One-Dimensional Inelastic Collisions with Impulsive Interactions

Abstract

A linear oscillator interacting impulsively with a free particle incident along its line of oscillation will in general suffer a change in state; and this model may be taken to represent an inelastic molecular collision. It is pointed out that this picture is substantially equivalent to that used by Castellan and Hulburt, with one modification. The problem is formulated for both quantum and classical mechanics. Explicit transition probabilities are calculated for two extreme cases: (i) the mass and momentum of the incident particle are small, while its energy is fixed; (ii) the mass of the incident particle is small and its energy is large, while its momentum is fixed. The quantum form of approximation (i) is essentially the approximate theory of Castellan and Hulburt. In the present paper the emphasis is on approximation (ii), for which it is shown that in both the classical and quantum theories the average energy transferred to an oscillator of mass m by an incident particle of momentum p is 2p2/m, whatever the law of force of the oscillator and whatever its initial state. It is also shown in this approximation that the classical transition probability for the harmonic oscillator may be recast in a pseudoquantum form so as to allow direct comparison with the quantum mechanical transition probability over the whole range of momenta of the incident particle.