A high-order accuracy method for numerical solving of the time-dependent Schrodinger equation

A generalization of the Crank-Nicolson algorithm to higher orders for the t ime-dependent Schrodinger equation is proposed to improve the accuracy of t he time approximation. The implicit difference schemes are obtained in term s of the Magnus expansion for the evolution operator and its further factor ization with the help of diagonal Pade approximations. Stability of the sch emes and conservation of the approximated solution norm are provided by the fact that the Magnus expansion of the evolution operator preserves its uni tarity in any order with respect to a time step tau. As an example, a compa rison between the numerical and analytical solutions of a model problem for the oscillator with an explicitly time-dependent frequency was performed f or the schemes O(tau(4)) and O(tau(6)) to demonstrate accuracy, efficiency and adequate convergence of the method. (C) 1999 Elsevier Science B.V. All rights reserved.