§34.7 Basic Properties: $9j$ Symbol

Permalink:: http://dlmf.nist.gov/34.7

§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

Notes:: See Edmonds (1974, pp. 105–106), de-Shalit and Talmi (1963, p. 517).
Keywords:: $9j$ symbols
Permalink:: http://dlmf.nist.gov/34.7.i

34.7.1 $\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{21}}&j_{{22}}&j_{{13}}\\ j_{{31}}&j_{{31}}&0\end{Bmatrix}=\frac{(-1)^{{j_{{12}}+j_{{21}}+j_{{13}}+j_{{31}}}}}{((2j_{{13}}+1)(2j_{{31}}+1))^{{\frac{1}{2}}}}\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{22}}&j_{{21}}&j_{{31}}\end{Bmatrix}.$

Symbols:: $\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{21}}&j_{{22}}&j_{{23}}\\ j_{{31}}&j_{{32}}&j_{{33}}\end{Bmatrix}$ : $9j$ symbol, $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.7.E1
Encodings:: TeX, pMML, png

§34.7(ii) Symmetry

Keywords:: $9j$ symbols
Permalink:: http://dlmf.nist.gov/34.7.ii

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

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

§34.7(iii) Recursion Relations

Keywords:: $9j$ symbols
Permalink:: http://dlmf.nist.gov/34.7.iii

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

§34.7(iv) Orthogonality

Notes:: See Edmonds (1974, p. 103).
Keywords:: $9j$ symbols
Permalink:: http://dlmf.nist.gov/34.7.iv

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:: $\delta _{{j,k}}$ : Kronecker delta, $\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{21}}&j_{{22}}&j_{{23}}\\ j_{{31}}&j_{{32}}&j_{{33}}\end{Bmatrix}$ : $9j$ symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.7.E2
Encodings:: TeX, pMML, png

§34.7(v) Generating Functions

Keywords:: $9j$ symbols
Permalink:: http://dlmf.nist.gov/34.7.v

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

§34.7(vi) Sums

Notes:: See Edmonds (1974, pp. 103–104), de-Shalit and Talmi (1963, pp. 127, 518).
Keywords:: $9j$ symbols
Permalink:: http://dlmf.nist.gov/34.7.vi

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:: $\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{21}}&j_{{22}}&j_{{23}}\\ j_{{31}}&j_{{32}}&j_{{33}}\end{Bmatrix}$ : $9j$ symbol and $j,j_{r}$ : nonnegative integer
Referenced by:: §34.9
Permalink:: http://dlmf.nist.gov/34.7.E3
Encodings:: TeX, pMML, png

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

34.7.4 $\begin{pmatrix}j_{{13}}&j_{{23}}&j_{{33}}\\ m_{{13}}&m_{{23}}&m_{{33}}\end{pmatrix}\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{21}}&j_{{22}}&j_{{23}}\\ j_{{31}}&j_{{32}}&j_{{33}}\end{Bmatrix}=\sum _{{m_{{r1}},m_{{r2}},r=1,2,3}}\begin{pmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ m_{{11}}&m_{{12}}&m_{{13}}\end{pmatrix}\begin{pmatrix}j_{{21}}&j_{{22}}&j_{{23}}\\ m_{{21}}&m_{{22}}&m_{{23}}\end{pmatrix}\*\begin{pmatrix}j_{{31}}&j_{{32}}&j_{{33}}\\ m_{{13}}&m_{{23}}&m_{{33}}\end{pmatrix}\begin{pmatrix}j_{{11}}&j_{{21}}&j_{{31}}\\ m_{{11}}&m_{{21}}&m_{{31}}\end{pmatrix}\begin{pmatrix}j_{{12}}&j_{{22}}&j_{{32}}\\ m_{{12}}&m_{{22}}&m_{{32}}\end{pmatrix}.$

Symbols:: $\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{21}}&j_{{22}}&j_{{23}}\\ j_{{31}}&j_{{32}}&j_{{33}}\end{Bmatrix}$ : $9j$ symbol, $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol, $j,j_{r}$ : nonnegative integer and $r$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.7.E4
Encodings:: TeX, pMML, png

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:: $\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{21}}&j_{{22}}&j_{{23}}\\ j_{{31}}&j_{{32}}&j_{{33}}\end{Bmatrix}$ : $9j$ symbol, $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.7.E5
Encodings:: TeX, pMML, png

§34.7 Basic Properties: Symbol

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