A Laguerre Polynomial Orthogonality and the Hydrogen Atom

TL;DR

This paper provides an elementary proof of the orthogonality of hydrogen atom wave functions involving Laguerre polynomials and explores related orthogonality properties of Meixner polynomials, highlighting their mathematical significance.

Contribution

It offers a new elementary proof of Laguerre polynomial orthogonality in quantum mechanics and reveals a novel orthogonality relation for Meixner polynomials.

Findings

Orthogonality of hydrogen wave functions established

Unique natural orthogonality for Laguerre polynomials identified

Analogous orthogonality relation derived for Meixner polynomials

Abstract

The radial part of the wave function of an electron in a Coulomb potential is the product of a Laguerre polynomial and an exponential with the variable scaled by a factor depending on the degree. This note presents an elementary proof of the orthogonality of wave functions with differing energy levels. It is also shown that this is the only other natural orthogonality for Laguerre polynomials. By expanding in terms of the usual Laguerre polynomial basis an analogous strange orthogonality is obtained for Meixner polynomials.