Classical Limit of a Hydrogen Atom

Sergey A. Rashkovskiy

Tasko P. Grozdanov, Е. А. Соловьев

Andrea Chaney, James M. Espinosa, James R. Woodyard

S Nandi, C. S. Shastry

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.

Michael Nauenberg, Carlos R. Stroud, John A. Yeazell

Debajyoti Bhaumik, Binayak Dutta-Roy, Gautam Kumar Ghosh

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.