34 3j, 6j, 9j Symbols Properties 34.6 Definition:

\mathit{9j}

Symbol 34.8 Approximations for Large Parameters

§34.7 Basic Properties: $\mathit{9j}$ Symbol

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Permalink:: http://dlmf.nist.gov/34.7
See also:: Annotations for Ch.34

§34.7(i) Special Case
§34.7(ii) Symmetry
§34.7(iii) Recursion Relations
§34.7(iv) Orthogonality
§34.7(v) Generating Functions
§34.7(vi) Sums

§34.7(i) Special Case

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Keywords:: $\mathit{9j}$ symbols, special case
Notes:: See Edmonds (1974, pp. 105–106), de-Shalit and Talmi (1963, p. 517).
Permalink:: http://dlmf.nist.gov/34.7.i
See also:: Annotations for §34.7 and Ch.34

34.7.1		$\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{13}\\ j_{31}&j_{31}&0\end{Bmatrix}=\frac{(-1)^{j_{12}+j_{21}+j_{13}+j_{31}}}{((2j_{1% 3}+1)(2j_{31}+1))^{\frac{1}{2}}}\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{22}&j_{21}&j_{31}\end{Bmatrix}.$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.7.E1 Encodings: TeX, pMML, png See also: Annotations for §34.7(i), §34.7 and Ch.34

§34.7(ii) Symmetry

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Keywords:: $\mathit{9j}$ symbols, symmetry
Permalink:: http://dlmf.nist.gov/34.7.ii
See also:: Annotations for §34.7 and Ch.34

The $\mathit{9j}$ symbol has symmetry properties with respect to permutation of columns, permutation of rows, and transposition of rows and columns; these relate 72 independent $\mathit{9j}$ symbols. Even (cyclic) permutations of either columns or rows, as well as transpositions, leave the $\mathit{9j}$ symbol unchanged. Odd permutations of columns or rows introduce a phase factor $(-1)^{R}$ , where $R$ is the sum of all arguments of the $\mathit{9j}$ symbol.

For further symmetry properties of the $\mathit{9j}$ symbol see Edmonds (1974, pp. 102–103) and Varshalovich et al. (1988, §10.4.1).

§34.7(iii) Recursion Relations

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Keywords:: $\mathit{9j}$ symbols, recursion relations
Permalink:: http://dlmf.nist.gov/34.7.iii
See also:: Annotations for §34.7 and Ch.34

For recursion relations see Varshalovich et al. (1988, §10.5).

§34.7(iv) Orthogonality

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Keywords:: $\mathit{9j}$ symbols, orthogonality
Notes:: See Edmonds (1974, p. 103).
Permalink:: http://dlmf.nist.gov/34.7.iv
See also:: Annotations for §34.7 and Ch.34

34.7.2		$\sum_{j_{12}\,j_{34}}(2j_{12}+1)(2j_{34}+1)(2j_{13}+1)(2j_{24}+1)\begin{% Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j_{13}&j_{24}&j\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j^{\prime}_{13}&j^{\prime}_{24}&j\end{Bmatrix}=\delta_{j_{13},j^{\prime}_{13}}% \delta_{j_{24},j^{\prime}_{24}}.$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.7.E2 Encodings: TeX, pMML, png See also: Annotations for §34.7(iv), §34.7 and Ch.34

§34.7(v) Generating Functions

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Keywords:: generating functions, $\mathit{9j}$ symbols
Permalink:: http://dlmf.nist.gov/34.7.v
See also:: Annotations for §34.7 and Ch.34

For generating functions for the $\mathit{9j}$ symbol see Biedenharn and van Dam (1965, p. 258, eq. (4.37)).

§34.7(vi) Sums

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Keywords:: addition theorem, $\mathit{9j}$ symbols, sum rule, sums
Notes:: See Edmonds (1974, pp. 103–104), de-Shalit and Talmi (1963, pp. 127, 518).
Permalink:: http://dlmf.nist.gov/34.7.vi
See also:: Annotations for §34.7 and Ch.34

34.7.3		$\sum_{j_{13}\,j_{24}}(-1)^{2j_{2}+j_{24}+j_{23}-j_{34}}(2j_{13}+1)(2j_{24}+1)% \begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{3}&j_{4}&j_{34}\\ j_{13}&j_{24}&j\end{Bmatrix}\begin{Bmatrix}j_{1}&j_{3}&j_{13}\\ j_{4}&j_{2}&j_{24}\\ j_{14}&j_{23}&j\end{Bmatrix}=\begin{Bmatrix}j_{1}&j_{2}&j_{12}\\ j_{4}&j_{3}&j_{34}\\ j_{14}&j_{23}&j\end{Bmatrix}.$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Referenced by: §34.9 Permalink: http://dlmf.nist.gov/34.7.E3 Encodings: TeX, pMML, png See also: Annotations for §34.7(vi), §34.7 and Ch.34

This equation is the sum rule. It constitutes an addition theorem for the $\mathit{9j}$ symbol.

34.7.4		$\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix}\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{m_{r1},m_{r2},r=1,2,3}\begin{pmatrix}j% _{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\*\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}.$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $r$ : nonnegative integer Referenced by: Erratum (V1.0.10) for Equation (34.7.4) Permalink: http://dlmf.nist.gov/34.7.E4 Encodings: TeX, pMML, png Errata (effective with 1.0.10): Originally the third $\mathit{3j}$ symbol in the summation was written incorrectly as $\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix}.$ Reported 2015-01-19 by Yan-Rui Liu See also: Annotations for §34.7(vi), §34.7 and Ch.34

34.7.5		$\sum_{j^{\prime}}(2j^{\prime}+1)\begin{Bmatrix}j_{11}&j_{12}&j^{\prime}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}\begin{Bmatrix}j_{11}&j_{12}&j^{\prime}\\ j_{23}&j_{33}&j\end{Bmatrix}={(-1)^{2j}}\begin{Bmatrix}j_{21}&j_{22}&j_{23}\\ j_{12}&j&j_{32}\end{Bmatrix}\begin{Bmatrix}j_{31}&j_{32}&j_{33}\\ j&j_{11}&j_{21}\end{Bmatrix}.$
ⓘ Symbols: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.7.E5 Encodings: TeX, pMML, png See also: Annotations for §34.7(vi), §34.7 and Ch.34

\mathit{9j}