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Optics and Spectroscopy

Radial matrix elements and the angular momentum technique

Dmitry A. Varshalovich, A. V. Karpova

Published Jan 1, 2015• 4 citations
Publisher

Abstract

For hydrogen wave functions, it is shown that radial matrix elements of type 〈nl|r k |nl′〉 can be expressed in terms of the Clebsch-Gordan coefficients, which characterize the relation between angular momenta l and l′, but in a nonphysical region of their arguments.
DOI
10.1134/s0030400x15010233

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The non-compact group O (2, 1) is used in an investigation of hydrogenic radial wavefunctions. These radial functions are shown to form bases for infinite dimensional representations of the algebra of O (2, 1). It is found that rN and r-N (N positive) transform as tensors with respect to this algebra. Analysis of matrix elements of rN and r-N shows that the Wigner-Eckart theorem is valid for this group and that the pertinent Clebsch-Gordan coefficients are proportional to the familiar R(3) Clebsch-Gordan coefficients. This proportionality provides simple explanations of the selection rules for hydrogenic radial matrix elements noted by Pasternack and Sternheimer, and the proportionality of hydrogenic expectation values of rN and r-N to 3 - j symbols.
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.
Formulas are obtained which give the matrix representation, relative to a product basis, of the projection operator for the total angular momentum of a system. If the individual angular momenta are not too large, the matrix elements depend upon a small number of parameters, independent of the number of angular momenta coupled. Recurrence relations between elements and symmetry properties of elements are derived. These results enable one to perform the vector coupling of a large number of angular momenta in a relatively simple fashion. The connection between the matrix elements and vector coupling coefficients is discussed. Several important special cases are treated in detail.
This book offers a concise introduction to the angular momentum, one of the most fundamental quantities in all of quantum mechanics. Beginning with the quantization of angular momentum, spin angular momentum, and the orbital angular momentum, the author goes on to discuss the Clebsch-Gordan coefficients for a two-component system. After developing the necessary mathematics, specifically spherical tensors and tensor operators, the author then investigates the 3- j , 6- j , and 9- j symbols. Throughout, the author provides practical applications to atomic, molecular, and nuclear physics. These include partial-wave expansions, the emission and absorption of particles, the proton and electron quadrupole moment, matrix element calculation in practice, and the properties of the symmetrical top molecule.