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The Journal of Chemical Physics

Semiclassical calculation of quantum-mechanical wave functions for a two-dimensional scattering system

Stephen K. Knudson, J. B. Delos, B. Bloom

Published Dec 1, 1985• 36 citations
Publisher

Abstract

The semiclassical theory developed by Maslov and Fedoriuk is used to calculate the wave function for two-dimensional scattering from a Morse potential. The characteristic function S and the density Jacobian J are computed in order to obtain the primitive wave function. The incident part shows distorted plane-wave behavior and the scattered part shows radially outgoing behavior. A uniform approximation gives a wave function that is well-behaved near the caustic.

DOI
10.1063/1.449645

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Physical Review Letters•Mar 18, 1996•42 citations
The semiclassical theory developed by Maslov and Fedoriuk is used to calculate the wave function for two-dimensional scattering from a Morse potential. The characteristic function S and the density Jacobian J are computed in order to obtain the primitive wave function. The incident part shows distorted plane-wave behavior and the scattered part shows radially outgoing behavior. A uniform approximation gives a wave function that is well-behaved near the caustic.
The semiclassical theory developed by Maslov and Fedoriuk is used to calculate the wave function for a two-dimensional bound state system. We investigate in detail an eigenstate of a coupled anharmonic oscillator system. The primitive semiclassical wave function is obtained from the characteristic function S and the density function J. Each of these functions consists of four branches corresponding to the four possible directions of motion of the classical trajectory through any point. The interference from the four branches determines the basic structure of the wave function. A uniform approximation gives a wave function which is well behaved along each caustic and which is in good agreement with the fully quantal wave function.
A new technique is developed to generate semiclassical wave functions. The method uses only information already available from a standard semiclassical quantization of a system. Linear superpositions of Gaussian coherent states that lie along quantizing classical trajectories are used, with phases given by the action integrals plus a Maslov-type correction. Wave functions generated in this way suffer from none of the problems with caustics that primitive semiclassical wave functions encounter. The semiclassical wave functions are convenient for subsequent use in applications, e.g., molecular spectra. By generating wave functions for several simple systems, we show that under most circumstances these wave functions are very accurate approximations to the true quantum states.
Abstract The Maslov correction to the wave function is the jump of (−π2) \left( { - \frac{\pi } {2}} \right) (−2π​) in the phase when the system passes through a caustic. This can be explained by studying the second variation and the geometry of paths, as conveniently seen in Feynman’s path integral framework. The results can be extended to any system using the semiclassical approximation. The 1-dimensional harmonic oscillator is used to illustrate the different derivations reviewed here.
Initial-value phase-space integral representations for a time-dependent propagator are obtained in the coordinate and momentum representations. To do so we first derive nonuniform semiclassical propagators for the various representations, obtaining the global-time semiclassical phase indices (Maslov indices) due to caustics. Results include readily implementable general phase index formulas for any type of caustic, including cases where the Morse index theorem is inapplicable. The method of obtaining the indices is general and based simply on concatenating uniform short-time propagators which also gives rise to alternative path-integral forms. Initial-value integral representations are then derived by introducing a method of extending short-time initial-value propagator formulas to global times via a simple stationary-phase asympotoic-equivalence approach. The integrals reduce to the nonuniform semiclassical propagators within the stationary-phase approximation, are uniform about caustics, and have integrand phases which properly account for the global-time phases in terms of appropriate Maslov indices. The initial-value integrals are also consistently derived via a canonical mapping procedure on the coordinate-space path integral. Initial-value integrals for time-dependent wave-function propgation are also given. Evaluation of the initial-value integral expressions do not require trajectory root searches for propagation.