Direct solution of the time-dependent Schroedinger equation for proton-hydrogen collisions in two-dimensional Cartesian space

The wave function for the collision of a proton with a hydrogen atom is obtained by direct solution of the time-dependent Schroedinger equation in a model two-dimensional Cartesian space using lattice techniques. The dimensional reduction from the full three-dimensional (3D) space allows the consideration of huge collision ''volumes'' with minimal computational expense, while still preserving the idea of an impact parameter. The probabilities for charge exchange and ionization are estimated for different projectile energies ranging from 10 to 100 keV/amu. Comparisons with full 3D calculations in smaller collision volumes show that the time-dependent behavior of the probability densities is quite similar for the 2D soft-core model interaction chosen here. The ionization dynamics are further analyzed in regard to the fate of that fraction of the probability density located near the saddle point between projectile and target potentials