Quantum Treatment of Continuum Electrons in the Fields of Moving Charges

An ab initio, three-dimensional quantum mechanical calculation has been performed for the time-evolution of continuum electrons in the fields of moving charges. Here the essential singularity associated with the diverging phase factor in the continuum wave function is identified and removed analytically. As a result, the continuum components of the regularized wave function are slowly varying with time. Therefore, one can propagate continuum electrons to asymptotically large times and obtain numerically stable, well-converged ejected electron momentum spectra with very low numerical noise. As a consequence, our approach resolves outstanding controversies concerning structures in electron momentum distributions. The main conclusions are general and are illustrated here for ionization of atomic hydrogen by proton impact. Our results show that in order to obtain correct long-time free-particle propagation, the essential singularity identified here should be removed from the continuum components of solutions to the time-dependent Schrodinger equation.