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In a recent paper [J.-C. Pain, Opt. Spectrosc. ${\bf 218}$, 1105-1109(2020)], we discussed the link between expectation values of powers of $r$ andClebsch-Gordan coefficients. In this short note we provide additionalinformation, reminding that such a connection is a direct consequence of grouptheory. The hydrogenic radial wavefunctions form bases for infinite dimensionalrepresentations of the algebra of the non-compact group $O(2,1)$ and theexpectation values $r^p$ and $r^{-p}$ ($p$ being positive) transform as tensorswith 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 ofmatrix elements of $r^p$ and $r^{-p}$ reveals that the Wigner-Eckart theorem isvalid for this group and that the corresponding Clebsch-Gordan coefficients areproportional to the usual $SO(3)$ Clebsch-Gordan coefficients. Thisproportionality provides simple explanations of the selection rules forhydrogenic 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.

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