3-j symbol

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{2 j_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 $j 1 - j 2 - m 3$ is an integer and making the substitution $m 3 \to - m 3$ :

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

Note that the exponent of the sign factor is always an integer, therefore it remains the same under inversion.

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 3-j symbol is zero unless all these conditions are satisfied:

\begin{align} &m_1 + m_2 + m_3 = 0 \\ &|j_1 - j_2| \le j_3 \le j_1 + j_2 \\ &|m_i| \le j_i \\ &(j_1 + j_2 + j_3) \text{ is an integer (and, moreover, an even integer if } m_1 = m_2 = m_3 = 0 \text{)} \\ &(j_i - m_i) \text{ is an integer} \\ \end{align}

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

References

↑ Regge, T. (1958). "Symmetry Properties of Clebsch-Gordan Coefficients". Nuovo Cimento 10: 544. doi:10.1007/BF02859841.
↑ Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients". SIAM J. Sci. Comput. 25 (4): 1416–1428. doi:10.1137/s1064827503422932.
↑ Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients". SIAM J. Sci. Comput. 25 (4): 1416–1428. doi:10.1137/s1064827503422932.

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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)
Johansson, H.T. "(FASTWIGXJ)". (fast lookup, accurate; C, fortran)

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