§ 34.5. Basic Properties: $6j$ Symbol

Permalink:: http://dlmf.nist.gov/34.5

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

§ 34.5(i). Special Cases

Notes:: See Edmonds (1974, pp. 98, 130–132), de-Shalit and Talmi (1963, p. 520).
Permalink:: http://dlmf.nist.gov/34.5.SS1

In the following equations it is assumed that the triangle inequalities are satisfied and that $J$ is again defined by (34.3.4).

If any lower argument in a $6j$ symbol is 0, $\tfrac{1}{2}$ , or 1, then the $6j$ symbol has a simple algebraic form. Examples are provided by:

34.5.1 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 0&j_{3}&j_{2}\end{Bmatrix}=\frac{(-1)^{J}}{\left((2j_{2}+1)(2j_{3}+1)\right)^{{\frac{1}{2}}}},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E1
Encodings:: TeX, pMathML, png

34.5.2 $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ \frac{1}{2}&j_{{3}}-\frac{1}{2}&j_{{2}}+\frac{1}{2}\end{Bmatrix}=(-1)^{{J}}\left(\frac{(j_{1}+j_{3}-j_{2})(j_{1}+j_{2}-j_{3}+1)}{(2j_{2}+1)(2j_{2}+2)2j_{3}(2j_{3}+1)}\right)^{{\frac{1}{2}}},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E2
Encodings:: TeX, pMathML, png

34.5.3 $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ \frac{1}{2}&j_{{3}}-\frac{1}{2}&j_{{2}}-\frac{1}{2}\end{Bmatrix}=(-1)^{{J}}\left(\frac{(j_{{2}}+j_{{3}}-j_{{1}})(j_{{1}}+j_{{2}}+j_{{3}}+1)}{2j_{{2}}(2j_{{2}}+1)2j_{{3}}(2j_{{3}}+1)}\right)^{{\frac{1}{2}}},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E3
Encodings:: TeX, pMathML, png

34.5.4 $\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ 1&j_{3}-1&j_{2}-1\end{Bmatrix}=(-1)^{J}\left(\frac{J(J+1)(J-2j_{1})(J-2j_{1}-1)}{(2j_{2}-1)2j_{2}(2j_{2}+1)(2j_{3}-1)2j_{3}(2j_{3}+1)}\right)^{{\frac{1}{2}}},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E4
Encodings:: TeX, pMathML, png

34.5.5 $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ 1&j_{{3}}-1&j_{{2}}\end{Bmatrix}=(-1)^{{J}}\left(\frac{2(J+1)(J-2j_{{1}})(J-2j_{{2}})(J-2j_{{3}}+1)}{2j_{{2}}(2j_{{2}}+1)(2j_{{2}}+2)(2j_{{3}}-1)2j_{{3}}(2j_{{3}}+1)}\right)^{{\frac{1}{2}}},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E5
Encodings:: TeX, pMathML, png

34.5.6 $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ 1&j_{{3}}-1&j_{{2}}+1\end{Bmatrix}=(-1)^{{J}}\left(\frac{(J-2j_{{2}}-1)(J-2j_{{2}})(J-2j_{{3}}+1)(J-2j_{{3}}+2)}{(2j_{{2}}+1)(2j_{{2}}+2)(2j_{{2}}+3)(2j_{{3}}-1)2j_{{3}}(2j_{{3}}+1)}\right)^{{\frac{1}{2}}},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E6
Encodings:: TeX, pMathML, png

34.5.7 $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ 1&j_{{3}}&j_{{2}}\end{Bmatrix}=(-1)^{{J+1}}\frac{2(j_{{2}}(j_{{2}}+1)+j_{{3}}(j_{{3}}+1)-j_{{1}}(j_{{1}}+1))}{\left(2j_{{2}}(2j_{{2}}+1)(2j_{{2}}+2)2j_{{3}}(2j_{{3}}+1)(2j_{{3}}+2)\right)^{{\frac{1}{2}}}}.$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $J$ : sum
Permalink:: http://dlmf.nist.gov/34.5.E7
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§ 34.5(ii). Symmetry

