Abstract
A new approach based upon the Taylor series method is proposed for propagating solutions of the time-dependent Schrödinger equation. Replacing the spatial derivative of the wave function with finite difference formulas, we derive a recursive formula for the evaluation of Taylor coefficients. The automatic differentiation technique is used to recursively calculate the required Taylor coefficients. We also develop an implicit scheme for the recursive evaluation of these coefficients. We then advance the solution in time using a Taylor series expansion. Excellent computational results are obtained when this method is applied to a one-dimensional reflectionless potential and a two-dimensional barrier transmission problem. © 2010 Wiley Periodicals, Inc.