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Numerical methods for the Dirac equation in the non-relativistic limit regime

Banff International Research Station for Mathematical Innovation and Discovery

Dirac equation, proposed by Paul Dirac in 1928, is a relativistic version of the Schroedinger equation for quantum mechanics. It describes the evolution of spin-1/2 massive particles, e.g. electrons. Due to its applications in graphene and 2D materials, Dirac equations has drawn considerable interests recently. We are concerned with the numerical methods for solving the Dirac equation in the non-relativistic limit regime, involving a small parameter inversely proportional to the speed of light. We begin with commonly used numerical methods in literature, including finite difference time domain and time splitting spectral, which need very small time steps to solve the Dirac equation in the non-relativistic limit regime. We then propose and analyze a multi-scale time integrator pseudospectral method for the Dirac equation, and prove its uniform convergence in the non-relativistic limit regime. We will extend the study to the nonlinear Dirac equation case.

Mathematics; Quantum theory; Optics, electromagnetic theory; Mathematical physics

video/mp4

Author affiliation: Beijing Computational Science Research Center

2018-02-16

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/64644

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/