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Acta Physica Sinica

ANALYSIS OF WAVE FUNCTIONS FOR HYDROGEN ATOMS IN CLASSICAL LIMIT

HUANG XIANG-YOU, 北京大学物理系,北京,100871;中国高等科学技术中心(世界实验室)

Published Jan 1, 1991• 1 citation
Publisher

Abstract

Basing on the classical trajectory motion for an electron circling an atomic nucleus and averaging the motion over initial coordinate parameters of ellipse trajectories for the electron, we get a density distribution in space. This distribution is consistent with probability density of stationary state for hydrogen atom in the classical limit. This comparison shows that a wave function of hydrogen atom in stationary state does not describe an individual atom but an ensemble.

DOI
10.7498/aps.40.1553

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Basing on the classical trajectory motion for an electron circling an atomic nucleus and averaging the motion over initial coordinate parameters of ellipse trajectories for the electron, we get a density distribution in space. This distribution is consistent with probability density of stationary state for hydrogen atom in the classical limit. This comparison shows that a wave function of hydrogen atom in stationary state does not describe an individual atom but an ensemble.
The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.
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.