The Classical Limit of an Atom

Z. D. Gaeta

Ángel S. Sanz, F. Borondo, Salvador Miret-Artés

The transition to the classical limit in atom-surface diffraction is studied using the de Broglie-Bohm causal formalism. In particular, we focus on rainbow scattering, which is a well-defined effect in classical mechanics and has a clear counterpart in quantum mechanics. In order to achieve this limit, we consider the scattering of particles with increasing masses off a Cu(110) surface. Although the classical limit seems strictly unreachable, quantum trajectories mimic the characteristics of the classical intensity distribution, and their use allows to unveil the mechanism by which the quantum rainbow condition takes place.

A. O. Bolivar

Aleksandra Pitak, Jan Mostowski

Abstract In this work we discuss Bohr’s correspondence principle in a particular physical system, namely the hydrogen atom. It is shown that while diagonal elements of the position operator are equal to zero, off-diagonal elements between two states of different but close energies have a classical limit. The classical limit is reached when both principal quantum numbers, and hence both energies, are large but their difference is small. In this case the matrix elements of the position operator tend to the Fourier component of a classical trajectory, whose parameters are found. The classical limit is proved by finding numerical values of the matrix elements and also by exact treatment of the analytical formulas. This discussion provides a deeper understanding of the correspondence between quantum and classical mechanics and contributes to a better understanding of hydrogen atom wave functions.

Todd A. Oliynyk

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.