3j-symbol

In physics and mathematics, Wigner 3-jm symbols, also called 3j symbols, are related to the Clebsch-Gordan coefficients of the groups SU(2) and SO(3) through

\begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} \equiv \frac{(-1)^{j_1-j_2-m_3}}{\sqrt{2j_3+1}} \langle j_1 m_1 j_2 m_2 | j_3 \, {-m_3} \rangle. $$ The 3j symbols show more symmetry in permutation of the labels than the corresponding Clebsch-Gordan coefficients. Exactly as is true for the Clebsch-Gordan coefficients, the j-values are positive and either integral: (0, 1, 2,..) or half-integral: (1/2, 3/2, 5/2, ...).

Note on phases
All 3j symbols are real, which means that the value of n in overall phase factors of the kind:

(-1)^n = e^{i\pi n}\; $$ must be integral, otherwise exp[i&pi;n] would not be on the real axis in the complex plane. For half-integral n the phase factor is purely imaginary and a 3j symbol containing the factor would be too, so that half-integral values do not appear as powers of &minus;1. The overall powers of &minus;1 are odd or even integral numbers. The following relation (n integral) seems  not to be known to everyone contributing to the corresponding article on Wikipedia:

(-1)^n = (-1)^{-n}= \frac{1}{(-1)^n}= \begin{cases} \;\;1 &\hbox{if}\quad n\quad \hbox{even}\\ -1   &\hbox{if}\quad n\quad \hbox{odd}\\ \end{cases} $$ Note that the expression

j \pm m\quad\hbox{with}\quad -j \le m \le j $$ is necessarily integral, since m runs in unit steps and j&minus;mmax =  j+mmin =0. Likewise j1&minus;j2&plusmn;m3 is integral [&plusmn;m3 = &plusmn;(&minus;m1&minus;m2)].

Inverse relation
The inverse relation&mdash;the Clebsch-Gordan coefficient given by a 3j symbol&mdash;can be found by noting that j1 - j2 - m3 is an integral number and making the substitution $$ m_3 \rightarrow -m_3 $$

\langle j_1 m_1 j_2 m_2 | j_3 m_3 \rangle = (-1)^{j_1-j_2+m_3}\sqrt{2j_3+1} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & -m_3 \end{pmatrix}. $$

Symmetry properties
The symmetry properties of 3j symbols are more convenient than those of Clebsch-Gordan coefficients. A 3j symbol is invariant under an even permutation of its columns:

\begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = \begin{pmatrix} j_2 & j_3 & j_1\\ m_2 & m_3 & m_1 \end{pmatrix} = \begin{pmatrix} j_3 & j_1 & j_2\\ m_3 & m_1 & m_2 \end{pmatrix}. $$ An odd permutation of the columns gives a phase factor:

\begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = (-1)^{j_1+j_2+j_3} \begin{pmatrix} j_2 & j_1 & j_3\\ m_2 & m_1 & m_3 \end{pmatrix} = (-1)^{j_1+j_2+j_3} \begin{pmatrix} j_1 & j_3 & j_2\\ m_1 & m_3 & m_2 \end{pmatrix}. $$ Changing the sign of the $$m$$ quantum numbers also gives a phase:

\begin{pmatrix} j_1 & j_2 & j_3\\ -m_1 & -m_2 & -m_3 \end{pmatrix} = (-1)^{j_1+j_2+j_3} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix}. $$

Selection rules
The Wigner 3j is zero unless $$m_1+m_2+m_3=0$$, $$j_1+j_2 + j_3$$ is integer, $$|m_i| \le j_i$$ and $$|j_1-j_2|\le j_3 \le j_1+j_2$$.

Scalar invariant
The contraction of the product of three rotational states with a 3j symbol,

\sum_{m_1=-j_1}^{j_1} \sum_{m_2=-j_2}^{j_2} \sum_{m_3=-j_3}^{j_3} |j_1 m_1\rangle |j_2 m_2\rangle |j_3 m_3\rangle \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix}, $$ is invariant under rotations.

Orthogonality Relations
$$ (2j+1)\sum_{m_1 m_2} \begin{pmatrix} j_1 & j_2 & j\\ m_1 & m_2 & m \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j'\\ m_1 & m_2 & m' \end{pmatrix} =\delta_{j j'}\delta_{m m'}. $$

$$ \sum_{j m} (2j+1) \begin{pmatrix} j_1 & j_2 & j\\ m_1 & m_2 & m \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j\\ m_1' & m_2' & m \end{pmatrix} =\delta_{m_1 m_1'}\delta_{m_2 m_2'}. $$