A Proof of Generalized Kramers-Pasternack Relation using Hyper-Radial Equation

TL;DR

This paper provides a rigorous proof of the generalized Kramers-Pasternack relation using the hyper-radial equation, extending its applicability to arbitrary dimensions and offering new insights into quantum mechanical recurrence relations.

Contribution

It introduces a novel proof method for the generalized Kramers-Pasternack relation via hyper-radial equations, broadening its theoretical foundation in quantum mechanics.

Findings

Extended the Kramers-Pasternack relation to arbitrary dimensions.

Provided a systematic derivation using hyper-radial equations and integrations by parts.

Enhanced understanding of recurrence relations for radial matrix elements.

Abstract

We present a proof of the generalized Kramers-Pasternack relation using the hyper-radial equation approach. Following Kramers' method, we manipulate the radial equation by multiplying it with an expression closely related to terms in the hyper-virial theorem. Through successive integrations by parts, we systematically derive the second Pasternack formula, extending its validity to arbitrary dimensions. This approach provides deeper insights into the algebraic structure of recurrence relations for diagonal radial matrix elements in quantum mechanics.