Quantum charged spinning massless particles in 2 + 1 dimensions

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, from which 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