Factorization of Dirac equation in two space dimensions

Abstract

We present a systematic approach for the separation of variables for the two-dimensional Dirac equation in polar coordinates. The three vector potential, which couple to the Dirac spinor via minimal coupling, along with the scalar potential are chosen to have angular dependence which emanate the Dirac equation to complete separation of variables. Exact solutions are obtained for a class of solvable potentials along with their relativistic spinor wavefunctions. Particular attention is paid to the situation where the potentials are confined to a quantum dot region and are of scalar, vector and pseudo-scalar type. The study of a single charged impurity embedded in a 2D Dirac equation in the presence of a uniform magnetic field was treated as a particular case of our general study.

Gate-tunable graphene quantum dot and Dirac oscillator

“…In this case we arrive, after applying the boundary conditions and finding both ratios, at the following characteristic equation for ξ > 0 (27) and for ξ < 0…”

“…which can also be used to derive (31) from (27) in similar way as before. Moreover, the bound-state levels for negative effective magnetic field can be found by using the symmetry…”

“…We note that any other choice for the pair of Pauli matrices can be obtained from the present one through a unitary transformation, hence leaving the physics of the problem unaltered. A 2 × 2 matrix that is defined by [26,27] Λ…”

“…We combine different approaches to achieve our goal. Indeed, based on [26,27] we set the Hamiltonian system of our problem where a similarity transformation is used to simplify the process for obtaining the solutions. Later on, we define the QD by gates introducing an electrostatic confining potential.…”

Factorization of the Dirac equation and a graphene quantum dot

“…Very recently, we have presented a systematic approach for the separation of variables for the two-dimensional Dirac equation in polar coordinates [9]. The three vector potential, which couple to the Dirac spinor via minimal coupling, along with the scalar potential were chosen to have angular dependence which emanate the Dirac equation to complete separation of variables.…”