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11: 16.24 Physical Applications

… ►

§16.24(iii) $\mathit{3j}$ , $\mathit{6j}$ , and $\mathit{9j}$ Symbols

… ►The coefficients of transformations between different coupling schemes of three angular momenta are related to the Wigner

\mathit{6j}

symbols. …

12: 34.7 Basic Properties: $\mathit{9j}$ Symbol

… ►

34.7.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{13}\\ j_{31}&j_{31}&0\end{Bmatrix}=\frac{(-1)^{j_{12}+j_{21}+j_{13}+j_{31}}}{((2j_{1% 3}+1)(2j_{31}+1))^{\frac{1}{2}}}\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{22}&j_{21}&j_{31}\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.7(i), §34.7 and Ch.34

… ►

34.7.5

\sum_{j^{\prime}}(2j^{\prime}+1)\begin{Bmatrix}j_{11}&j_{12}&j^{\prime}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}\begin{Bmatrix}j_{11}&j_{12}&j^{\prime}\\ j_{23}&j_{33}&j\end{Bmatrix}={(-1)^{2j}}\begin{Bmatrix}j_{21}&j_{22}&j_{23}\\ j_{12}&j&j_{32}\end{Bmatrix}\begin{Bmatrix}j_{31}&j_{32}&j_{33}\\ j&j_{11}&j_{21}\end{Bmatrix}.

ⓘ

Symbols:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol, $\begin{Bmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{l_{1}}&\NVar{l_{2}}&\NVar{l_{3}}\end{Bmatrix}$ : $\mathit{6j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Permalink:: http://dlmf.nist.gov/34.7.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §34.7(vi), §34.7 and Ch.34

13: 16.7 Relations to Other Functions

… ►For

\mathit{3j}

,

\mathit{6j}

,

\mathit{9j}

symbols see Chapter 34. …

14: 34.3 Basic Properties: $\mathit{3j}$ Symbol

… ►Similar conventions apply to all subsequent summations in this chapter. …

15: 34 3j, 6j, 9j Symbols

Chapter 34 $\mathit{3j},\mathit{6j},\mathit{9j}$ Symbols

…

16: 16.4 Argument Unity

… ►See Raynal (1979) for a statement in terms of

\mathit{3j}

symbols (Chapter 34). … ►These series contain

\mathit{6j}

symbols as special cases when the parameters are integers; compare §34.4. …

17: Errata

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Section 34.1

The relation between Clebsch-Gordan and $\mathit{3j}$ symbols was clarified, and the sign of $m_{3}$ was changed for readability. The reference Condon and Shortley (1935) for the Clebsch-Gordan coefficients was replaced by Edmonds (1974) and Rotenberg et al. (1959) and the references for $\mathit{3j}$ , $\mathit{6j}$ , $\mathit{9j}$ symbols were made more precise in §34.1.

… ►

Section 34.1

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.

…

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