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The Journal of Chemical Physics

Bound state semiclassical wave functions

Stephen K. Knudson, J. B. Delos, D. W. Noid

Published Jun 15, 1986• 33 citations
Publisher

Abstract

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.

DOI
10.1063/1.450693

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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.
The wave function for a state of definite angular momentum of the three-dimensional isotropic harmonic oscillator is expressed exactly in terms of the corresponding classical trajectories. In particular, the three-dimensional wave function as well as the radial wave function and the spherical harmonics are obtained as integrals over quantities determined entirely by the classical motion. The expression for the harmonic-oscillator radial wave function is shown also to yield the exact radial wave function for the free particle. The expressions are cast in forms suitable for use as uniform semiclassical approximations for wave functions of other systems. Numerical examples confirm that such wave functions obey boundary conditions appropriate for spherical coordinates and that they are free of caustic singularities. The wave functions obtained by this technique can be quite accurate even for low-energy states where semiclassical approximations are expected to be poor.
In this Master's thesis, we give a pedagogical introduction to the semiclassical analysis of wave dynamics in single-band and multi-band systems. We show how the semiclassical evolutions of Wigner functions can approximate the dynamics of wave packet centers and wave packet deformations. Three systems of increasing complexity are considered: the quantum harmonic oscillator and the quantum anharmonic oscillator as single-band systems, and the Dirac oscillator as a two-band system. We explain the origins of the first-order semiclassical corrections (Moyal, Littlejohn-Flynn, and Berry) in the equations of motion and examine their effects on predictive accuracy. Another focus is set on the conditions for validity of the semiclassical analysis. For both the quantum anharmonic and the Dirac oscillators, a dimensionless parameter controlling the accuracy of the approximation is identified. In multi-band systems, the formalism applies only to states within a single adiabatic subspace, which, for the Dirac oscillator, corresponds to an energy branch. This motivates the construction of single-branch almost coherent states: approximately Gaussian wave packets formed by superpositions of states from a single energy branch. Their centers undergo oscillations with an aperiodic envelope. These states enable comparison between semiclassical Wigner function evolution and the commonly used Sundaram-Niu formalism. We show that semiclassical Wigner function evolution captures deformation-related dynamics, including the initial decay of oscillations, which the Sundaram-Niu formalism cannot describe by construction. Additionally, exact quantum evolution reveals partial relocalization of these single-branch almost coherent states, a phenomenon that cannot be captured by the semiclassical evolution.
Using an analytic approximation to the quantum momentum previously derived to calculate single particle reduced density matrices in the semiclassical approximation, we have been able to construct equations of motion with which to perform trajectory calculations. Information calculated along these trajectories is used to calculate single particle reduced density matrices for one-dimensional systems. Charge densities for N fermions in a harmonic oscillator potential calculated using this quantum momentum approximation are in excellent agreement with the exact quantum mechanical calculations.