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Physical Review A

Time-independent variational approach to inelastic collisions of a particle with a harmonic oscillator

Yasutami Takada

Published Jul 1, 1988• 2 citations
Publisher

Abstract

A variational approach is applied to collisions of a particle having a mass m with a harmonic oscillator having a mass M in one dimension. We have used trial functions which become exact in the limit of either m/M\ensuremath{\gg}1 or m/M\ensuremath{\ll}1. We have derived exact expressions in the limit m/M\ensuremath{\rightarrow}0 for inelastic as well as elastic collisions to leading order in the parameter m/M for any impenetrable interaction potential.

DOI
10.1103/physreva.38.98

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A variational approach is applied to collisions of a particle having a mass m with a harmonic oscillator having a mass M in one dimension. We have used trial functions which become exact in the limit of either m/M\ensuremath{\gg}1 or m/M\ensuremath{\ll}1. We have derived exact expressions in the limit m/M\ensuremath{\rightarrow}0 for inelastic as well as elastic collisions to leading order in the parameter m/M for any impenetrable interaction potential.
Variational calculations play an important role in quantum mechanics, particularly in determining the ground-state energy of physical systems. Didactic examples in which one can see how the exact energy and wave function are approached by using a family of trial functions are not very numerous. An example was given in a recent book [M. Moshinsky, The Harmonic Oscillator in Modern Physics: From Atoms to Quarks (Gordon and Breach, New York, 1969)] which discusses the ground state of the hydrogen atom using as trial wave function a linear combination of harmonic-oscillator states. In the present paper we carry out a similar analysis for the ground state of a particle of mass m in a three-dimensional square-well potential. We discuss not only the approach to the exact energy when we vary the frequency and the number of oscillator states but also analyze the overlap of the exact and variational wave functions and compare the exact and approximate form factors.
A density matrix evolution (DME) method (Berendsen, H. J. C., and Mavri, J., 1993, J. phys. Chem., 97, 13464) to simulate the dynamics of quantum systems embedded in a classical environment is applied to study the inelastic collisions of a classical particle with a five level quantum harmonic oscillator. The results are compared with the exact quantum calculations of Secrest, D., and Johnson, B. R., 1966, J. chem. Phys., 45, 4556, and with the results of classical mechanics. The DME results are between the results of the full quantum treatment and classical mechanics but closer to the latter. Furthermore, we demonstrate the time reversibility of the DME equations of motion and discuss the importance for the time reversibility of the quantum phase described by the off-diagonal elements of the density matrix.