34 3j, 6j, 9j Symbols Properties 34.3 Basic Properties:

\mathit{3j}

\mathit{6j}

Symbol

§34.4 Definition: $\mathit{6j}$ Symbol

ⓘ

Keywords:: definition, finite sum of $\mathit{3j}$ symbols, generalized hypergeometric functions, representation as, $\mathit{6j}$ symbols
Notes:: See Varshalovich et al. (1988, §9.2.4), de-Shalit and Talmi (1963, p. 131, Eq. (15.12)), Edmonds (1974, p. 99).
Referenced by:: §16.4(iii)
Permalink:: http://dlmf.nist.gov/34.4
See also:: Annotations for Ch.34

The $\mathit{6j}$ symbol is defined by the following double sum of products of $\mathit{3j}$ symbols:

34.4.1		$\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\sum_{m_{r}m^{\prime}_{s}}(-1)^{l_{1}+m^{\prime% }_{1}+l_{2}+m^{\prime}_{2}+l_{3}+m^{\prime}_{3}}\*\begin{pmatrix}j_{1}&j_{2}&j% _{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}\begin{pmatrix}j_{1}&l_{2}&l_{3}\\ m_{1}&m^{\prime}_{2}&-m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&j_{2}&l_% {3}\\ -m^{\prime}_{1}&m_{2}&m^{\prime}_{3}\end{pmatrix}\begin{pmatrix}l_{1}&l_{2}&j_% {3}\\ m^{\prime}_{1}&-m^{\prime}_{2}&m_{3}\end{pmatrix},$
ⓘ Defines: $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol Symbols: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half. and $r$ : nonnegative integer Referenced by: §34.10, §34.4, §34.9 Permalink: http://dlmf.nist.gov/34.4.E1 Encodings: TeX, pMML, png See also: Annotations for §34.4 and Ch.34

where the summation is taken over all admissible values of the $m$ ’s and $m^{\prime}$ ’s for each of the four $\mathit{3j}$ symbols; compare (34.2.2) and (34.2.3).

Except in degenerate cases the combination of the triangle inequalities for the four $\mathit{3j}$ symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths $j_{1},j_{2},j_{3},l_{1},l_{2},l_{3}$ ; see Figure 34.4.1.

See accompanying text — Figure 34.4.1: Tetrahedron corresponding to $\mathit{6j}$ symbol. Magnify

The $\mathit{6j}$ symbol can be expressed as the finite sum

34.4.2		$\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}=\Delta(j_{1}j_{2}j_{3})\Delta(j_{1}l_{2}l_{3})% \Delta(l_{1}j_{2}l_{3})\Delta(l_{1}l_{2}j_{3})\\sum_{s}\frac{(-1)^{s}(s+1)!}{% (s-j_{1}-j_{2}-j_{3})!(s-j_{1}-l_{2}-l_{3})!(s-l_{1}-j_{2}-l_{3})!(s-l_{1}-l_{% 2}-j_{3})!}\\frac{1}{(j_{1}+j_{2}+l_{1}+l_{2}-s)!(j_{2}+j_{3}+l_{2}+l_{3}-s)!% (j_{3}+j_{1}+l_{3}+l_{1}-s)!},$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half. and $\Delta(j_{1}j_{2}j_{3})$ : product Referenced by: Erratum (V1.0.7) for Equation (34.4.2) Permalink: http://dlmf.nist.gov/34.4.E2 Encodings: TeX, pMML, png Errata (effective with 1.0.7): Originally the factor $\Delta(j_{1}j_{2}j_{3})\Delta(j_{1}l_{2}l_{3})\Delta(l_{1}j_{2}l_{3})\Delta(l_% {1}l_{2}j_{3})$ was missing in this equation. Reported 2012-12-31 by Yu Lin See also: Annotations for §34.4 and Ch.34

where the summation is over all nonnegative integers $s$ such that the arguments in the factorials are nonnegative.

Equivalently,

34.4.3		$\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix}={(-1)^{j_{1}+j_{3}+l_{1}+l_{3}}}\frac{\Delta(j_% {1}j_{2}j_{3})\Delta(j_{2}l_{1}l_{3})(j_{1}-j_{2}+l_{1}+l_{2})!(-j_{2}+j_{3}+l% _{2}+l_{3})!(j_{1}+j_{3}+l_{1}+l_{3}+1)!}{\Delta(j_{1}l_{2}l_{3})\Delta(j_{3}l% _{1}l_{2})(j_{1}-j_{2}+j_{3})!(-j_{2}+l_{1}+l_{3})!(j_{1}+l_{2}+l_{3}+1)!(j_{3% }+l_{1}+l_{2}+1)!}\*{{}_{4}F_{3}}\left({-j_{1}+j_{2}-j_{3},j_{2}-l_{1}-l_{3},-% j_{1}-l_{2}-l_{3}-1,-j_{3}-l_{1}-l_{2}-1\atop-j_{1}+j_{2}-l_{1}-l_{2},j_{2}-j_% {3}-l_{2}-l_{3},-j_{1}-j_{3}-l_{1}-l_{3}-1};1\right),$
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized hypergeometric function, $!$ : factorial (as in $n!$ ), $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $l,l_{r}$ : non-negative integers or non-negative integers plus one half. and $\Delta(j_{1}j_{2}j_{3})$ : product Referenced by: §18.38(iii) Permalink: http://dlmf.nist.gov/34.4.E3 Encodings: TeX, pMML, png See also: Annotations for §34.4 and Ch.34

where ${{}_{4}F_{3}}$ is defined as in §16.2.

For alternative expressions for the $\mathit{6j}$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${{}_{4}F_{3}}$ of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).

34.3 Basic Properties:

\mathit{3j}

Symbol 34.5 Basic Properties:

\mathit{6j}

Symbol

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§34.4 Definition: 6⁢j Symbol

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§34.4 Definition: $\mathit{6j}$ Symbol