Volume 5, Issue 5
A Finite Element Method Solver for Time-dependent and Stationary Schrödinger Equations with a Generic Potential

A. Soba

Commun. Comput. Phys., 5 (2009), pp. 914-927.

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  • Abstract

A general finite element solution of the Schro¨dinger 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 external magnetic 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 Schro¨dinger 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. 

  • History

Published online: 2009-05

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