Free Oscillator Realization of the Laguerre Polynomial

TL;DR

This paper presents a novel operator-based approach to relate harmonic oscillator eigenfunctions to radial oscillator eigenfunctions, revealing a connection between Hermite and Laguerre polynomials through free oscillator realization.

Contribution

It introduces an operator that maps harmonic oscillator eigenfunctions to radial oscillator eigenfunctions, establishing a new link between Hermite and Laguerre polynomials.

Findings

Constructed an operator linking harmonic and radial oscillators.

Derived a polynomial operator mapping Hermite to Laguerre polynomials.

Revealed a new perspective on oscillator eigenfunctions using free oscillator realization.

Abstract

We revisit the radial oscillator from the free oscillator realization point of view. By using a free oscillator, namely the creation/annihilation operators of the harmonic oscillator, we construct an operator that maps the eigenfunctions of the harmonic oscillator to those of the radial oscillator. As a polynomial part of this relation, we obtain an operator that maps the Hermite polynomials to the Laguerre polynomials.