  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.
