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Journal of Physics A Mathematical and General

Classical limit of the hydrogen atom

Debajyoti Bhaumik, Binayak Dutta-Roy, Gautam Kumar Ghosh

Published Jun 1, 1986• 40 citations
Publisher

Abstract

A wave packet which travels on an elliptic trajectory is constructed for the hydrogen atom. This is achieved by mapping the Schrodinger equation for the hydrogen atom into the equation for a four-dimensional oscillator with a constraint. A set of coherent states for the constrained oscillator are then shown to have at high average energy the classical limit properties as obtained for planetary motion, to a good approximation.

DOI
10.1088/0305-4470/19/8/017

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Acta Physica Sinica•Jan 1, 1991•1 citation
A wave packet which travels on an elliptic trajectory is constructed for the hydrogen atom. This is achieved by mapping the Schrodinger equation for the hydrogen atom into the equation for a four-dimensional oscillator with a constraint. A set of coherent states for the constrained oscillator are then shown to have at high average energy the classical limit properties as obtained for planetary motion, to a good approximation.
We explore the classical limit of the hydrogen atom by constructing a minimum-uncertainty wave packet that travels along a Kepler orbit. The dynamics of the wave packet display both classical and quantum-mechanical properties. Over a limited period of time, during which the dynamics of the wave packet may be considered linear, the motion of the wave packet can be described by classical equations of motion. After this time, the quantum dynamics, whose most prominent features are decays and revivals of the wave packet, becomes dominant. We discuss decays and revivals of the wave packet in detail.
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.
We construct wave packets for the hydrogen atom labeled by the classical action-angle variables with the following properties. (i) The time evolution is exactly given by classical evolution of the angle variables. (The angle variable corresponding to the position on the orbit is now noncompact and we do not get exactly the same state after one period; however, the gross features do not change. In particular, the wave packet remains peaked around the labels.) (ii) The resolution of the identity using this overcomplete set involves exactly the classical phase-space measure. (iii) The semiclassical limit is related to Bohr-Sommerfeld quantization. (iv) They are almost minimum uncertainty wave packets in position and momentum.