On the Numerical Solution of the Time-Dependent Schrodinger Equation with Time-Dependent Potentials

We discuss some procedures to improve the accuracy of the numerical solution of the time-dependent Schrodinger equation with a time-dependent potential. In particular, a spatial discretization by an exponentially fitted Numerov formula leading to a higher order approximation of the second spatial derivative in the Hamiltonian (compared to the standard finite difference) along with a better description of oscillating or exponential behavior, a propagation scheme of order 4 with respect to the time step, based on the Magnus expansion and artificial boundary conditions aimed to reduce the reflections of the wave packet at the numerical boundaries are presented. The procedures are illustrated on a test equation with analytic solution.