Algebraic derivation of Kramers-Pasternack relations based on the Schrodinger factorization method

TL;DR

This paper presents an algebraic derivation of the Kramers-Pasternack relations for hydrogenic atoms using the Schrödinger factorization method, simplifying the calculation of moments of the radial coordinate.

Contribution

It introduces a purely algebraic approach to derive the recurrence relations and the second inverse moment, avoiding the need for Feynman-Hellman theorem or brute-force integration.

Findings

Derived the Kramers-Pasternack relations algebraically.

Provided an algebraic method to compute the second inverse moment.

Simplified the pedagogical understanding of the relations.

Abstract

The Kramers-Pasternack relations are used to compute the moments of r (both positive and negative) for all radial energy eigenfunctions of hydrogenic atoms. They consist of two algebraic recurrence relations, one for positive powers and one for negative. Most derivations employ the Feynman-Hellman theorem or a brute-force integration to determine the second inverse moment, which is needed to complete the recurrence relations for negative moments. In this work, we show both how to derive the recurrence relations algebraically and how to determine the second inverse moment algebraically, which removes the pedagogical confusion associated with differentiating the Hamiltonian with respect to the angular momentum quantum number l in order to find the inverse second moment.