Non-dispersive analytical solutions to the Dirac equation

TL;DR

This paper introduces new geometric algebra-based analytical solutions to the Dirac equation that describe shape-preserving, dispersionless electron wavepacket motion along arbitrary trajectories, enhancing understanding of relativistic quantum dynamics.

Contribution

It presents the first analytic solutions demonstrating non-dispersive, shape-preserving wavepacket translation in the Dirac equation using geometric algebra methods.

Findings

Gaussian wavepackets can move along elliptical and circular paths without dispersion.

Electromagnetic field configurations enabling dispersionless motion are identified.

Solutions connect quantum relativistic dynamics with Lorentz group geometry.

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.