Wigner 3-j symbols

In quantum mechanics, the Wigner 3-j symbols, also called 3j or 3-jm symbols, are related to Clebsch–Gordan coefficients through

$\begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} \equiv \frac{(-1)^{j_1-j_2-m_3}}{\sqrt{2j_3%2B1}} \langle j_1 m_1 j_2 m_2 | j_3 \, {-m_3} \rangle.$

1 Inverse relation
2 Symmetry properties
3 Selection rules
4 Scalar invariant
5 Orthogonality relations
6 Relation to spherical harmonics
7 Relation to integrals of spin-weighted spherical harmonics
8 Recursion relations
9 Asymptotic expressions
10 Other properties
11 See also
12 References
13 External links

Inverse relation

The inverse relation can be found by noting that j₁ - j₂ - m₃ is an integer number and making the substitution $m_3 \rightarrow -m_3$

$\langle j_1 m_1 j_2 m_2 | j_3 m_3 \rangle = (-1)^{-j_1%2Bj_2-m_3}\sqrt{2j_3%2B1} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & -m_3 \end{pmatrix}.$

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:

$\begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = (-1)^{j_1%2Bj_2%2Bj_3} \begin{pmatrix} j_2 & j_1 & j_3\\ m_2 & m_1 & m_3 \end{pmatrix} = (-1)^{j_1%2Bj_2%2Bj_3} \begin{pmatrix} j_1 & j_3 & j_2\\ m_1 & m_3 & m_2 \end{pmatrix}.$

Changing the sign of the $m$ quantum numbers also gives a phase:

$\begin{pmatrix} j_1 & j_2 & j_3\\ -m_1 & -m_2 & -m_3 \end{pmatrix} = (-1)^{j_1%2Bj_2%2Bj_3} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix}.$

Regge symmetries also give

$\begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = \begin{pmatrix} j_1 & \frac{j_2%2Bj_3-m_1}{2} & \frac{j_2%2Bj_3%2Bm_1}{2}\\ j_3-j_2 & \frac{j_2-j_3-m_1}{2}-m_3 & \frac{j_2-j_3%2Bm_1}{2}%2Bm_3 \end{pmatrix}.$

$\begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = (-1)^{j_1%2Bj_2%2Bj_3} \begin{pmatrix} \frac{j_2%2Bj_3%2Bm_1}{2} & \frac{j_1%2Bj_3%2Bm_2}{2} & \frac{j_1%2Bj_2%2Bm_3}{2}\\ j_1 - \frac{j_2%2Bj_3-m_1}{2} & j_2 - \frac{j_1%2Bj_3-m_2}{2} & j_3-\frac{j_1%2Bj_2-m_3}{2} \end{pmatrix}.$

Selection rules

The Wigner 3j is zero unless all these conditions are satisfied:

$m_1%2Bm_2%2Bm_3=0\,$

$j_1%2Bj_2 %2B j_3\text{ is an integer} \, \text{(or an even integer if} \,m_1=m_2=m_3=0)\,$

$|m_i| \le j_i \,$

$|j_1-j_2|\le j_3 \le j_1%2Bj_2. \,$

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

$(2j%2B1)\sum_{m_1 m_2} \begin{pmatrix} j_1 & j_2 & j\\ m_1 & m_2 & m \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j'\\ m_1 & m_2 & m' \end{pmatrix} =\delta_{j j'}\delta_{m m'}.$

$\sum_{j m} (2j%2B1) \begin{pmatrix} j_1 & j_2 & j\\ m_1 & m_2 & m \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j\\ m_1' & m_2' & m \end{pmatrix} =\delta_{m_1 m_1'}\delta_{m_2 m_2'}.$

Relation to spherical harmonics

The 3jm symbols give the integral of the products of three spherical harmonics

$\begin{align} & {} \quad \int Y_{l_1m_1}(\theta,\varphi)Y_{l_2m_2}(\theta,\varphi)Y_{l_3m_3}(\theta,\varphi)\,\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\varphi \\ & = \sqrt{\frac{(2l_1%2B1)(2l_2%2B1)(2l_3%2B1)}{4\pi}} \begin{pmatrix} l_1 & l_2 & l_3 \\[8pt] 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end{pmatrix} \end{align}$

with $l_1$ , $l_2$ and $l_3$ integers.

Relation to integrals of spin-weighted spherical harmonics

$\begin{align} & {} \quad \int d{\mathbf{\hat n}} {}_{s_1} Y_{j_1 m_1}({\mathbf{\hat n}}) {}_{s_2} Y_{j_2m_2}({\mathbf{\hat n}}) {}_{s_3} Y_{j_3m_3}({\mathbf{\hat n}}) \\[8pt] & = \sqrt{\frac{(2j_1%2B1)(2j_2%2B1)(2j_3%2B1)}{4\pi}} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j_3\\ -s_1 & -s_2 & -s_3 \end{pmatrix} \end{align}$

Recursion relations

$\begin{align} & {} \quad -\sqrt{(l_3\mp s_3)(l_3\pm s_3%2B1)} \begin{pmatrix} l_1 & l_2 & l_3\\ s_1 & s_2 & s_3\pm 1 \end{pmatrix} \\ & = \sqrt{(l_1\mp s_1)(l_1\pm s_1%2B1)} \begin{pmatrix} l_1 & l_2 & l_3\\ s_1 \pm 1 & s_2 & s_3 \end{pmatrix} %2B\sqrt{(l_2\mp s_2)(l_2\pm s_2%2B1)} \begin{pmatrix} l_1 & l_2 & l_3\\ s_1 & s_2 \pm 1 & s_3 \end{pmatrix} \end{align}$

Asymptotic expressions

For $l_1\ll l_2,l_3$ a non-zero 3-j symbol has

$\begin{pmatrix} l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end{pmatrix} \approx (-1)^{l_3%2Bm_3} \frac{ d^{l_1}_{m_1, l_3-l_2}(\theta)}{\sqrt{2l_3%2B1}}$

where $\cos(\theta) = -2m_3/(2l_3%2B1)$ and $d^l_{mn}$ is a Wigner function. Generally a better approximation obeying the Regge symmetry is given by

$\begin{pmatrix} l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end{pmatrix} \approx (-1)^{l_3%2Bm_3} \frac{ d^{l_1}_{m_1, l_3-l_2}(\theta)}{\sqrt{l_2%2Bl_3%2B1}}$

where $\cos(\theta) = (m_2-m_3)/(l_2%2Bl_3%2B1)$ .

Other properties

$\sum_m (-1)^{j-m} \begin{pmatrix} j & j & J\\ m & -m & 0 \end{pmatrix} = \sqrt{2j%2B1}~ \delta_{J0}$

$\frac{1}{2} \int_{-1}^1 P_{l_1}(x)P_{l_2}(x)P_{l}(x) \, dx = \begin{pmatrix} l & l_1 & l_2 \\ 0 & 0 & 0 \end{pmatrix} ^2$

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External links

Stone, Anthony. "Wigner coefficient calculator". http://www-stone.ch.cam.ac.uk/wigner.shtml. (Gives exact answer)
Volya, A.. "Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator". http://www.volya.net/vc/vc.php. (Numerical)
Stevenson, Paul. Clebsch-O-Matic. Bibcode 2002CoPhC.147..853S. doi:10.1016/S0010-4655(02)00462-9. http://personal.ph.surrey.ac.uk/~phs3ps/cleb.html.
369j-symbol calculator at the Plasma Laboratory of Weizmann Institute of Science (Numerical)
Frederik J Simons: Matlab software archive, the code THREEJ.M