Factorization of Dirac Equation in Two Space Dimensions

TL;DR

This paper develops a systematic method for separating variables in the two-dimensional Dirac equation with various potentials, providing exact solutions relevant for quantum dots and impurity problems in 2D materials.

Contribution

It introduces a novel approach for variable separation in 2D Dirac equations with angular-dependent potentials, enabling exact solutions for complex quantum systems.

Findings

Exact solutions for a class of solvable potentials in 2D Dirac equation.

Application to quantum dots with scalar, vector, and pseudo-scalar potentials.

Analysis of a charged impurity in a magnetic field within the Dirac framework.

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.