# Fourier transform method for the classical trajectory problem

## Abstract

By introducing an impulse potential at an external boundary, or by reflecting the scattering potential at such a boundary, collision dynamics is converted into an equivalent bound-state problem. Recent theoretical developments in classical bound-state dynamics, which make use of the geometrical structure in phase space known as an invariant toroid, can then be applied to collision problems. Fast Fourier transform techniques are used to construct the invariant toroid, which is the locus of an indefinitely extended nonperiodic classical trajectory. This approach to collision dynamics is applied to the collinear atom-oscillator problem modelling He-H2 vibrational excitation. Results are compared to previous classical and quantum dynamical studies. The proposed method obtains in a single calculation the complete ensemble of trajectories that correspond to a semiclassical wave function. At given total energy and for given global action integral values all trajectories are parallel straight lines when expressed in the coordinate space of true angle variables obtained by the Fourier transform calculation. These aspects of the formalism reduce the computation of inelastic transition probabilities and action integrals to simple geometric exercises. © 1979 American Institute of Physics.