§34.1 Special Notation

Defines:: $\left(j_{1}m_{1}j_{2}m_{2}|j_{1}j_{2}j_{3}-m_{3}\right)$ : Clebsch–Gordan coefficient
Keywords:: Clebsch–Gordan coefficients, Wigner $3j,6j,9j$ symbols, angular momentum coupling coefficients, $9j$ symbols, $6j$ symbols, $3j$ symbols
Permalink:: http://dlmf.nist.gov/34.1

(For other notation see Notation for the Special Functions.)

$2j_{{1}},2j_{{2}},2j_{{3}},2l_{1},2l_{2},2l_{3}$	nonnegative integers.
$r,s,t$	nonnegative integers.

The main functions treated in this chapter are the Wigner $3j,6j,9j$ symbols, respectively,

$\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix},$

$\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix},$

$\begin{Bmatrix}j_{{11}}&j_{{12}}&j_{{13}}\\ j_{{21}}&j_{{22}}&j_{{23}}\\ j_{{31}}&j_{{32}}&j_{{33}}\end{Bmatrix}.$

The most commonly used alternative notation for the $3j$ symbol is the Clebsch–Gordan coefficient

$\left(j_{{1}}m_{{1}}j_{{2}}m_{{2}}|j_{{1}}j_{{2}}j_{{3}}-m_{{3}}\right)=(-1)^{{j_{{1}}-j_{{2}}-m_{{3}}}}(2j_{3}+1)^{{\frac{1}{2}}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix};$

see Condon and Shortley (1935). For other notations see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).