Notes:: See Edmonds (1974, pp. 94, 95).
Permalink:: http://dlmf.nist.gov/34.5.SS2

The $6j$ symbol is invariant under interchange of any two columns and also under interchange of the upper and lower arguments in each of any two columns, for example,

34.5.8 $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}=\begin{Bmatrix}j_{{2}}&j_{{1}}&j_{{3}}\\ l_{{2}}&l_{{1}}&l_{{3}}\end{Bmatrix}=\begin{Bmatrix}j_{{1}}&l_{{2}}&l_{{3}}\\ l_{{1}}&j_{{2}}&j_{{3}}\end{Bmatrix}.$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Referenced by:: §34.5(ii)
Permalink:: http://dlmf.nist.gov/34.5.E8
Encodings:: TeX, pMathML, png

Next,

34.5.9 $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}=\begin{Bmatrix}j_{{1}}&\frac{1}{2}(j_{{2}}+l_{{2}}+j_{{3}}-l_{{3}})&\frac{1}{2}(j_{{2}}-l_{{2}}+j_{{3}}+l_{{3}})\\ l_{{1}}&\frac{1}{2}(j_{{2}}+l_{{2}}-j_{{3}}+l_{{3}})&\frac{1}{2}(-j_{{2}}+l_{{2}}+j_{{3}}+l_{{3}})\end{Bmatrix},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Referenced by:: §34.5(ii)
Permalink:: http://dlmf.nist.gov/34.5.E9
Encodings:: TeX, pMathML, png

34.5.10 $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}=\begin{Bmatrix}\frac{1}{2}(j_{{2}}+l_{{2}}+j_{{3}}-l_{{3}})&\frac{1}{2}(j_{{1}}-l_{{1}}+j_{{3}}+l_{{3}})&\frac{1}{2}(j_{{1}}+l_{{1}}+j_{{2}}-l_{{2}})\\ \frac{1}{2}(j_{{2}}+l_{{2}}-j_{{3}}+l_{{3}})&\frac{1}{2}(-j_{{1}}+l_{{1}}+j_{{3}}+l_{{3}})&\frac{1}{2}(j_{{1}}+l_{{1}}-j_{{2}}+l_{{2}})\end{Bmatrix}.$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Referenced by:: §34.5(ii)
Permalink:: http://dlmf.nist.gov/34.5.E10
Encodings:: TeX, pMathML, png

Equations (34.5.9) and (34.5.10) are called Regge symmetries. Additional symmetries are obtained by applying (34.5.8) to (34.5.9) and (34.5.10). See Srinivasa Rao and Rajeswari (1993, pp. 102–103) and references given there.

§ 34.5(iii). Recursion Relations

Permalink:: http://dlmf.nist.gov/34.5.SS3

In the following equation it is assumed that the triangle conditions are satisfied.

34.5.11 ${(2j_{1}+1)\left((J_{3}+J_{2}-J_{1})(L_{3}+L_{2}-J_{1})-2(J_{3}L_{3}+J_{2}L_{2}-J_{1}L_{1})\right)\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}}\\ =j_{1}E(j_{1}+1)\begin{Bmatrix}j_{1}+1&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}+(j_{1}+1)E(j_{1})\begin{Bmatrix}j_{1}-1&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer, $l,l_{r}$ : nonnegative integer, $J_{r}$ , $L_{r}$ and $E(j)$
Permalink:: http://dlmf.nist.gov/34.5.E11
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where

34.5.12

$J_{r}=j_{r}(j_{r}+1),$

$L_{r}=l_{r}(l_{r}+1),$

Defines:: $J_{r}$ and $L_{r}$
Symbols:: $j,j_{r}$ : nonnegative integer, $l,l_{r}$ : nonnegative integer and $r$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.5.E12
Encodings:: TeX, TeX, pMathML, pMathML, png, png

34.5.13 $E(j)=\left((j^{2}-(j_{2}-j_{3})^{2})((j_{2}+j_{3}+1)^{2}-j^{2})(j^{2}-(l_{2}-l_{3})^{2})((l_{2}+l_{3}+1)^{2}-j^{2})\right)^{{\frac{1}{2}}}.$

Defines:: $E(j)$
Symbols:: $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.5.E13
Encodings:: TeX, pMathML, png

For further recursion relations see Varshalovich et al. (1988, §9.6) and Edmonds (1974, pp. 98–99).

