Optimization of explicit symplectic schemes for time-dependent schrödinger equations

In this paper, in order to conserve the discrete squared norm of the wave function, we propose a condition for optimizing the n-stage and n-order explicit symplectic schemes, which are applied to solving finite-dimensional canonical equations obtained by discretizing the time-dependent Schrödinger equations. It is showed that the 'half unitary' (i.e., symmetric) conditions proposed by Gray and Manolopoulos can automatically satisfy the optimal condition proposed in this paper for even n but not for odd n. In particular, two-order or four-order optimized explicit symplectic schemes are obtained. Calculations and comparisons with three kinds of explicit symplectic schemes are presented for a model in quantum systems.