3-j symbol

In quantum mechanics, the Wigner 3-jm symbols, also called 3j 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 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 all these conditions are satisfied:

$m_{1}+m_{2}+m_{3}=0\,$

$j_{1}+j_{2}+j_{3}\,$ is integer

$|m_{i}|\leq j_{i}$

$|j_{1}-j_{2}|\leq j_{3}\leq j_{1}+j_{2}$.

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

$(2j+1)\sum _{m_{1}m_{2}}{\begin{pmatrix}j_{1}&j_{2}&j\\m_{1}&m_{2}&m\end{pmatrix}}{\begin{pmatrix}j_{1}&j_{2}&j'\\m_{1}&m_{2}&m'\end{pmatrix}}=\delta _{jj'}\delta _{mm'}.$

$\sum _{jm}(2j+1){\begin{pmatrix}j_{1}&j_{2}&j\\m_{1}&m_{2}&m\end{pmatrix}}{\begin{pmatrix}j_{1}&j_{2}&j\\m_{1}'&m_{2}'&m\end{pmatrix}}=\delta _{m_{1}m_{1}'}\delta _{m_{2}m_{2}'}.$

Relation to spherical harmonics

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

$\int Y_{l_{1}m_{1}}(\theta ,\varphi )Y_{l_{2}m_{2}}(\theta ,\varphi )Y_{l_{3}m_{3}}(\theta ,\varphi )\,\sin \theta \,\mathrm {d} \theta \,\mathrm {d} \varphi ={\sqrt {\frac {(2l_{1}+1)(2l_{2}+1)(2l_{3}+1)}{4\pi }}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\0&0&0\end{pmatrix}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}$

with $l_{1}$, $l_{2}$ and $l_{3}$ integers.

Relation to integrals of spin-weighted spherical harmonics

$\int d{\mathbf {\hat {n}} }{}_{s_{1}}Y_{j_{1}m_{1}}({\mathbf {\hat {n}} }){}_{s_{2}}Y_{j_{2}m_{2}}({\mathbf {\hat {n}} }){}_{s_{3}}Y_{j_{3}m_{3}}({\mathbf {\hat {n}} })=(-1)^{m_{1}+s_{1}}{\sqrt {\frac {(2j_{1}+1)(2j_{2}+1)(2j_{3}+1)}{4\pi }}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\-s_{1}&-s_{2}&-s_{3}\end{pmatrix}}$

This should be checked for phase conventions of the harmonics.

Other properties

$\sum _{m}(-1)^{j+m}{\begin{pmatrix}j&j&J\\m&-m&0\end{pmatrix}}={\sqrt {\frac {2j+1}{2J+1}}}\delta _{J0}$

${\frac {1}{2}}\int _{-1}^{1}dxP_{l_{1}}(x)P_{l_{2}}(x)P_{l}(x)={\begin{pmatrix}l&l_{1}&l_{2}\\0&0&0\end{pmatrix}}^{2}$

See also

• Clebsch-Gordan coefficients
• Spherical harmonics
• 6-j symbol
• 9-j symbol
• 12-j symbol
• 15-j symbol

References

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• Maximon, Leonard C. (2010), "3j,6j,9j Symbols", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
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• Bickerstaff, R. P.; Butler, P. H.; Butts, M. B.; Haase, R. w.; Reid, M. F. (1982). "3jm and 6j tables for some bases of SU₆ and SU₃". J. Phys. A. 15: 1087. doi:10.1088/0305-4470/15/4/014.
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External links

• Anthony Stone’s Wigner coefficient calculator (Gives exact answer)
• Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator (Numerical)
• 369j-symbol calculator at the Plasma Laboratory of Weizmann Institute of Science (Numerical)