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Physics Letters A

A semiclassical approximation for the Wigner distribution function. Application to planar channeling

V.G. Novitzky

Published Apr 1, 1983• 2 citations
Publisher
DOI
10.1016/0375-9601(83)90848-4

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A new semiclassical theory is formulated by introducing a new distribution function to be propagated in phase space. Primarily this theory is intended to describe molecular collisions such as reactive scattering, but is also applicable to intramolecular processes including bound state problems. This new distribution function is a kind of characteristic function in the terminology of the phase-space-quantum theory developed by Weyl, Wigner, Moyal and others, and is called a dynamical characteristic function (DCF). The equation of motion and some properties of the DCF are presented. The essential feature of our theory is that the semiclassical DCF can be propagated using classical trajectories. This distribution function carried information not only on the density, but also on the phase (action integral). In this sense this theory can be regarded as a kind of phase-space-path-integral formalism. The semiclassical approximation is analyzed from the viewpoint of path integrals. The Wigner distribution function is also discussed in the context of our analysis. As a theoretical application, the semiclassical DCF is shown to reproduce the expression of the periodic orbit theory for bound states. Also, as a simple numerical application, a propagation of the DCF on a nonquadratic (Morse) potential is presented.

The Journal of Chemical Physics•Mar 15, 1985•14 citations
The mapping of the Wigner distribution function (WDF) for a given bound state onto a semiclassical distribution function (SDF) satisfying the Liouville equation introduced previously by us is applied to the ground state of the Morse oscillator. The purpose of the present work is to obtain values of the potential parameters represented by the number of levels in the case of the Morse oscillator, for which the SDF becomes a faithful approximation of the corresponding WDF. We find that for a Morse oscillator with one level only, the agreement between the WDF and the mapped SDF is very poor but for a Morse oscillator of ten levels it becomes satisfactory. We also discuss the limit h --> 0 for fixed potential parameters.
A new semiclassical theory is formulated by introducing a new distribution function to be propagated in phase space. Primarily this theory is intended to describe molecular collisions such as reactive scattering, but is also applicable to intramolecular processes including bound state problems. This new distribution function is a kind of characteristic function in the terminology of the phase-space-quantum theory developed by Weyl, Wigner, Moyal and others, and is called a dynamical characteristic function (DCF). The equation of motion and some properties of the DCF are presented. The essential feature of our theory is that the semiclassical DCF can be propagated using classical trajectories. This distribution function carried information not only on the density, but also on the phase (action integral). In this sense this theory can be regarded as a kind of phase-space-path-integral formalism. The semiclassical approximation is analyzed from the viewpoint of path integrals. The Wigner distribution function is also discussed in the context of our analysis. As a theoretical application, the semiclassical DCF is shown to reproduce the expression of the periodic orbit theory for bound states. Also, as a simple numerical application, a propagation of the DCF on a nonquadratic (Morse) potential is presented.