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Exact solutions of the Dirac equation, a system of four partial differentialequations, are rare. The vast majority of them are for highly symmetricstationary systems. Moreover, only a handful of solutions for time dependentdynamics exists. Given the growing number of applications of high energyelectron beams interacting with a variety of quantum systems in laser fields,novel methods for finding exact solutions to the Dirac equation are called for.We present a method for building up solutions to the Dirac equation employing arecently introduced approach for the description of spinorial fields and theirdriving electromagnetic fields in terms of geometric algebras. We illustratethe method by developing several stationary as well as non-stationary solutionsof the Dirac equation with well defined orbital angular momentum along theelectron's propagation direction. The first set of solutions describe freeelectron beams in terms of Bessel functions as well as stationary solutions forboth a homogeneous and an inhomogeneous magnetic field. The second set ofsolutions are new and involve a plane electromagnetic wave combined with agenerally inhomogeneous longitudinal magnetic field. Moreover, the developedtechnique allows us to derive general physical properties of the dynamics insuch field configurations, as well as provides physical predictions on theself-consistent electromagnetic fields induced by the dynamics.

Construction of Dirac spinors for electron vortex beams in background electromagnetic fields