Factorization Method for d-Dimensional Isotropic Harmonic Oscillator and the Generalized Laguerre Polynomials

TL;DR

This paper applies the factorization method to the d-dimensional isotropic harmonic oscillator, deriving ladder operators and expressing eigenstates with generalized Laguerre polynomials, also extending to the Morse oscillator.

Contribution

It introduces a novel application of the factorization method to higher-dimensional oscillators and connects ladder operators with generalized Laguerre polynomials.

Findings

Eigenstates expressed in terms of generalized Laguerre polynomials

Ladder operators derived for the d-dimensional case

Bound states of the Morse oscillator obtained

Abstract

The factorization method of Infeld and Hull is applied to the radial Schr\"{o}dinger equation for $d$-dimensional isotropic harmonic oscillator and various ladder operators are defined. The radial energy eigenstates are expressed in terms of the generalized Laguerre polynomials and their properties are shown to follow from the expressions involving the ladder operators. In the same way as the harmonic oscillator we also obtain the bound energy eigenstates of the Morse oscillator.