2021/01/19 by Jean‐Christophe Pain, Jean-Christophe Pain, Pain, Jean-Christophe
Chemistry · Physics and Astronomy · #Advanced Chemical Physics Studies #Atomic Physics (physics.atom-ph) #Atomic and Molecular Physics #Chemical Thermodynamics and Molecular Structure #FOS: Physical sciences #Quantum Physics (quant-ph) #physics.atom-ph #quant-ph
paper · pdf · doi:10.48550/arxiv.2101.07872
arxiv created 2021/01/19 · openalex publication_date 2021/01/19 · arxiv updated 2021/01/21 · openalex created_date 2021/02/01 · openalex updated_date 2026/07/28
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 rp 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 rp 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 rp and r-p to 3jm symbols.