Efficient explicit numerical solutions of the time-dependent Schrödinger equation

Explicit numerical solutions of the time-dependent Schrödinger equation are more efficient than those obtained by commonly used implicit approaches. They are more practical, especially for a system with higher spatial dimensions. To that end, we introduce a generalization of an explicit three-level method to obtain solutions with spatial and temporal errors of the order of O[(Δx)^{2r}] and O[(Δt)^{2M+3}], where Δx and Δt are the spatial and temporal grid elements, and r and M are positive integers. Sample calculations illustrate the efficacy and stability of the algorithm.