34 3j, 6j, 9j Symbols Notation 34 3j, 6j, 9j Symbols 34.2 Definition:

\mathit{3j}

§34.1 Special Notation

ⓘ

Keywords:: Clebsch–Gordan coefficients, Wigner $\mathit{3j},\mathit{6j},\mathit{9j}$ symbols, angular momentum coupling coefficients, $\mathit{9j}$ symbols, notation, $\mathit{6j}$ symbols, $\mathit{3j}$ symbols
Referenced by:: Erratum (V1.0.14) for Section 34.1, Erratum (V1.0.15) for Section 34.1, Erratum (V1.0.15) for Section 34.1
Permalink:: http://dlmf.nist.gov/34.1
Clarification (effective with 1.0.14):: The reference for Clebsch-Gordan coefficients, Condon and Shortley (1935), was replaced by Edmonds (1974) and Rotenberg et al. (1959). The references for $\mathit{3j}$ , $\mathit{6j}$ , $\mathit{9j}$ symbols were made more precise.
See also:: Annotations for Ch.34

(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 $\mathit{3j},\mathit{6j},\mathit{9j}$ symbols, respectively,

	$\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix},$
	$\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\ l_{1}&l_{2}&l_{3}\end{Bmatrix},$
	$\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}.$

An often used alternative to the $\mathit{3j}$ symbol is the Clebsch--Gordan coefficient

34.1.1		$\left(j_{1}\;m_{1}\;j_{2}\;m_{2}\|j_{1}\;j_{2}\;j_{3}\,\,m_{3}\right)=(-1)^{j_{% 1}-j_{2}+m_{3}}(2j_{3}+1)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&-m_{3}\end{pmatrix};$
ⓘ 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 and $j,j_{r}$ : non-negative integers or non-negative integers plus one half. Sources: Edmonds (1974, p. 46, Eq. (3.7.3)); Rotenberg et al. (1959, p. 1, Eq. (1.1a)) Permalink: http://dlmf.nist.gov/34.1.E1 Encodings: TeX, pMML, png Clarification (effective with 1.0.15): A wording change reflects that the Clebsch–Gordan coefficients are an alternative formulation of angular momentum problems, rather than alternative notation for the $\mathit{3j}$ . The sign of $m_{3}$ was changed for clarity. See also: Annotations for §34.1 and Ch.34

see Edmonds (1974, p. 46, Eq. (3.7.3)) and Rotenberg et al. (1959, p. 1, Eq. (1.1a)). For other notations for $\mathit{3j}$ , $\mathit{6j}$ , $\mathit{9j}$ symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).

\mathit{3j}

Symbol