维格纳 3-j 符号

维格纳 3-j 符号也称3-j符号，与量子力学中的克莱布希－高登系数有密切关系。

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\equiv {\frac {(-1)^{j_{1}-j_{2}-m_{3}}}{\sqrt {2j_{3}+1}}}\langle j_{1}m_{1}j_{2}m_{2}|j_{3}\,{-m_{3}}\rangle .}$

反演关系

${\displaystyle \langle j_{1}m_{1}j_{2}m_{2}|j_{3}m_{3}\rangle =(-1)^{-j_{1}+j_{2}-m_{3}}{\sqrt {2j_{3}+1}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&-m_{3}\end{pmatrix}}.}$ 

对称性

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}={\begin{pmatrix}j_{2}&j_{3}&j_{1}\\m_{2}&m_{3}&m_{1}\end{pmatrix}}={\begin{pmatrix}j_{3}&j_{1}&j_{2}\\m_{3}&m_{1}&m_{2}\end{pmatrix}}.}$ 

位相

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}=(-1)^{j_{1}+j_{2}+j_{3}}{\begin{pmatrix}j_{2}&j_{1}&j_{3}\\m_{2}&m_{1}&m_{3}\end{pmatrix}}=(-1)^{j_{1}+j_{2}+j_{3}}{\begin{pmatrix}j_{1}&j_{3}&j_{2}\\m_{1}&m_{3}&m_{2}\end{pmatrix}}.}$ 

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\-m_{1}&-m_{2}&-m_{3}\end{pmatrix}}=(-1)^{j_{1}+j_{2}+j_{3}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}.}$ 

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}={\begin{pmatrix}j_{1}&{\frac {j_{2}+j_{3}-m_{1}}{2}}&{\frac {j_{2}+j_{3}+m_{1}}{2}}\\j_{3}-j_{2}&{\frac {j_{2}-j_{3}-m_{1}}{2}}-m_{3}&{\frac {j_{2}-j_{3}+m_{1}}{2}}+m_{3}\end{pmatrix}}.}$ 

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}=(-1)^{j_{1}+j_{2}+j_{3}}{\begin{pmatrix}{\frac {j_{2}+j_{3}+m_{1}}{2}}&{\frac {j_{1}+j_{3}+m_{2}}{2}}&{\frac {j_{1}+j_{2}+m_{3}}{2}}\\j_{1}-{\frac {j_{2}+j_{3}-m_{1}}{2}}&j_{2}-{\frac {j_{1}+j_{3}-m_{2}}{2}}&j_{3}-{\frac {j_{1}+j_{2}-m_{3}}{2}}\end{pmatrix}}.}$ 

里奇对称性包括72类对称性 symmetries.^{[1] [2]}

${\displaystyle R={\begin{array}{|ccc|}\hline -j_{1}+j_{2}+j_{3}&j_{1}-j_{2}+j_{3}&j_{1}+j_{2}-j_{3}\\j_{1}-m_{1}&j_{2}-m_{2}&j_{3}-m_{3}\\j_{1}+m_{1}&j_{2}+m_{2}&j_{3}+m_{3}\\\hline \end{array}}}$ 

参考文献

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