The Angular Momentum Operator in the Dirac Equation.pdf

a r X i v : h e p - t h / 9 4 0 5 0 3 3 v 1 5 M a y 1 9 9 4 CENTRE DE PHYSIQUE THEORIQUE - CNRS - Luminy, Case 907 F-13288 Marseille Cedex 9 - France Unite? Propre de Recherche 7061 The angular momentum operator in the Dirac equation V??ctor M. Villalba1 Abstract The Dirac equation in spherically symmetric fields is separated in two different tetrad frames. One is the standard cartesian (fixed) frame and the second one is the diagonal (rotating) frame. After separating variables in the Dirac equation in spherical coordinates, and solving the corresponding eingenvalues equations associated with the angular operators, we obtain that the spinor solution in the rotating frame can be expressed in terms of Jacobi polynomials, and it is related to the standard spherical harmonics, which are the basis solution of the angular momentum in the Cartesian tetrad, by a similarity transformation Key-Words : 3.65, 11.10, 4.90. April 1994 CPT-94/P.3027 anonymous ftp or gopher: cpt.univ-mrs.fr 1permanent address: Centro de F??sica, Instituto Venezolano de Investigaciones Cient??ficas IVIC, Apdo. 21827, Caracas 1020-A Venezuela e-mail address: villalba@cpt.univ-mrs.fr, villalba@dino.conicit.ve 1 Introduction The Dirac equation is a system of four coupled partial differential equation which describes the relativistic electron and other spin 1/2 particles. Despite the remarkable effort made during the last decades in order to find exact solutions for the relativistic electron in the presence of external fields, the amount of solvable problems is relatively scarce, being the Coulomb problem [1], the plane wave [2], and some electromagnetic configurations [3] the most representative examples. The problem of finding exact solutions of the Dirac equation is closely related to the possibility of accomplishing a complete separation of variables of the Dirac equation. During the last decades there have been different approaches to tackling the problem of separating variables in the Dirac