Three-dimensional Finite Difference-Time Domain Solution of Dirac Equation

The Dirac equation is solved using three-dimensional Finite DifferenceTime Domain (FDTD) method. Zitterbewegung and the dynamics of a well-localized electron are used as examples of FDTD application to the case of free electrons. PACS numbers: 03.65.Pm, 02.60.-x, 02.60.Lj § Correspondence should be addressed to Louisiana Tech University, PO Box 10348, Ruston, LA 71272, Tel: +1.318.257.3591, Fax: +1.318.257.4228, E-mail: [email protected] ar X iv :0 81 2. 18 07 v1 [ ph ys ic s. co m pph ] 9 D ec 2 00 8 Three-dimensional Finite Difference-Time Domain Solution of Dirac Equation 2 In this letter, for the first time, full three-dimensional FDTD method to solve the Dirac equation is described. The Finite Difference-Time Domain (FDTD) method is a fast growing numerical method originally introduced by Kane Yee [1] to solve Maxwell’s equations. In the last two decades the method was intensively developed primarily in the field of electrodynamics [2, 3, 4, 5], but was, at a much lower scale, also extended to other fields of applications such as acoustics and elastodynamics. When applied to solving Maxwell’s equations, the FDTD method is relatively simple and numerically robust, has almost no limit in the description of geometrical and dispersive properties of the material, and is appropriate for the computer technology of today. As an example, we have applied the FDTD method to calculate the exposure of complex biological tissues to non-ionizing ultra-wide band (UWB) radiation using high-resolution description of the geometry and realistic physical properties of exposed material over a broad frequency range [6, 7]. In the case of electrodynamics, the FDTD method was able to solve problems with complexity that was far beyond allowing analytical solutions and was fundamental to the advancement of electrical engineering [5]. In the same sense, we expect that the application of FDTD method in quantum mechanics, in this particular case for the solution of the Dirac equation, will become a stepping stone for the advancement of modern physics. Similarities between Maxwell’s equations and the Dirac equation are obvious if the Dirac equation is written, in a standard notation, as a system of two first-order equations [8] [ı~σ · ∇ − ı c ∂ ∂t ]Φ = −mcΦ, [−ı~σ · ∇ − ı c ∂ ∂t ]Φ = −mcΦ. (1) Two component wave functions Φ and Φ couple in these equations as the electric ~ E and magnetic ~ B fields couple in Maxwell’s equations [8]. While the FDTD scheme can be applied to Eq. (1), it is applied here to the form originally written by Dirac. In the case when the electromagnetic field described by the four-potential A = {A0(x), ~ A(x)} is minimally coupled to the particle, the Dirac equation can be written as ı~ ∂Ψ ∂t = (Hfree +Hint)Ψ, (2) where Hfree = −ıc~α · ∇+ βmc, (3) Hint = −eα · ~ A+ eA0, (4)