§34.2 Definition: $3j$ Symbol

Notes:: See Edmonds (1974, pp. 44–45).
Keywords:: angular momenta, generalized hypergeometric functions, projective quantum numbers, $3j$ symbols, triangle conditions
Permalink:: http://dlmf.nist.gov/34.2

The quantities $j_{{1}},j_{{2}},j_{{3}}$ in the $3j$ 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:: $j,j_{r}$ : nonnegative integer, $r\in 1,2,3$ , $t\in 1,2,3$ 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

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:: $j,j_{r}$ : nonnegative integer and $r\in 1,2,3$
Referenced by:: §34.4
Permalink:: http://dlmf.nist.gov/34.2.E2
Encodings:: TeX, pMML, png

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 Figure 34.2.1 for a schematic representation.

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

Symbols:: $j,j_{r}$ : nonnegative integer and $r\in 1,2,3$
Referenced by:: §34.2
Permalink:: http://dlmf.nist.gov/34.2.F1
Encodings:: pdf, png

If either of the conditions (34.2.1) or (34.2.3) is not satisfied, then the $3j$ symbol is zero. When both conditions are satisfied the $3j$ 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:: $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol
Symbols:: $!$ : $n!$ : factorial, $j,j_{r}$ : nonnegative integer, $\Delta(j_{1}j_{2}j_{3})$ : product and $s$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.2.E4
Encodings:: TeX, pMML, png

where

34.2.5 $\Delta(j_{{1}}j_{{2}}j_{{3}})=\left(\frac{(j_{{1}}+j_{{2}}-j_{{3}})!(j_{{1}}-j_{{2}}+j_{{3}})!(-j_{{1}}+j_{{2}}+j_{{3}})!}{(j_{{1}}+j_{{2}}+j_{{3}}+1)!}\right)^{{\frac{1}{2}}},$

Symbols:: $!$ : $n!$ : factorial, $j,j_{r}$ : nonnegative integer and $\Delta(j_{1}j_{2}j_{3})$ : product
Permalink:: http://dlmf.nist.gov/34.2.E5
Encodings:: TeX, pMML, png

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

Equivalently,

34.2.6 $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}={(-1)^{{j_{{2}}-m_{{1}}+m_{{3}}}}}\frac{(j_{{1}}+j_{{2}}+m_{{3}})!(j_{{2}}+j_{{3}}-m_{{1}})!}{\Delta(j_{{1}}j_{{2}}j_{{3}})(j_{{1}}+j_{{2}}+j_{{3}}+1)!}\left(\frac{(j_{{1}}+m_{{1}})!(j_{{3}}-m_{{3}})!}{(j_{{1}}-m_{{1}})!(j_{{2}}+m_{{2}})!(j_{{2}}-m_{{2}})!(j_{{3}}+m_{{3}})!}\right)^{{\frac{1}{2}}}\*{\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left(-j_{{1}}-j_{{2}}-j_{{3}}-1,-j_{{1}}+m_{{1}},-j_{{3}}-m_{{3}};-j_{{1}}-j_{{2}}-m_{{3}},-j_{{2}}-j_{{3}}+m_{{1}};1\right)},$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $!$ : $n!$ : factorial, $\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}$ : $3j$ symbol, $j,j_{r}$ : nonnegative integer and $\Delta(j_{1}j_{2}j_{3})$ : product
Permalink:: http://dlmf.nist.gov/34.2.E6
Encodings:: TeX, pMML, png

where $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits$ is defined as in §16.2.

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

§34.2 Definition: Symbol

§34.2 Definition: $3j$ Symbol