Orateur
Description
We study two-dimensional fermion motion with circular symmetry using
both 3+1 and 2+1 Dirac equations with a general Lorentz structure. Using
a different approach than usual, we fully develop the formalism for
these equations using cylindrical coordinates and discuss the quantum
numbers, spinors and differential equations in both cases when there is
circular symmetry. Although there is no spin quantum number in the 2+1
case, we find that, as remarked already by other authors, in this
case the spin projection $s$ in the direction perpendicular to the plane
of motion can be emulated by a parameter preserving the
anti-commutation relations between the Dirac matrices. The formalism
developed allowed us to recognize an equivalence between a pure vector
potential and a pure tensor potential under circular symmetry, if the
former is multiplied by $s$, for any functional form of these
potentials. We apply the formalism, both in the 3+1 and 2+1 cases, to
the problem of a uniform magnetic field perpendicular to the plane of
motion. We fully discuss its solutions, their properties, including the energy spectra, compare them to the
relativistic Landau problem and obtain the non-relativistic
limit as well. This calculation enabled us to clarify the physical meaning of
the $s$ parameter, representing the spin quantum number in the 3+1 case and
just a parameter in the Hamiltonian in the 2+1 case.
