{
  "id": "2101.07872",
  "version": "v1",
  "published": "2021-01-19T21:48:16.000Z",
  "updated": "2021-01-19T21:48:16.000Z",
  "title": "Group theory and the link between expectation values of powers of $r$ and Clebsch-Gordan coefficients",
  "authors": [
    "Jean-Christophe Pain"
  ],
  "categories": [
    "quant-ph",
    "physics.atom-ph"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-19T21:48:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "expectation values",
      "clebsch-gordan coefficients",
      "group theory",
      "hydrogenic radial wavefunctions form bases",
      "hydrogenic radial matrix elements"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}