Impulse Approximation for One-Dimensional Atom-Oscillator Collisions. I

Abstract

Translational-vibrational energy transfer in collinear collisions of an atom and an harmonic oscillator has been treated in the impulse approximation of Chew. Although transition probabilities for a soft exponential interaction potential compared poorly with exact quantum-mechanical results, for sufficiently small incident masses good results were obtained for a hard impulsive interaction. This is understood in terms of the ratio of the collision time to the oscillator period; the impulse approximation is expected to be valid when this ratio is small. For the smaller incident masses, qualitative agreement with exact results was obtained over a wide energy range for the "hard-sphere" collisions. This represents the first successful approximate treatment of impulsive collisions for this model. In contrast to the familiar perturbative methods, the quality of the approximation does not decrease for multiple-quantum transitions and improves as the energy increases. The one-dimensional impulse-approximation transition probabilities $P_{\mathrm{if}}$ are ill behaved at threshold; however, most computed probabilities remained well behaved to within about $\frac{1}{4}\hslash \omega$ of threshold. The approximation fails to conserve probability. It was found that the sums of probabilities $\Sigma f^{}{P}_{\mathrm{if}}$ for the various initial states $i$ provided reliable measures of confidence in the approximation. For a given $i$, if the probability sum was nearly constant with changing energy, the $P_{\mathrm{if}}$ curves were qualitatively good; on the contrary, the curves were in less good agreement with exact results if the probability sums were less stable with changing energy. When the curves for a given $i$ were qualitatively good, a simple renormalization yielded quantitative agreement with exact results and minimized the effects of threshold misbehavior.