen atom shares with the harmonic oscillator the eigenspaces of negative energies. The four-dimensional conformal hydrogen atom reduces to the three-dimensional ordinary hydrogen atom. The symmetry group SO(4) of the hydrogen atom is brought out from the symmetry subgroup of the harmonic oscillator. The conformal hydrogen atom gives an example of those quantum systems of which the configuration spaces are curved Riemannian spaces with nonconstant scalar curvatures and of which the Hamiltonian operators depend on the scalar curvatures.
Fock studied the hydrogen atom problem in momentum space by projecting the space into a 4-dimensional hyperspherical space. He found that as a consequence of the symmetry of the problem in this space the eigenfunctions are the R4 spherical harmonics and that the eigenvalues are determined only by the principal quantum number n. In this paper we note that if his method is applied to the 2-dimensional Kepler problem in momentum space, the eigenfunctions are R3 spherical harmonics, Ylm, and the eigenvalues are determined only by the quantum number l. These facts enable one to give a visualizable geometrical discussion of the dynamical degeneracy.
In the problem of inverse-square force, there exists a hidden SO(4) symmetry, which is manifested as a conserved quantity the Laplace-Runge-Lenz vector in the classical Kepler problem, and as the degeneracy of hydrogen atom energy levels beyond the three-dimensional rotational symmetry in quantum mechanics. This paper presents a more intuitive representation of this hidden symmetry through geometric imagery. Additionally, the paper derives in detail the relevant formulas for the raising and lowering operators corresponding to this symmetry in quantum mechanics.
The problem of degeneracy in quantum mechanics is related to the existence of groups of contact transformations under which the Hamiltonian is invariant. The correspondence between transformations in classical and quantum theories is developed. The Fock-Bargmann treatment of the symmetry group of the hydrogenic atom comes under this theory. The symmetry group of the 2-dimensional Kepler problem is found to be the 3-dimensional rotation group; that of the nnn-dimensional isotropic oscillator is isomorphic to the unimodular unitary group in nnn dimensions. The 2-dimensional anisotropic oscillator has the same symmetry as the isotropic oscillator in classical mechanics, but the quantum-mechanical problem presents complications which leave its symmetry group in doubt.
The aim of this chapter is to present a construction of a set of coherent states for the hydrogen atom proposed by Villegas-Blas (The Laplacian on the n-sphere, the hydrogen atom and the Bargmann space representation. Ph.D. thesis, Mathematics Department, University of Virginia, 1996, [251]), Thomas and Villegas-Blas (Commun Math Phys 187:623–645, 1997, [244]). We show that in a semiclassical sense they concentrate essentially around the Kepler orbits (in configuration space) of the classical motion. A suitable unitary transformation (the Fock operator) maps the pure point subspace of the hydrogen atom Hamitonian onto the Hilbert space L2(S3)L^2({\mathbb S}^3)L2(S3) for the S3\mathbb S^3S3 sphere. We study the coherent states for the S3\mathbb S^3S3 sphere (as introduced by Uribe (J Funct Anal 59:535–556, 1984, [247]) and show that the action of the group SO(4) is irreducible in the space generated by the spherical harmonics of a given degree. Note that coherent states for the hydrogen atom have been extensively studied by J. Klauder and his school. We have chosen not to present them here and refer the interested reader to Klauder and Skagerstam (Coherent states, Worlds Scientific Publishing, Singapore, 1985, [167]).