Eur. Phys. J. Plus (2016) 131: 127
https://doi.org/10.1140/epjp/i2016-16127-2

Regular Article

The role of shape invariance potentials in the relativistic quantum mechanics

¹ Department of Physics, Faculty of Basic Sciences, Shahed University, 18151/159, Tehran, Iran
² Department of Physics, University of Guilan, 51335-1914, Rasht, Iran

^* e-mail: z.bakhshi@shahed.ac.ir

Received: 23 May 2015
Revised: 5 March 2016
Accepted: 14 March 2016
Published online: 4 May 2016

Abstract

The point canonical transformation in non-relativistic quantum mechanics is applied as an algebraic method to obtain the solutions of the Dirac equation with spherical symmetry electromagnetic potentials. We want to show that some of the non-relativistic solvable potentials with shape-invariant symmetry can be related to the radial Dirac equation. Using this method, the idea of supersymmetry and shape invariance can be expanded to the relativistic quantum mechanics. The spinor wave functions for some of the obtained four-component electromagnetic potential are given in terms of special functions such as Jacobi, generalized Laguerre and Hermite polynomials. The relativistic bound-states spectrum for each case is also calculated in terms of the bound-states spectrum of the solvable potentials.