Nondispersive analytical solutions to the Dirac equation

Andre G Campos,Renan Cabrera

doi:10.1103/physrevresearch.2.013051

Andre G Campos, Renan Cabrera

Open Access

https://doi.org/10.1103/physrevresearch.2.013051

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Abstract

This paper presents new analytic solutions to the Dirac equation employing a recently introduced method that is based on the formulation of spinorial fields and their driving electromagnetic fields in terms of geometric algebras. A first family of solutions describe the shape-preserving translation of a wavepacket along any desired trajectory in the x-y plane. In particular, we show that the dispersionless motion of a Gaussian wavepacket along both elliptical and circular paths can be achieved with rather simple electromagnetic field configurations. A second family of solutions involves a plane electromagnetic wave and a combination of generally inhomogeneous electric and magnetic fields. The novel analytical solutions of the Dirac equation given here provide important insights into the connection between the quantum relativistic dynamics of electrons and the underlying geometry of the Lorentz group.

Highlights

In this work we further expand upon the recently developed framework of relativistic dynamical inversion (RDI) [1] whose purpose is to solve the following problem
The method can be used to assess for attainable dynamics
The Dirac equation is commonly expressed as γ μ[ich∂μ − ceAμ]ψ = mc2ψ, (1)

Summary

INTRODUCTION

In this work we further expand upon the recently developed framework of relativistic dynamical inversion (RDI) [1] whose purpose is to solve the following problem. Given an arbitrary (desired) spinorial space-time wave packet ψ, find an electromagnetic vector potential Aμ such that the Dirac equation is satisfied. This is accomplished by RDI in two steps. RDI fulfils all these needs by providing stationary as well as time-dependent exact solutions in two and three dimensions RDI is used to construct electromagnetic fields that move a given Dirac spinor along any desired trajectory in the x-y plane without spreading. General solutions for a combination of plane electromagnetic waves and electric and magnetic fields along the wave’s propagation direction with arbitrary perpendicular profiles are constructed. The analytical solutions of the Dirac equation given here provide important insights into the relativistic dynamics of electrons

METHODOLOGY OF RELATIVISTIC DYNAMICAL INVERSION

A GENERAL SOLUTION FOR MOTION CONFINED TO THE x-y PLANE

OUTLOOK