A Finite Element Method Solver for Time-Dependent and Stationary Schrodinger Equations with a Generic Potential
A. Soba 1*1 Computer Applications in Science and Engineering, Barcelona Supercomputing Center, c/Jordi Girona, 31, 08034, Barcelona, Spain.
Received 29 May 2008; Accepted (in revised version) 14 July 2008
Available online 29 September 2008
A general finite element solution of the Schrodinger equation for a one-dimensional 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 spin-magnetic 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 Schrodinger 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.
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PACS: 02.70.Dh; 03.67.Lx; 75.40.Mg
Key words: One dimensional finite element methods, time dependent Schrodinger equation, periodic boundary conditions, quantum computer simulation.
Email: email@example.com (A. Soba)