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On an exact path integral of the hydrogen atom

Kazuo Fujikawa

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 1The 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

Feb 15, 19960 citations

Path-integral solution for pseudoharmonic potential

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We obtained the path-integral solution of a three-dimensional pseudoharmonic potential by applying a suitably chosen point canonical transformation.Received 28 August 1987DOI:https://doi.org/10.1103/PhysRevA.37.2687©1988 American Physical Society

Physical Review AApr 1, 198827 citations

Exact path integral treatment of the hydrogen atom

Frank Steiner

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Alternative exact-path-integral treatment of the hydrogen atom

Akira Inomata

Physics Letters AApr 1, 198464 citations

Confluence of Bound-Free Coherences in Laser-Induced Autoionization

Kazimierz Rza̧żewski, Joseph H. Eberly

We solve exactly the problem of double configuration mixing (Coulombic and radiative) in a "Fano model" atom, and predict a new confluence of coherences in laser-induced autoionized electron spectra.Received 6 April 1981DOI:https://doi.org/10.1103/PhysRevLett.47.408©1981 American Physical Society

Physical Review LettersAug 10, 1981233 citations

Path integrals for hydrogen atom

Nikom Choosiri, Virulh Sa-yakanit

Jan 1, 19830 citations

Exact-Path-Integral Treatment of the Hydrogen Atom

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