The discrete variable method for the solution of the time-dependent Schrِdinger equation

A solution of the time-dependent Schrِdinger equation (TDSE) is required in a variety of problems in physics and chemistry. These include atoms and molecules in time dependent electromagnetic fields, a variety of atomic collision problems and in describing the behavior of materials subjected to internal and external forces. In many situations, especially at the nanoscale level, the evolution of the physical system occurs over multiple length and time scales and a computationally efficient approach is a necessity to make progress. Here we explore and compare, for both accuracy and efficiency, a number of methods to discretize and to propagate solutions to the Schrِdinger equation.