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Title: Exact supersymmetry in the relativistic hydrogen atom in general dimensions--supercharge and the generalized Johnson-Lippmann operator

Abstract

A Dirac particle in general dimensions moving in a 1/r potential is shown to have an exact supersymmetry, for which the two supercharge operators are obtained in terms of (a D-dimensional generalization of) the Johnson-Lippmann operator, an extension of the Runge-Lenz-Pauli vector that relativistically incorporates spin degrees of freedom. So the extra symmetry (S(2)) in the quantum Kepler problem, which determines the degeneracy of the levels, is so robust as to accommodate the relativistic case in arbitrary dimensions.

Authors:
;  [1]
  1. Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033 (Japan)
Publication Date:
OSTI Identifier:
20768736
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 47; Journal Issue: 3; Other Information: DOI: 10.1063/1.2173173; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ATOMS; DEGREES OF FREEDOM; HYDROGEN; POTENTIALS; RELATIVISTIC RANGE; SPIN; SUPERSYMMETRY; VECTORS

Citation Formats

Katsura, Hosho, and Aoki, Hideo. Exact supersymmetry in the relativistic hydrogen atom in general dimensions--supercharge and the generalized Johnson-Lippmann operator. United States: N. p., 2006. Web. doi:10.1063/1.2173173.
Katsura, Hosho, & Aoki, Hideo. Exact supersymmetry in the relativistic hydrogen atom in general dimensions--supercharge and the generalized Johnson-Lippmann operator. United States. doi:10.1063/1.2173173.
Katsura, Hosho, and Aoki, Hideo. Wed . "Exact supersymmetry in the relativistic hydrogen atom in general dimensions--supercharge and the generalized Johnson-Lippmann operator". United States. doi:10.1063/1.2173173.
@article{osti_20768736,
title = {Exact supersymmetry in the relativistic hydrogen atom in general dimensions--supercharge and the generalized Johnson-Lippmann operator},
author = {Katsura, Hosho and Aoki, Hideo},
abstractNote = {A Dirac particle in general dimensions moving in a 1/r potential is shown to have an exact supersymmetry, for which the two supercharge operators are obtained in terms of (a D-dimensional generalization of) the Johnson-Lippmann operator, an extension of the Runge-Lenz-Pauli vector that relativistically incorporates spin degrees of freedom. So the extra symmetry (S(2)) in the quantum Kepler problem, which determines the degeneracy of the levels, is so robust as to accommodate the relativistic case in arbitrary dimensions.},
doi = {10.1063/1.2173173},
journal = {Journal of Mathematical Physics},
number = 3,
volume = 47,
place = {United States},
year = {Wed Mar 15 00:00:00 EST 2006},
month = {Wed Mar 15 00:00:00 EST 2006}
}
  • The relativistic Kepler problem is discussed, with emphasis on the exact supersymmetry of the problem. It is shown that the supersymmetry is generated by the Johnson-Lippmann operator. Two related operators are found to generate new supersymmetries in an extended function space. Each of these supersymmetries may be disguised as radial supersymmetries. The radial supersymmetries are discussed and it is shown that each of them defines a normal-mode representation of the hydrogen-atom radial functions. Thus, one obtains two different, but equivalent, analytical expressions for these functions. The expressions are well known, but are rederived here in the light of the newmore » understanding. Finally, the nonrelativistic image of the relativistic supersymmetry is constructed and its generators shown to be identical with those recently presented in the literature. 24 refs., 1 fig.« less
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  • In this paper, we work out a most general and the simplest relationship between the energy eigenstates of a [ital d]-dimensional hydrogen atom and those of a [ital D]-dimensional harmonic oscillator in terms of the su(1,1) algebra.
  • We require that there exist some additional symmetry of the Dirac Hamiltonian in an arbitrary central potential which describes the degeneracy of spectrum with respect to the two signs of eigenvalues of famous Dirac's K operator. It is proved that this happens only for the Coulomb potential. In this particular case symmetry operator reduces to the well-known Johnson-Lippmann operator, which plays a role of supercharge in the Dirac-Coulomb problem and generates a Witten's superalgebra.
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