§ 34.5(iv). Orthogonality

Notes:: See Edmonds (1974, p. 96).
Permalink:: http://dlmf.nist.gov/34.5.SS4

34.5.14 $\sum _{{j_{{3}}}}(2j_{{3}}+1)(2l_{{3}}+1)\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l^{{\prime}}_{{3}}\end{Bmatrix}=\delta _{{l_{3},l^{{\prime}}_{{3}}}}.$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.5.E14
Encodings:: TeX, pMathML, png

§ 34.5(v). Generating Functions

Permalink:: http://dlmf.nist.gov/34.5.SS5

For generating functions for the $6j$ symbol see Biedenharn and van Dam (1965, p. 255, eq. (4.18)).

§ 34.5(vi). Sums

Notes:: See de-Shalit and Talmi (1963, pp. 517–518), Varshalovich et al. (1988, §9.8), Edmonds (1974, pp. 95–97), Dunlap and Judd (1975).
Permalink:: http://dlmf.nist.gov/34.5.SS6

34.5.15 $\sum _{{j}}(-1)^{{j+j^{{\prime}}+j^{{\prime\prime}}}}(2j+1)\begin{Bmatrix}j_{{1}}&j_{{2}}&j\\ j_{{3}}&j_{{4}}&j^{{\prime}}\end{Bmatrix}\begin{Bmatrix}j_{{1}}&j_{{2}}&j\\ j_{{4}}&j_{{3}}&j^{{\prime\prime}}\end{Bmatrix}=\begin{Bmatrix}j_{{1}}&j_{{4}}&j^{{\prime}}\\ j_{{2}}&j_{{3}}&j^{{\prime\prime}}\end{Bmatrix},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol and $j,j_{r}$ : nonnegative integer
Referenced by:: §34.5(vi), §34.9
Permalink:: http://dlmf.nist.gov/34.5.E15
Encodings:: TeX, pMathML, png

34.5.16 $(-1)^{{j_{{1}}+j_{{2}}+j_{{3}}+j_{{1}}^{{\prime}}+j_{{2}}^{{\prime}}+l_{1}+l_{2}}}\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}\begin{Bmatrix}j_{{1}}^{{\prime}}&j_{{2}}^{{\prime}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}^{{\prime}}\end{Bmatrix}\\ =\sum _{{j}}(-1)^{{l_{{3}}+l_{{3}}^{{\prime}}+j}}(2j+1)\begin{Bmatrix}j_{{1}}&j_{{1}}^{{\prime}}&j\\ j_{{2}}^{{\prime}}&j_{{2}}&j_{{3}}\end{Bmatrix}\begin{Bmatrix}l_{{3}}&l_{{3}}^{{\prime}}&j\\ j_{{1}}^{{\prime}}&j_{{1}}&l_{{2}}\end{Bmatrix}\begin{Bmatrix}l_{{3}}&l_{{3}}^{{\prime}}&j\\ j_{{2}}^{{\prime}}&j_{{2}}&l_{{1}}\end{Bmatrix}.$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Referenced by:: §34.5(vi)
Permalink:: http://dlmf.nist.gov/34.5.E16
Encodings:: TeX, pMathML, png

Equations (34.5.15) and (34.5.16) are the sum rules. They constitute addition theorems for the $6j$ symbol.

