Numerical methods for solving time-dependent quantum-mechanical problems with applications

Two numerical techniques which can be applied to time-dependent quantum-mechanical problems are described and compared with a predictor-corrector-type method. The first method, the piecewise Magnus solution, provides an approximate solution with the exact Hamiltonian by using the first Magnus approximation over many time intervals. The second approach, the piecewise analytical solution, produces the analytical solution to an approximate Hamiltonian obtained by ignoring off-diagonal elements within each small time interval. Several illustrative model problems are reported in which the speed and accuracy of these procedures were compared with a standard Gear package. Included in these examples is the physically interesting problem of using the stimulated Raman effect to produce selectively excited molecules. Within this problem the quality of the rotating wave approximation is tested. In cases with highly oscillatory wave functions, it was found that the piecewise solution methods performed well while the Gear program was incapable of providing a reliable solution.