SUSYQM AND SWKB APPROACH TO THE DIRAC EQUATION WITH A COULOMB POTENTIAL IN 2+1 DIMENSIONS

Abstract

The exact solutions of quantum systems play an important role in quantum mechanics. For example, the exact solutions of the Schrödinger equation for a hydrogen atom and for a harmonic oscillator in three dimensions [11, 215, 297, 537] were important achievements at the beginning stage of quantum mechanics , which provided a strong evidence in favor of the theory being correct. In the comprehensive paper [538], Kolsrud fully exploited Hylleraas' approach [539-543] to propose two different methods to study the bound and continuous states of the Dirac equation with a Coulomb potential. On the other hand, Waldenstrøm [544, 545] directly solved the single second-order radial differential equation, which was derived by Dirac himself [546], where the same technique employed for solving the Schrödinger equation with a Coulomb potential was used. The relativistic Dirac-oscillator potential introduced in Refs. [547, 548] has been studied by adding an off-diagonal linear radial term to the Dirac operator. Recently, the Schrödinger equation with the Coulomb and oscillator problems in arbitrary dimensions have been studied [549, 550]. On the other hand, due to recent interest in the lower-dimensional field theory and condensed matter physics, the two-dimensional case, which seems physically relevant, has been investigated. With this spirit, the study of quantum scattering theory of the relativistic particle by the Coulomb field has been discussed in two dimensions [551-556]. In addition, some papers are devoted to the studies of the non-relativistic equation with the Coulomb potential in one dimension [557-560] and in two dimensions [561-566] as well as those of the relativistic equation in one dimension [567]. The Dirac equation with the Coulomb potential in two dimensions was investigated by the series methods [568, 569]. The Dirac 187