Employing an operator form of the Rodrigues formula to calculate wavefunctions without differential equations

TL;DR

This paper introduces a generalized operator-based Rodrigues formula approach to compute quantum wavefunctions directly from energy eigenstates, avoiding differential equations for various quantum systems.

Contribution

It extends the Rodrigues formula method to a representation-independent form applicable to multiple quantum problems, simplifying wavefunction calculations.

Findings

Successfully applied to harmonic oscillator and Coulomb problems

Provides a unified, algebraic method for wavefunction determination

Suitable for educational use in quantum mechanics courses

Abstract

The factorization method of Schrodinger shows us how to determine the energy eigenstates without needing to determine the wavefunctions in position or momentum space. A strategy to convert the energy eigenstates to wavefunctions is well known for the one-dimensional simple harmonic oscillator by employing the Rodrigues formula for the Hermite polynomials in position or momentum space. In this work, we illustrate how to generalize this approach in a representation-independent fashion to find the wavefunctions of other problems in quantum mechanics that can be solved by the factorization method. We examine three problems in detail: (i) the one-dimensional simple harmonic oscillator; (ii) the three-dimensional isotropic harmonic oscillator; and (iii) the three-dimensional Coulomb problem. This approach can be used in either undergraduate or graduate classes in quantum mechanics.