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Physical Review A

Hydrogen atom in the phase-space formulation of quantum mechanics

José M. Gracia-Bondı́a

Published Aug 1, 1984• 59 citations
Publisher

Abstract

Using a coordinate transformation which regularizes the classical Kepler problem, we show that the hydrogen-atom case may be analytically solved via the phase-space formulation of nonrelativistic quantum mechanics. The problem is essentially reduced to that of a four-dimensional oscillator whose treatment in the phase-space formulation is developed. Furthermore, the method allows us to calculate the Green's function for the H atom in a surprisingly simple way.

DOI
10.1103/physreva.30.691

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Journal of Mathematical Physics•Jun 1, 1982•28 citations
Using a coordinate transformation which regularizes the classical Kepler problem, we show that the hydrogen-atom case may be analytically solved via the phase-space formulation of nonrelativistic quantum mechanics. The problem is essentially reduced to that of a four-dimensional oscillator whose treatment in the phase-space formulation is developed. Furthermore, the method allows us to calculate the Green's function for the H atom in a surprisingly simple way.
The Hamiltonian of the three-dimensional hydrogen atom is reduced, in parabolic coordinates, to the Hamiltonians of two bidimensional harmonic oscillators, by doing several space-time transformations,separating the movement along the three parabolic directions (ξ,η,φ), and introducing two auxiliary angular variables ψ and ψ′, 0≤ψ, ψ′≤2π. The Green’s function is developed into partial Green’s functions, and expressed in terms of two Green’s functions that describe the movements along both the ξ and η axes. Introducing auxiliary Hamiltonians allows one to calculate the Green’s function in the configurational space, via the phase-space evolution function of the two-dimensional harmonic oscillator. The auxiliary variables ψ and ψ′ are eliminated by projection. The thus-obtained Green’s function, save for a multiplicating factor, coincides with that calculated following the path-integral formalism.
A coordinate transformation which exhibits the rotational invariance of the hydrogen atom in four-dimensional Hilbert space is introduced. The coordinates are shown to be directly related to the spherical polar and parabolic coordinates in position space. With the use of the transformation, the Schr\"odinger equation for the hydrogen atom left-multiplied by 4r4r4r is transformed into one for a four-dimensional harmonic oscillator. Solutions are obtained and related to the hydrogenic wave functions. Group-theoretical implications of the transformation and its application to the hydrogen Stark problem are briefly discussed.
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.