  Motivated by the conduction properties of graphene discovered and studied in
the last decades, we consider the quantum dynamics of a massless, charged, spin
1/2 relativistic particle in three dimensional space-time, in the presence of
an electrostatic field in various configurations such as step or barrier
potentials and generalizations of them. The field is taken as parallel to the y
coordinate axis and vanishing outside of a band parallel to the x axis. The
classical theory is reviewed, together with its canonical quantization leading
to the Dirac equation for a 2-component spinor. Stationary solutions are
numerically found for each of the field configurations considered, fromwhich we
calculate the mean quantum trajectories of the particle and compare them with
the corresponding classical trajectories, the latter showing a classical
version of the Klein phenomenon. Transmission and reflection probabilities are
also calculated, confirming the Klein phenomenon.
