The linearly driven parametric oscillator: Application to collisional energy transfer

Abstract

The time-evolution operator is explicitly constructed for a general linearly driven parametric quantum oscillator, equivalent to a harmonic oscillator driven by linear plus quadratic potentials. The method is based on an algebra of operators which are bilinear in the position and momentum operators, and form a closed set with respect to commutation. The obtained result requires only integrals over time and the solution of two coupled first order linear differential equations related to the classical equations of motion. The model is used to obtain vibration-translation probabilities in a collinear collision of an atom with a diatomic molecule. Numerical calculations have been performed for systems with several mass combinations and potential parameters. Approximation methods are compared, and criteria are established to determine when it is necessary to go beyond the popular linearly driven harmonic oscillator.