Efficient Finite Difference Solutions to the Time-Dependent Schrödinger Equation

The matrix elements of the exponential of a finite difference realization of the one-dimensional Laplacian are found exactly. This matrix is used to formulate an efficient algorithm for the numerical solution to the time-dependent quantum mechanical scattering of a single particle from a time-independent potential in one-space and one-time dimension. The method generalizes to higher spatial dimensions, as well as to multiparticle problems.