Explicit Magnus expansions for solving the time-dependent Schrödinger equation

Abstract

The symmetric implicit operator-difference multi-layer schemes for solving the time-dependent Schrödinger equation based on decomposition of the evolution operator via the explicit Magnus expansion up to the sixth order of accuracy with respect to the time step are presented. Reduced schemes for solving the set of coupled time-dependent Schrödinger equations with respect to the hyper-radial variable are devised by using the Kantorovich expansion of the wave packet over a set of appropriate parametric basis angular functions. Further discretization of the resulting problem with symmetric operators is implemented by means of the finite-element method. The convergence and efficiency of the numerical schemes are demonstrated in benchmark calculations of the exactly solvable models of a one-dimensional time-dependent oscillator, a two-dimensional oscillator in time-dependent electric field by using the conventual angular basis, and the inexactly solvable model of a three-dimensional kicked hydrogen atom in a magnetic field by using a parametric basis of the angular oblate spheroidal functions developed in our previous paper (Chuluunbaatar O et al 2007 J. Phys. A: Math. Theor. 40 11485–524).