High-fidelity numerical solution of the time-dependent Dirac equation

Authors:
Martin Almquist;Ken Mattsson;Tomas Edvinsson
Affiliations:
Uppsala University, Department of Information Technology, Lägerhyddsvägen 2, 752 37 Uppsala, Sweden;Uppsala University, Department of Information Technology, Lägerhyddsvägen 2, 752 37 Uppsala, Sweden;Uppsala University, Department of Chemistry - íngström Laboratory, Lägerhyddsvägen 1, 751 21 Uppsala, Sweden
Venue:
Journal of Computational Physics
Year:
2014

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Abstract

A stable high-order accurate finite difference method for the time-dependent Dirac equation is derived. Grid-convergence studies in 1-D and 3-D corroborate the analysis. The method is applied to time-resolved quantum tunneling where a comparison with the solution to the time-dependent Schrodinger equation in 1-D illustrates the differences between the two equations. In contrast to the conventional tunneling probability decay predicted by the Schrodinger equation, the Dirac equation exhibits Klein tunneling. Solving the time-dependent Dirac equation with a step potential in 3-D reveals that particle spin affects the tunneling process. The observed spin-dependent reflection allows for a new type of spin-selective measurements.