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The Journal of Chemical Physics

Semiclassical calculation of bound states of a multidimensional system

Walter Eastes, R. A. Marcus

Published Nov 15, 1974• 176 citations
Publisher

Abstract

The semiclassical quantum conditions for bound states are studied for a nonseparable system of two coupled oscillators which possesses a collection of quasiperiodic trajectories. The semiclassical energy levels are found to be in excellent agreement with the exact quantum levels. This treatment differs from previous work in that periodic trajectories are not needed and in that the potential surface is smooth and nonseparable in all regions.

DOI
10.1063/1.1681734

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The semiclassical quantum conditions for bound states are studied for a nonseparable system of two coupled oscillators which possesses a collection of quasiperiodic trajectories. The semiclassical energy levels are found to be in excellent agreement with the exact quantum levels. This treatment differs from previous work in that periodic trajectories are not needed and in that the potential surface is smooth and nonseparable in all regions.
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.
We present a semiclassical study of the spectrum of a few-body system consisting of two short-range interacting bosonic particles in one dimension, a particular case of a general class of integrable many-body systems where the energy spectrum is given by the solution of algebraic transcendental equations. By an exact mapping between δ-potentials and boundary conditions on the few-body wave functions, we are able to extend previous semiclassical results for single-particle systems with mixed boundary conditions to the two-body problem. The semiclassical approach allows us to derive explicit analytical results for the smooth part of the two-body density of states that are in excellent agreement with numerical calculations. It further enables us to include the effect of bound states in the attractive case. Remarkably, for the particular case of two particles in one dimension, the discrete energy levels obtained through a requantization condition of the smooth density of states are essentially in perfect agreement with the exact ones.
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.
Dynamics based on quantum trajectories with approximate quantum potential is generalized to nonadiabatic systems and its semiclassical properties are discussed. The formulation uses the mixed polar-coordinate space representation of a wave function. The polar part describes the overall time evolution of the wave-function components semiclassically using the single-surface approximate quantum potential. The coordinate part represents a complex``population'' amplitude, which in case of localized coupling can be solved for quantum mechanically in an efficient manner. In the high-energy regime this is accomplished by using a small basis determined by the coupling between surfaces. An illustration is given for a typical curve-crossing problem. The energy-resolved probabilities obtained from the time evolution of two wave packets for a wide range of energies are in excellent agreement with exact results for energies above the threshold of the diabatic reaction, including the case of total nonadiabatic transition.