The classical limits for two sequences of probability distributions for hydrogenic atoms with constant energy are derived. The corresponding classical ensembles are described.
In an earlier paper appropriate limiting procedure is discussed in a general way for quantum mechanical energy eigenfunctions (one-dimensional bound states) — a single interpretational postulate leading smoothly to entire compatible classical objective description without facing any contradiction. The method is consistently extended to two- and three-dimensional cases and it is interesting to note that results of earlier study on the classical limit of the radial distribution function of hydrogen atom are easily obtained as special cases of our analysis.
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.