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International Journal of Quantum Chemistry

Some remarks on the hydrogen atom

Horacio Grinberg, Julio Marañón, H. Vucetich

Published Feb 1, 1983• 9 citations
Publisher

Abstract

Abstract The path integral for the Green's function involving the Coulomb potential in combination with the Kustaanheimo‐Stiefel transformation is used to generate the atomic orbitals of the nonrelativistic hydrogen atom as various combinations of the product of one‐dimensional isotropic harmonic oscillator wave functions. The use of the transformation is justified, by connecting the homogeneous space with the quotient space in the Feynman quantization formalism.

DOI
10.1002/qua.560230206

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Journal of Physics A Mathematical and General•May 7, 1990•7 citations
Abstract The path integral for the Green's function involving the Coulomb potential in combination with the Kustaanheimo‐Stiefel transformation is used to generate the atomic orbitals of the nonrelativistic hydrogen atom as various combinations of the product of one‐dimensional isotropic harmonic oscillator wave functions. The use of the transformation is justified, by connecting the homogeneous space with the quotient space in the Feynman quantization formalism.
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.
The path integral evaluation of the hydrogen atom by Duru and Kleinert is studied by recognizing it as a special case of the general treatment of the separable Hamiltonian of Liouville-type. The basic dynamical principle involved is identified as the Jacobi’s principle of least action for given energy which is reparametrization invariant, and thus the appearance of a gauge freedom is naturally understood. Unlike the conventional Feynman path integral,which deals with a space-time picture of particle motion, the path integral on the basis of the Jacobi’s principle sums over orbits in space. We illustrate these properties by evaluating an exact path integral of the hydrogen atom in parabolic coordinates. In 3-dimensions the parabolic coordinates do not quite lead to a Liouville-type Hamiltonian, but we can devise asimple trick which converts it to a Liouville-type Hamiltonian and thus avoiding the use of the Kustaanheimo-Stiefel transformation. In the present formulation, the Hamiltonian for Stark effect is converted to the one for anharmonic oscillators with an unstable quartic coupling. We also study the hydrogen atom path integral from a view point of one-dimensional quantum gravity coupled to matter fields representing the electron coordinates. A simple BRST analysis of the problem is presented. 1 1
The authors show how explicit use of the constraint equation that arises when the Kustaanheimo-Stiefel transformation (1965) is applied to the hydrogen atom in R 3 enables them to determine the wavefunctions for the constrained harmonic oscillator in R 4 with the correct restriction on the quantum numbers in a direct and natural manner. They then highlight and resolve a discrepancy between the R 4 oscillator wavefunctions produced by different authors by considering the measure to be used in R 4 and the operator ordering in the quantised Hamiltonian.