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Acta Physica Sinica

KEPLER ORBITS FOLLOWED BY HYDROGEN ATOMS IN COHERENT STATES

Xu Bo-wei, GU WEI-HUA, 上海交通大学应用物理系,上海200030

Published Jan 1, 1993• 2 citations
Publisher

Abstract

Transforming the Coulomb problem to harmonic osillator, we construct nonspreading coherent states for the 2D and 3D hydrogen atom and show that the expectation values of the observables in Q. M. follow the classical Kepler orbits in physical space. The use of a fictitious time variable in the associated auxiliary osillator problem plays a crucial role.

DOI
10.7498/aps.42.1050

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Journal of Physics A Mathematical and General•Apr 21, 1989•14 citations
Transforming the Coulomb problem to harmonic osillator, we construct nonspreading coherent states for the 2D and 3D hydrogen atom and show that the expectation values of the observables in Q. M. follow the classical Kepler orbits in physical space. The use of a fictitious time variable in the associated auxiliary osillator problem plays a crucial role.
The Kustaanheino-Stiefel transformation is used to transform the Coulomb problem to a four-dimensional isotropic harmonic oscillator with a constraint. Ordinary coherent states are introduced over this oscillator and are shown to evolve in a fictitious time. When projected back into the physical space, the states follow the classical Kepler orbits in physical time. The resulting quasiclassical picture is obtained without the Bohr correspondence principle of taking only contributions from high principal quantum numbers.
Using the harmonic oscillator representation of the hydrogen atom and constructing the appropriate coherent state corresponding to the minimum uncertainty product, the classical limits of the two-dimensional and the three-dimensional hydrogen atom are examined. The method adopted is similar to that used by Bhaumik, Dutta Roy and Ghosh (1986). The authors deduce the classical limit by requiring that the expectation value (r) of the radial variable is large for the two-dimensional hydrogen atom. The resulting trajectory is an ellipse satisfying the conditions of the Kepler orbit in classical mechanics. In order to obtain the classical limit of the three-dimensional hydrogen atom, besides the condition of large (r), they make the uncertainty Delta r=((r 2 )-(r) 2 ) 1/2 a minimum with respect to certain parameters of the coherent state. Imposition of only the first condition leads to an elliptical orbit, which is not a Kepler orbit in general. Imposition of both conditions leads to an orbit identical with that of the two-dimensional hydrogen atom. However, the time evolution of the expectation values of position variables are consistent with the corresponding results of the Kepler problem only for small values of eccentricity, but not in general.