Quantum-Mechanical Calculation of Harmonic Oscillator Transition Probabilities in a One-Dimensional Impulsive Collision

Abstract

Quantum-mechanical vibrational transition probabilities Pi→f(ε) for harmonic oscillators, undergoing impulsive hard sphere collisions along the line of centers with an incident atom with relative kinetic energy ε, have been computed by a machine (IBM-704) solution of the relevant Schrödinger equation. Curves for Pi→f(ε) over a range of ε are presented for initial (i) and final (f) vibrational oscillator states i, f=0, 1, 2, and 3. It is shown that this model of an inelastic collision gives rise to appreciable vibrational transitions v(i)→v(f) with | Δv |>1 (in addition to | Δv |=1) in contrast to the Landau-Teller-Herzfeld adiabatic, first-order perturbation treatment which permits only transitions with | Δv |=1. This result is discussed in relation to the dissociation of diatomic molecules and to the adsorption of atoms on solids. Averaged transition probabilities P̄i→f(T) are computed for an incident beam of particles with a Maxwellian velocity distribution. It is pointed out that such averaged transition probabilities may give a misleading impression of the efficiency of translational-vibrational energy transfer if the Pi→f(ε) show a resonance type of behavior, i.e., a strong dependence of Pi→f(ε) on ε over a small interval of ε.