Normalizing the hydrogenic polar solutions $$\Theta _{\ell m}(\theta )$$ without Associated Legendre polynomials

Abstract

The normalization of the polar functions \(\Theta _{\ell , m} (\theta )\) for the solution of Schrödinger’s equation for the Coulomb potential usually proceeds by appealing to the properties of Associated Legendre polynomials. We show how to achieve the normalization directly from the overall form of the solution and the recursion relation for its series part. When combined with a previous such normalization for the radial part of the solution, the entire hydrogen atom solution can be normalized without having to invoke any properties of special functions.