34.5.17 $\sum _{{j}}(2j+1)\begin{Bmatrix}j_{{1}}&j_{{2}}&j\\ j_{{1}}&j_{{2}}&j^{{\prime}}\end{Bmatrix}=(-1)^{{2(j_{{1}}+j_{{2}})}},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.5.E17
Encodings:: TeX, pMathML, png

34.5.18 $\sum _{{j}}(-1)^{{j_{{1}}+j_{{2}}+j}}(2j+1)\begin{Bmatrix}j_{{1}}&j_{{2}}&j\\ j_{{2}}&j_{{1}}&j^{{\prime}}\end{Bmatrix}=\sqrt{(2j_{{1}}+1)(2j_{{2}}+1)}\,\delta _{{j^{{\prime}},0}},$

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol and $j,j_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.5.E18
Encodings:: TeX, pMathML, png

34.5.19 $\sum _{{l}}\begin{Bmatrix}j_{{1}}&j_{{2}}&l\\ j_{{2}}&j_{{1}}&j\end{Bmatrix}=0,$ $2\mu-j$ odd, $\mu=\min(j_{{1}},j_{{2}})$ ,

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.5.E19
Encodings:: TeX, pMathML, png

34.5.20 $\sum _{{l}}(-1)^{{l+j}}\begin{Bmatrix}j_{{1}}&j_{{2}}&l\\ j_{{1}}&j_{{2}}&j\end{Bmatrix}=\frac{(-1)^{{2\mu}}}{2j+1},$ $\mu=\min(j_{{1}},j_{{2}})$ ,

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.5.E20
Encodings:: TeX, pMathML, png

34.5.21 $\sum _{{l}}(-1)^{{l+j+j_{1}+j_{2}}}\begin{Bmatrix}j_{{1}}&j_{{2}}&l\\ j_{{2}}&j_{{1}}&j\end{Bmatrix}=\frac{1}{2j+1}\left(\frac{(2j_{{1}}-j)!(2j_{{2}}+j+1)!}{(2j_{{2}}-j)!(2j_{{1}}+j+1)!}\right)^{{\frac{1}{2}}},$ $j_{{2}}\le j_{{1}}$ ,

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.5.E21
Encodings:: TeX, pMathML, png

34.5.22 $\sum _{{l}}(-1)^{{l+j+j_{1}+j_{2}}}\frac{1}{l(l+1)}\begin{Bmatrix}j_{{1}}&j_{{2}}&l\\ j_{{2}}&j_{{1}}&j\end{Bmatrix}=\frac{1}{j_{{1}}(j_{{1}}+1)-j_{{2}}(j_{{2}}+1)}\left(\frac{(2j_{{1}}-j)!(2j_{{2}}+j+1)!}{(2j_{{2}}-j)!(2j_{{1}}+j+1)!}\right)^{{\frac{1}{2}}},$ $j_{{2}}<j_{{1}}$ .

Symbols:: $\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}$ : $6j$ symbol, $j,j_{r}$ : nonnegative integer and $l,l_{r}$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.5.E22
Encodings:: TeX, pMathML, png

34.5.23 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{Bmatrix}j_{{1}}&j_{{2}}&j_{{3}}\\ l_{{1}}&l_{{2}}&l_{{3}}\end{Bmatrix}=\sum _{{m^{{\prime}}_{{1}}m^{{\prime}}_{{2}}m^{{\prime}}_{{3}}}}(-1)^{{l_{{1}}+l_{{2}}+l_{{3}}+m^{{\prime}}_{{1}}+m^{{\prime}}_{{2}}+m^{{\prime}}_{{3}}}}\begin{pmatrix}j_{1}&l_{2}&l_{3}\\ m_{1}&m^{{\prime}}_{2}&-m^{{\prime}}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&j_{2}&l_{3}\\ -m^{{\prime}}_{1}&m_{2}&m^{{\prime}}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&j_{3}\\ m^{{\prime}}_{1}&-m^{{\prime}}_{2}&m_{3}\end{pmatrix}.$