Group theory and the link between expectation values of powers of $r$ and Clebsch-Gordan coefficients

TL;DR

This paper explores how the expectation values of powers of the radial coordinate in hydrogenic atoms relate to Clebsch-Gordan coefficients through group theory, specifically using the non-compact group O(2,1), revealing proportionalities and selection rules.

Contribution

It demonstrates that the connection between radial expectation values and Clebsch-Gordan coefficients arises from group theory, extending previous results with new insights into the algebraic structure.

Findings

Expectation values transform as tensors under O(2,1)

Wigner-Eckart theorem applies to this group

Proportionality to $3jm$ symbols explains selection rules

Abstract

In a recent paper [J.-C. Pain, Opt. Spectrosc. ${\bf 218}$, 1105-1109 (2020)], we discussed the link between expectation values of powers of $r$ and Clebsch-Gordan coefficients. In this short note we provide additional information, reminding that such a connection is a direct consequence of group theory. The hydrogenic radial wavefunctions form bases for infinite dimensional representations of the algebra of the non-compact group $O(2,1)$ and the expectation values $r^p$ and $r^{-p}$ ($p$ being positive) transform as tensors with respect to this algebra. As shown a long time ago by Armstrong [L. Armstrong Jr., J . Phys. (Paris) Suppl. C 4 ${\bf 31}$, 17 (1970)], analysis of matrix elements of $r^p$ and $r^{-p}$ reveals that the Wigner-Eckart theorem is valid for this group and that the corresponding Clebsch-Gordan coefficients are proportional to the usual $SO(3)$ Clebsch-Gordan coefficients. This proportionality provides simple explanations of the selection rules for hydrogenic radial matrix elements pointed out by Pasternack and Sternheimer, and the proportionality of hydrogenic expectation values of $r^p$ and $r^{-p}$ to $3jm$ symbols.