34 3j, 6j, 9j Symbols Properties 34.1 Special Notation 34.3 Basic Properties:

\mathit{3j}

§34.2 Definition: $\mathit{3j}$ Symbol

ⓘ

Keywords:: $\mathit{3j}$ symbols, angular momenta, definition, finite sum of algebraic quantities, generalized hypergeometric functions, projective quantum numbers, representation as, $\mathit{3j}$ symbols, triangle conditions
Notes:: See Edmonds (1974, pp. 44–45).
Permalink:: http://dlmf.nist.gov/34.2
See also:: Annotations for Ch.34

The quantities $j_{1},j_{2},j_{3}$ in the $\mathit{3j}$ symbol are called angular momenta. Either all of them are nonnegative integers, or one is a nonnegative integer and the other two are half-odd positive integers. They must form the sides of a triangle (possibly degenerate). They therefore satisfy the triangle conditions

34.2.1		$\|j_{r}-j_{s}\|\leq j_{t}\leq j_{r}+j_{s},$
ⓘ Symbols: $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $r\in 1,2,3$ , $t\in 1,2,3$ and $s$ : nonnegative integer Referenced by: §34.10, §34.2, §34.3(iv) Permalink: http://dlmf.nist.gov/34.2.E1 Encodings: TeX, pMML, png See also: Annotations for §34.2 and Ch.34

where $r, s, t$ is any permutation of $1,2,3$ . The corresponding projective quantum numbers $m_{1},m_{2},m_{3}$ are given by

34.2.2		$m_{r}=-j_{r},-j_{r}+1,\dots,j_{r}-1,j_{r},$
$r=1,2,3$ ,
ⓘ Symbols: $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $r\in 1,2,3$ Referenced by: §34.4 Permalink: http://dlmf.nist.gov/34.2.E2 Encodings: TeX, pMML, png See also: Annotations for §34.2 and Ch.34

and satisfy

34.2.3		$m_{1}+m_{2}+m_{3}=0.$
ⓘ Referenced by: §34.10, §34.2, §34.3(iv), §34.4 Permalink: http://dlmf.nist.gov/34.2.E3 Encodings: TeX, pMML, png See also: Annotations for §34.2 and Ch.34

See Figure 34.2.1 for a schematic representation.

See accompanying text — Figure 34.2.1: Angular momenta $j_{r}$ and projective quantum numbers $m_{r}$ , $r=1,2,3$ . Magnify

If either of the conditions (34.2.1) or (34.2.3) is not satisfied, then the $\mathit{3j}$ symbol is zero. When both conditions are satisfied the $\mathit{3j}$ symbol can be expressed as the finite sum

34.2.4		$\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}={(-1)^{j_{1}-j_{2}-m_{3}}}\Delta(j_{1}j_{2}j_{3% })\left((j_{1}+m_{1})!(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})!(j_{3}+m_{3})!% (j_{3}-m_{3})!\right)^{\frac{1}{2}}\*\sum_{s}\frac{(-1)^{s}}{s!(j_{1}+j_{2}-j_% {3}-s)!(j_{1}-m_{1}-s)!(j_{2}+m_{2}-s)!(j_{3}-j_{2}+m_{1}+s)!(j_{3}-j_{1}-m_{2% }+s)!},$
ⓘ Defines: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol Symbols: $!$ : factorial (as in $n!$ ), $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $\Delta(j_{1}j_{2}j_{3})$ : product and $s$ : nonnegative integer Permalink: http://dlmf.nist.gov/34.2.E4 Encodings: TeX, pMML, png See also: Annotations for §34.2 and Ch.34

where

34.2.5		$\Delta(j_{1}j_{2}j_{3})=\left(\frac{(j_{1}+j_{2}-j_{3})!(j_{1}-j_{2}+j_{3})!(-% j_{1}+j_{2}+j_{3})!}{(j_{1}+j_{2}+j_{3}+1)!}\right)^{\frac{1}{2}},$
ⓘ Defines: $\Delta(j_{1}j_{2}j_{3})$ : product (locally) Symbols: $!$ : factorial (as in $n!$ ) and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Permalink: http://dlmf.nist.gov/34.2.E5 Encodings: TeX, pMML, png See also: Annotations for §34.2 and Ch.34

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

Equivalently,

34.2.6		$\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}={(-1)^{j_{2}-m_{1}+m_{3}}}\frac{(j_{1}+j_{2}+m_% {3})!(j_{2}+j_{3}-m_{1})!}{\Delta(j_{1}j_{2}j_{3})(j_{1}+j_{2}+j_{3}+1)!}\left% (\frac{(j_{1}+m_{1})!(j_{3}-m_{3})!}{(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})% !(j_{3}+m_{3})!}\right)^{\frac{1}{2}}\*{{{}_{3}F_{2}}\left(-j_{1}-j_{2}-j_{3}-% 1,-j_{1}+m_{1},-j_{3}-m_{3};-j_{1}-j_{2}-m_{3},-j_{2}-j_{3}+m_{1};1\right)},$
ⓘ 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 $n!$ ), $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half. and $\Delta(j_{1}j_{2}j_{3})$ : product Referenced by: §18.38(iii) Permalink: http://dlmf.nist.gov/34.2.E6 Encodings: TeX, pMML, png See also: Annotations for §34.2 and Ch.34

where ${{}_{3}F_{2}}$ is defined as in §16.2.

For alternative expressions for the $\mathit{3j}$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${{}_{3}F_{2}}$ of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).

34.1 Special Notation 34.3 Basic Properties:

\mathit{3j}

Symbol

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§34.2 Definition: 3⁢j Symbol

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§34.2 Definition: $\mathit{3j}$ Symbol