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arXiv:2003.14197 · Optics and Spectroscopy

Expression of Relativistic Expectation Values of Powers of r in Terms of Clebsch–Gordan Coefficients

Jean‐Christophe Pain

Published Aug 1, 2020• 3 citations
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Abstract

It is shown that, in the relativistic hydrogenic approximation, expectation values of the type $$\left\langle {n\ell j} \right|{{r}^{k}}\left| {n\ell j} \right\rangle $$ can be expressed in terms of Clebsch–Gordan coefficients or $$3jm$$ symbols. This generalizes the results obtained by Varshalovich and Karpova [Opt. Spectrosc. 118, 1–5 (2015); Opt. Spektrosk. 118, 3–7 (2015)] in the non-relativistic hydrogenic case.

DOI
10.1134/s0030400x20080275

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It is shown that, in the relativistic hydrogenic approximation, expectation values of the type ⟨nℓj∣rk∣nℓj⟩\left\langle {n\ell j} \right|{{r}^{k}}\left| {n\ell j} \right\rangle ⟨nℓj∣rk∣nℓj⟩ can be expressed in terms of Clebsch–Gordan coefficients or 3jm3jm3jm symbols. This generalizes the results obtained by Varshalovich and Karpova [Opt. Spectrosc. 118, 1–5 (2015); Opt. Spektrosk. 118, 3–7 (2015)] in the non-relativistic hydrogenic case.
A recent paper by Drake and Swainson [Phys. Rev. A 42, 1123 (1990)] presents a recursion relation that generates a set of coefficients needed to obtain general closed-form expressions for hydrogenic expectation values for rP{\mathit{r}}^{\mathit{P}}rP. By combining this recursion relation with an earlier semiclassical formulation of this calculation [J. Phys. B 14, 1373 (1981)], it is shown that the correct expectation values can also be obtained by a simple substitution of prescribed angular momentum operators into the classical Kepler formula, which provides a transparent connection to the correspondence limit.
The radial functions for the discrete levels of Dirac's relativistic hydrogenic atom are expressed in terms of the generalized Laguerre polynomials which have known properties. This leads to an easy method of evaluating the normalization constants and the average values of rq{r}^{q}rq.