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The Journal of Chemical Physics

Semiclassical treatment of bound state systems: Trajectory calculations using an analytic approximation to the quantum momentum

George Starkschall, John C. Light

Published Sep 15, 1973• 7 citations
Publisher

Abstract

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.

DOI
10.1063/1.1680444

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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.
A complete derivation is provided of the uniform semiclassical approximations to the particle and kinetic energy densities of N noninteracting bounded fermions in one dimension. The employed methodology allows the inclusion of non-perturbative quantum effects, including tunneling and quantum oscillations, via an infinite resummation of the Poisson summation formula. We explore the analytic behavior, physical meaning, and the relationship between the semiclassical uniform approximations for the fermionic kinetic energy and particle densities.
The single particle reduced density matrices for fermions in a linear potential in one and three dimensions are derived. An idempotent reduced density matrix for a nonuniform field with a single turning point is obtained. The density expression contains the contributions from the direct path and the path with one turning point. When applied to a potential well the corrections from the long-time paths are taken from the theory of Light and Yuan. Only classical quantities of momentum, time, and action enter in the final expression. The density distribution for the nonclassical region is more accurate than that given by the formula of Light and Yuan.
A new approach has been used to determine directly the reduced single particle density matrix for noninteracting fermions in a one-dimensional potential. A time evolution operator closely related to the classical limit of the Feynman path integral propagator is derived which has superior behavior in turning point and nonclassical regions. This is used to determine the reduced density matrix in terms of the phase and time of oscillation at the Fermi energy. The density so obtained has the correct quantum oscillations and nonclassical tails and is a distinct improvement over the Thomas-Fermi density. Two simple examples are presented and implications for electronic structure calculations are discussed.
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.