Implicit fast Fourier transform method for integration of the time dependent Schrodinger or diffusion equation

摘要

The authors have found that the conventional exponentiated split operator procedure is subject to difficulties in energy conservation when solving the time-dependent Schrodinger equation for Coulombic systems. By rearranging the kinetic and potential energy terms in the temporal propagator of the finite difference equations, one can find a propagation algorithm for three dimensions that looks much like the Crank-Nicholson and alternating direction implicit methods for one- and two-space-dimensional partial differential equations. They report comparisons of this novel implicit split operator procedure with the conventional exponentiated split operator procedure on hydrogen atom solutions. The results look promising for a purely numerical approach to certain electron quantum mechanical problems.