Digital Library of Mathematical Functions
About the Project
NIST
34 3j,6j,9j SymbolsProperties

§34.5 Basic Properties: 6j Symbol

Contents

§34.5(i) Special Cases

Show Annotations
Notes:
See Edmonds (1974, pp. 98, 130–132), de-Shalit and Talmi (1963, p. 520).
Keywords:
6j symbols
Permalink:
http://dlmf.nist.gov/34.5.i

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}}}},
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}}},
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}}},
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}}},
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}}},
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}}},
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}}}}.

§34.5(ii) Symmetry

Show Annotations
Notes:
See Edmonds (1974, pp. 94, 95).
Keywords:
Regge symmetries, 6j symbols
Permalink:
http://dlmf.nist.gov/34.5.ii

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}.

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},
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}.

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

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

where

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

§34.5(iv) Orthogonality

Show Annotations
Notes:
See Edmonds (1974, p. 96).
Keywords:
6j symbols
Permalink:
http://dlmf.nist.gov/34.5.iv

§34.5(v) Generating Functions

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

§34.5(vi) Sums

Show Annotations
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).
Keywords:
6j symbols
Permalink:
http://dlmf.nist.gov/34.5.vi
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},
Show Annotations
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, pMML, 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}.

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

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}.

Equation (34.5.23) can be regarded as an alternative definition of the 6j symbol.

For other sums see Ginocchio (1991).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available.