Solving PDEs with Spatial & Time Varying Coefficients: Dirac Wave Function Passes Through EM Wave

A. J. Kalinowski1
Published in 2020

The Dirac equation is employed in particle physics and historically gave the first combined unification of quantum mechanics and relativity theory by introducing a four component wave function Ψn n=1…4. This wave function describes the behavior of fermion type particles. The effect of a pre-existing EM traveling wave (via a combined magnetic vector potential {A} and scalar electric potential Φ) on the classical Dirac equations are treated by adding additional potential terms in the free field PDE’s [1]. COMSOL® is used for obtaining a simpler 2-D wave function [Ψ1(x,y,t),Ψ4(x,y,t) ] as a solution to the coupled time dependent Dirac equation having PDE coefficients that vary with both time and space. The probability density ρ(x,y,t) = |Ψ1|^2 + |Ψ4|^2 of a particle being at a spacial point is treated with the "wave function formulation" which involves solving the electromagnetic Dirac field PDEs for Ψn .

The Equation Based Modeling General-Form PDE interface with a time dependent study is employed in COMSOL Multiphysics®. When the wave vector k lies in the xy plane, the 4 component Ψn uncouples into two PDEs in terms of components for n=1&4. The distortion of the freely propagating Dirac waves by the presence of the pre-existing EM traveling wave is examined through a series of examples. A local steady state version of the governing PDEs (temporarily holding {x,y,t} constant in the PDE coefficients over a small neighborhood) provides a-priori information about the required spatial FEM mesh size.

A series of increasing complex models are examined starting with (a) a simple PW propagating down a wave guide where the EM wave length is much bigger than the Dirac wave length, (b) propagation of Dirac wave functions emanating from a single slit and propagating outward into the full spatial domain where cylindrical spreading of the wave function is experienced, (c) full 360 deg cylindrical waves passing through an EM traveling wave field and (d) two slit interference models where an incident PW enters an infinite domain having a pre-existing EM wave. In Figs.(1-3), waves emerging from the two slits interact, forming bands of constructive and destructive interference in curved blade patterns as opposed to Fig.(4) straight blade patterns when the EM field is off. In the examples, the same simulation is also run with the EM field off so the effect of the EM field can be observed.

Agreement between the alternative FEM solution validations were good. The effect of the traveling EM field on solutions resulted in wave functions having gradually varying spatial frequencies and amplitude shapes for Ψ1(x,y,t),Ψ4(x,y,t) and hence the probability distribution ρ(x,y) as well.