A finite element method solver for time-dependent and stationarySchrodinger equations with a generic potential

SOBA

COMMUNICATIONS IN COMPUTATIONAL PHYSICS

Año: 2009 vol. 5 p. 914 - 927

1815-2406

A general finite element solution of the Schr ¨odinger equation for a onedimensional problem is presented. The solver is applicable to both stationary and time-dependent cases with a general user-selected potential term. Furthermore, it is possible to include externalmagnetic or electric fields, as well as spin-orbital and spinmagnetic interactions. We use analytically soluble problems to validate the solver. The predicted numerical auto-states are compared with the analytical ones, and selected mean values are used to validate the auto-functions. In order to analyze the performance of the time-dependent Schr ¨odinger equation, a traveling wave package benchmark was reproduced. In addition, a problem involving the scattering of a wave packet over a double potential barrier shows the performance of the solver in cases of transmission and reflection of packages. Other general problems, related to periodic potentials, are treated with the same general solver and a Lagrange multiplier method to introduce periodic boundary conditions. Some simple cases of known periodic potential solutions are reported.