Spinors and Rodrigues representations associated with orthogonal polynomials

TL;DR

This paper develops a method to relate Dirac spinor solutions to orthogonal polynomials using Rodrigues representations, connecting relativistic quantum mechanics with classical orthogonal polynomial models.

Contribution

It introduces a novel approach to express Dirac spinor components via Rodrigues representations of orthogonal polynomials, bridging relativistic and non-relativistic quantum models.

Findings

Upper spinor wave functions are expressed in terms of Rodrigues representations.

Lower spinor components are also associated with Rodrigues representations.

The approach links relativistic solutions to well-known orthogonal polynomial models.

Abstract

An effective approach is presented to produce Schrodinger-like equation for the spinor components from Dirac equation. Considering electrostatic potential as a constant value yields a second-order differential equation that is comparable with the well-known solvable models in the non-relativistic quantum mechanics for the certain bound state energy spectrum and the well-known potentials. By this comparison, the gage field potential and the relativistic energy can be written by the non-relativistic models and the spinors will be related to the orthogonal polynomials. It has also shown that the upper spinors wave functions based on the orthogonal polynomials can be given in terms of the Rodrigues representations. Association with the Rodrigues representations of orthogonal polynomials have also been investigated in the lower spinor components, since they are related to the upper spinor components according to first-order differential equation that is attained from Dirac equation.