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 $\frac{1}{4}\ensuremath{\hbar}\ensuremath{\omega}$ of threshold. The approximation fails to conserve probability. It was found that the sums of probabilities $\ensuremath{\Sigma}{f}^{}{P}_{\mathrm{if}}$ 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.