Wigner 3j,6j,9j symbols

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1: 34.1 Special Notation

… ►The main functions treated in this chapter are the Wigner $\mathit{3}j,\mathit{6}j,\mathit{9}j$ symbols, respectively, … ►For other notations for $\mathit{3}j$, $\mathit{6}j$, $\mathit{9}j$ symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).

2: 34.12 Physical Applications

§34.12 Physical Applications

►The angular momentum coupling coefficients ($\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols) are essential in the fields of nuclear, atomic, and molecular physics. For applications in nuclear structure, see de-Shalit and Talmi (1963); in atomic spectroscopy, see Biedenharn and van Dam (1965, pp. 134–200), Judd (1998), Sobelman (1992, Chapter 4), Shore and Menzel (1968, pp. 268–303), and Wigner (1959); in molecular spectroscopy and chemical reactions, see Burshtein and Temkin (1994, Chapter 5), and Judd (1975). $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see Carlson and Rushbrooke (1950) and Judd (1976).

3: 16.24 Physical Applications

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§16.24(iii) $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ Symbols

►The $\mathit{3}j$ symbols, or Clebsch–Gordan coefficients, play an important role in the decomposition of reducible representations of the rotation group into irreducible representations. They can be expressed as ${}_3{}^{}F_2^{}$ functions with unit argument. The coefficients of transformations between different coupling schemes of three angular momenta are related to the Wigner $\mathit{6}j$ symbols. …Lastly, special cases of the $\mathit{9}j$ symbols are ${}_5{}^{}F_4^{}$ functions with unit argument. …

4: Joris Van der Jeugt

… ►His research interests are in the following areas: Group theoretical methods in physics; Representation theory of Lie algebras, Lie superalgebras and quantum groups with applications in mathematical physics; 3 $nj$-symbols and their relations to special functions and orthogonal polynomials; Quantum theory, finite quantum systems, quantum oscillator models, Wigner quantum systems; and Parabosons, parafermions and generalized quantum statistics. … ►His publications on Clebsch-Gordan coefficients, Racah coefficients, 3 $nj$-coefficients and their relation to hypergeometric series are considered as standard and a review is part of the volume on Multivariable Special Functions in the ongoing Askey–Bateman book project. … ►

34 3j, 6j, 9j Symbols

5: Bibliography R

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Some properties of Bernoulli’s numbers (J. Indian Math. Soc. 3 (1911), 219–234.). In Collected Papers,

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Referenced by:

§24.7(i).

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On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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External Links:

ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

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Calculation of $3n$-$j$ symbols by Labarthe’s method. International Journal of Quantum Chemistry 63 (1), pp. 57–64.

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Document

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§34.13.

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Another proof of the triple sum formula for Wigner $9j$-symbols. J. Math. Phys. 40 (12), pp. 6689–6691.

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ISSN 0022-2488, Document, MathReview (Michael Wüstner), ZentralBlatt

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§34.6.

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The $3$-$j$ and $6$-$j$ Symbols. The Technology Press, MIT, Cambridge, MA.

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ISBN 0-262-18004-9

Referenced by:

(34.1.1), §34.1, §34.1, §34.14, Erratum (V1.0.14) for Section 34.1, Erratum (V1.0.15) for Section 34.1.

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6: Bibliography W

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The surface of an ellipsoid. Quart. J. Math., Oxford Ser. 6, pp. 280–287.

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Document, ZentralBlatt

Referenced by:

§19.33(i).

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Asymptotic approximations for certain $6$-$j$ and $9$-$j$ symbols. J. Phys. A 32 (39), pp. 6901–6902.

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ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§34.8, §34.8.

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Surface Waves. In Handbuch der Physik, Vol. 9, Part 3, pp. 446–778.

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Online edition, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§11.12.

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Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Pure and Applied Physics. Vol. 5, Academic Press, New York.

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Notes:

Expanded and improved ed. Translated from the German by J. J. Griffin.

External Links:

ISBN 0-12-750550-4, MathReview (A. S. Wightman), ZentralBlatt

Referenced by:

§34.12.

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The Stokes set of the cusp diffraction catastrophe. J. Phys. A 13 (9), pp. 2913–2928.

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ISSN 0305-4470, Document, MathReview (I. Stewart), ZentralBlatt

Referenced by:

§36.5(ii).

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7: Bibliography L

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Exact computation of the $3$-$j$ and $6$-$j$ symbols. Comput. Phys. Comm. 61 (3), pp. 350–360.

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Notes:

A Fortran program is included.

External Links:

Document, ZentralBlatt

Referenced by:

§34.15.

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Exact computation of the $9$-$j$ symbols. Comput. Phys. Comm. 70 (3), pp. 544–556.

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Notes:

Double-precision Fortran

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ISSN 0010-4655, Document, Code, MathReview

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7th item.

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Statistical Physics, Part 2: Theory of the Condensed State. Pergamon Press, Oxford.

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Course of Theoretical Physics , Vol. 9. Translated from the Russian by J. B. Sykes and M. J. Kearsley.

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ISBN 0-08-023073-3; 0-08-023072-5, MathReview

Referenced by:

§25.17.

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Asymptotics of the first Appell function $F_1$ with large parameters. Integral Transforms Spec. Funct. 24 (9), pp. 715–733.

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ISSN 1065-2469, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§16.13, §16.13.

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Simplified recursive algorithm for Wigner $3j$ and $6j$ symbols. Phys. Rev. E 57 (6), pp. 7274–7277.

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Document

Referenced by:

§34.13, §34.3(iii).

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