  Exact solutions of the Dirac equation, a system of four partial differential
equations, are rare. The vast majority of them are for highly symmetric
stationary systems. Moreover, only a handful of solutions for time dependent
dynamics exists. Given the growing number of applications of high energy
electron 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 a
recently introduced approach for the description of spinorial fields and their
driving electromagnetic fields in terms of geometric algebras. We illustrate
the method by developing several stationary as well as non-stationary solutions
of the Dirac equation with well defined orbital angular momentum along the
electron's propagation direction. The first set of solutions describe free
electron beams in terms of Bessel functions as well as stationary solutions for
both a homogeneous and an inhomogeneous magnetic field. The second set of
solutions are new and involve a plane electromagnetic wave combined with a
generally inhomogeneous longitudinal magnetic field. Moreover, the developed
technique allows us to derive general physical properties of the dynamics in
such field configurations, as well as provides physical predictions on the
self-consistent electromagnetic fields induced by the dynamics.
