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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
\( \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} \equiv \frac{(-1)^{j_1-j_2-m_3}}{\sqrt{2j_3+1}} \langle j_1 m_1 j_2 m_2 | j_3 \, {-m_3} \rangle. \)
Inverse relation
The inverse relation can be found by noting that j1 − j2 − m3 is an integer 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:
\( \begin{pmatrix} j_1 & j_2 & j_3\\ -m_1 & -m_2 & -m_3 \end{pmatrix} = (-1)^{j_1+j_2+j_3} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix}. \)
Regge symmetries also give
\( \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = \begin{pmatrix} j_1 & \frac{j_2+j_3-m_1}{2} & \frac{j_2+j_3+m_1}{2}\\ j_3-j_2 & \frac{j_2-j_3-m_1}{2}-m_3 & \frac{j_2-j_3+m_1}{2}+m_3 \end{pmatrix}. \)
\( \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = (-1)^{j_1+j_2+j_3} \begin{pmatrix} \frac{j_2+j_3+m_1}{2} & \frac{j_1+j_3+m_2}{2} & \frac{j_1+j_2+m_3}{2}\\ j_1 - \frac{j_2+j_3-m_1}{2} & j_2 - \frac{j_1+j_3-m_2}{2} & j_3-\frac{j_1+j_2-m_3}{2} \end{pmatrix}. \)
Regge symmetries account for a total of 72 symmetries.[1] These are best displayed by the definition of a Regge symbol which is a one to one correspondence between it and a 3j symbol and assumes the properties of a semi-magic square[2]
\( R= \begin{array}{|ccc|} \hline -j_1+j_2+j_3 & j_1-j_2+j_3 & j_1+j_2-j_3\\ j_1-m_1 & j_2-m_2 & j_3-m_3\\ j_1+m_1 & j_2+m_2 & j_3+m_3\\ \hline \end{array} \)
whereby the 72 symmetries now correspond to 3! row and 3! column interchanges plus a transposition of the matrix. This can be used to devise an effective storage scheme.[3]
Selection rules
The Wigner 3j is zero unless all these conditions are satisfied:
\( m_1+m_2+m_3=0\, \)
\( j_1+j_2 + j_3\text{ is an integer} \, \text{(or an even integer if} \,m_1=m_2=m_3=0)\, \)
\( |m_i| \le j_i \, \)
\( |j_1-j_2|\le j_3 \le j_1+j_2. \, \)
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
\( (2j+1)\sum_{m_1 m_2} \begin{pmatrix} j_1 & j_2 & j\\ m_1 & m_2 & m \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j'\\ m_1 & m_2 & m' \end{pmatrix} =\delta_{j j'}\delta_{m m'}. \)
\( \sum_{j m} (2j+1) \begin{pmatrix} j_1 & j_2 & j\\ m_1 & m_2 & m \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j\\ m_1' & m_2' & m \end{pmatrix} =\delta_{m_{1} m_1'}\delta_{m_{2} m_2'}. \)
Relation to spherical harmonics
The 3jm symbols give the integral of the products of three spherical harmonics
\( \begin{align} & {} \quad \int Y_{l_1m_1}(\theta,\varphi)Y_{l_2m_2}(\theta,\varphi)Y_{l_3m_3}(\theta,\varphi)\,\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\varphi \\ & = \sqrt{\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}} \begin{pmatrix} l_1 & l_2 & l_3 \\[8pt] 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end{pmatrix} \end{align} \)
with \( l_1, l_2 \) and \( l_3 \) integers.
Relation to integrals of spin-weighted spherical harmonics
Similar relations exist for the spin-weighted spherical harmonics:
\( \begin{align} & {} \quad \int d{\mathbf{\hat n}}\,{}_{s_1} Y_{j_1 m_1}({\mathbf{\hat n}}) \,{}_{s_2} Y_{j_2m_2}({\mathbf{\hat n}})\, {}_{s_3} Y_{j_3m_3}({\mathbf{\hat n}}) \\[8pt] & = \sqrt{\frac{(2j_1+1)(2j_2+1)(2j_3+1)}{4\pi}} \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} \begin{pmatrix} j_1 & j_2 & j_3\\ -s_1 & -s_2 & -s_3 \end{pmatrix} \end{align} \)
Recursion relations
\( \begin{align} & {} \quad -\sqrt{(l_3\mp s_3)(l_3\pm s_3+1)} \begin{pmatrix} l_1 & l_2 & l_3\\ s_1 & s_2 & s_3\pm 1 \end{pmatrix} \\ & = \sqrt{(l_1\mp s_1)(l_1\pm s_1+1)} \begin{pmatrix} l_1 & l_2 & l_3\\ s_1 \pm 1 & s_2 & s_3 \end{pmatrix} +\sqrt{(l_2\mp s_2)(l_2\pm s_2+1)} \begin{pmatrix} l_1 & l_2 & l_3\\ s_1 & s_2 \pm 1 & s_3 \end{pmatrix} \end{align} \)
Asymptotic expressions
For \( l_1\ll l_2,l_3 a non-zero 3-j symbol has
\( \begin{pmatrix} l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end{pmatrix} \approx (-1)^{l_3+m_3} \frac{ d^{l_1}_{m_1, l_3-l_2}(\theta)}{\sqrt{2l_3+1}} \)
where \( \cos(\theta) = -2m_3/(2l_3+1) and d^l_{mn} \) is a Wigner function. Generally a better approximation obeying the Regge symmetry is given by
\( \begin{pmatrix} l_1 & l_2 & l_3\\ m_1 & m_2 & m_3 \end{pmatrix} \approx (-1)^{l_3+m_3} \frac{ d^{l_1}_{m_1, l_3-l_2}(\theta)}{\sqrt{l_2+l_3+1}} \)
where \( \cos(\theta) = (m_2-m_3)/(l_2+l_3+1). \)
Other properties
\( \sum_m (-1)^{j-m} \begin{pmatrix} j & j & J\\ m & -m & 0 \end{pmatrix} = \sqrt{2j+1}~ \delta_{J0} \)
\( \frac{1}{2} \int_{-1}^1 P_{l_1}(x)P_{l_2}(x)P_{l}(x) \, dx = \begin{pmatrix} l & l_1 & l_2 \\ 0 & 0 & 0 \end{pmatrix} ^2 \)
See also
Clebsch–Gordan coefficients
Spherical harmonics
6-j symbol
9-j symbol
References
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External links
Stone, Anthony. "Wigner coefficient calculator".
Volya, A. "Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator". (Numerical)
Stevenson, Paul. "Clebsch-O-Matic". Bibcode:2002CoPhC.147..853S. doi:10.1016/S0010-4655(02)00462-9.
369j-symbol calculator at the Plasma Laboratory of Weizmann Institute of Science (Numerical)
Frederik J Simons: Matlab software archive, the code THREEJ.M
Sage (mathematics software) Gives exact answer for any value of j, m
Johansson, H.T.; Forssén, C. "(WIGXJPF)". (accurate; C, fortran, python)
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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