The Angular Momentum Operator in the Dirac Equation

TL;DR

This paper investigates the angular momentum operator in the Dirac equation by analyzing solutions in different tetrad frames, revealing a relationship between solutions in rotating and fixed frames through Jacobi polynomials and spherical harmonics.

Contribution

It introduces a method to separate variables in the Dirac equation in spherical coordinates using two tetrad frames and relates solutions via similarity transformations involving Jacobi polynomials.

Findings

Solutions in the rotating frame are expressed with Jacobi polynomials.

Angular solutions relate to spherical harmonics through a similarity transformation.

The approach clarifies the role of tetrad choice in Dirac equation solutions.

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.