Angular momentum and the polar basis of harmonic oscillator

TL;DR

This paper develops a new method using the polar basis of harmonic oscillators to analyze angular momentum, deriving analytic expressions for passage matrices and generating functions, linking group theory and symmetries in quantum mechanics.

Contribution

Introduces a novel approach based on the polar basis of harmonic oscillators for angular momentum analysis, deriving new expressions and generating functions.

Findings

Derived analytic expressions for passage matrices in 4D polar basis.

Established connections between results and Laguerre polynomial group theory.

Found new generating functions for Clebsch-Gordan coefficients.

Abstract

In this paper we follow the Schwinger approach for angular momentum but with the polar basis of harmonic oscillator as a starting point. We derive by a new method two analytic expressions of the elements of passage matrix from the double polar basis to 4- dimensions polar basis of the harmonic oscillator. These expressions are functions of the modules of magnetic moments. The connection between our results and the results derived by the group theory of Laguerre polynomials is found. We determine a new expression for these elements in terms of magnetic moments in the general case. We deduce from these expressions the symmetries of 3j symbols. A new generating function of the Clebsh-Gordan coefficients, functions of the modules of magnetic moments are found. We prove that the generating function of recoupling coefficients 3nj for the polar basis are the same in the Schwinger's approach therefore the polar basis of harmonic oscillator may be a starting point to study the angular momentum.