Vignerov 3-j simbol, takođe zvan i 3j simbol ili 3-jm simbol povezan je sa Klebš-Gordanovim koeficijentima:
![{\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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51eec4610293bc18ac98f2ef4a7b24ad29970adf)
Vignerov 3-j simbol je invarijantan u slučaju parnih permutacija stupaca (kolona):
![{\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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66272830f76fd77b50b76b3b0be765b9bb90b75c)
U slučaju neparne permutacije kolona dobija se fazni faktor:
![{\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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/783d1bd3a55dd8401a76f9c345f5db5493b037c6)
Promjenom znaka
brojeva dobija se fazni faktor:
Postoje i 72 Regeove simetrije, koje daju:
![{\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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b4094f8a6641178061242908bddea01bece6b5)
![{\displaystyle (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'}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d7ce072285b977c5e9fa3e4f5834e2aec947420)
![{\displaystyle \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}'}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6702e86e215c9d44e068dd9dadc68a82ad0c6a6)
![{\displaystyle \sum _{m_{1}m_{2}m_{3}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd388507bd7412ca773e669d216f3ed9241cab52)
Inverzna relacija dobija se supstitucijom
:
![{\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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31334918e896fe6bec13b195d3a2843df3ff2e16)
Sledeći produkt tri rotaciona stanja sa 3-j simbolom je ivarijantan na rotacije:
![{\displaystyle \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}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f5692f1a06ff12601a0595253248f734df0c276)
Vignerov 3-j simbol nije jednak 0 samo ako su zadovoljena sledeća selekciona pravila:
![{\displaystyle m_{1}+m_{2}+m_{3}=0\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4ab0f71d428c4238df3569331ded8764fe4cc10)
celi broj
![{\displaystyle |m_{i}|\leq j_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b434136057041ccd0e71569c13621bc628b8136)
.
Integral tri sferna harmonika dat je preko 3-jm simbola:
![{\displaystyle {\begin{aligned}&{}\quad \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}\\[8pt]0&0&0\end{pmatrix}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b8b5fe21a4c05360c71ed4b066d9c40e3fb127d)
gde su
,
and
celi brojevi.
Sličan izraz postoji za spinske sferne harmonike:
![{\displaystyle {\begin{aligned}&{}\quad \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}} })\\[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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73769f2081a32507cf0a8e4c6f588dd48bb17779)
Rekurzivne relacije za
koeficijente:
![{\displaystyle {\begin{aligned}&{}\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25766e977593aa31de13fbf9f7c767a4e54e54cf)
Rekurzivne relacije za
koeficijente:
![{\displaystyle {\begin{aligned}&{}\quad (2j_{1}+1)\left((j_{2}(j_{2}+1)-j_{3}(j_{3}+1))m_{1}-j_{1}(j_{1}+1)(m_{3}-m_{2})\right){\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\\&=(j_{1}+1)\left(j_{1}^{2}-(j_{2}-j_{3})^{2}\right)^{\frac {1}{2}}\left((j_{2}+j_{3}+1)^{2}-j_{1}^{2}\right)^{\frac {1}{2}}\left(j_{1}^{2}-m_{1}^{2}\right)^{\frac {1}{2}}{\begin{pmatrix}j_{1}-1&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\\&+j_{1}\left((j_{1}+1)^{2}-(j_{2}-j_{3})^{2}\right)^{\frac {1}{2}}\left((j_{2}+j_{3}+1)^{2}-(j_{1}+1)^{2}\right)^{\frac {1}{2}}\left((j_{1}+1)^{2}-m_{1}^{2}\right)^{\frac {1}{2}}{\begin{pmatrix}j_{1}+1&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f00a8dbc90931826f766cecb0f1a7eabe9559fad)
Za
veće od nula 3-j simbol je:
![{\displaystyle {\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\approx (-1)^{l_{3}+m_{3}}{\frac {d_{m_{1},l_{3}-l_{2}}^{l_{1}}(\theta )}{\sqrt {2l_{3}+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b1c831c49ba1930aef60eef22a472e51d1ebdc)
gde je
i
je mala Vignerova funkcija. Bolja aproksimacija dobija se pomoću Rege simetrija:
![{\displaystyle {\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\approx (-1)^{l_{3}+m_{3}}{\frac {d_{m_{1},l_{3}-l_{2}}^{l_{1}}(\theta )}{\sqrt {l_{2}+l_{3}+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/242ea5a3def1d03bdc378cdfbd11aeb1b43e26a3)
gde je
.
![{\displaystyle \sum _{m}(-1)^{j-m}{\begin{pmatrix}j&j&J\\m&-m&0\end{pmatrix}}={\sqrt {2j+1}}~\delta _{J0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65ecc879e564bf6528043f476f61436a9385f545)
![{\displaystyle {\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99813fa0404af70b669824a8148974a21d9b8d95)
Opšti izraz za Vignerov 3-j simbol je podosta komplikovan:
![{\displaystyle {\begin{matrix}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\-m_{1}&-m_{2}&-m_{3}\end{pmatrix}}=&{\frac {{(j_{1}+j_{2}-j_{3})!}{(j_{1}-j_{2}+j_{3})!}{(-j_{1}+j_{2}+j_{3})!}}{(j_{1}+j_{2}+j_{3}+1)!}}\times \\&[{(j_{1}+m_{1})!}{(j_{1}-m_{1})!}{(j_{2}+m_{2})!}{(j_{2}-m_{2})!}{(j_{3}+m_{3})!}{(j_{3}-m_{3})!}]^{\frac {1}{2}}\times \\&\sum _{z=-\infty }^{\infty }{\frac {(-1)^{z+j_{1}+j_{2}-m_{3}}}{{z!}{(j_{1}+j_{2}-j_{3}-z)!}{(j_{1}-m_{1}-z)!}{(j_{2}-m_{2}-z)!}{(j_{3}-j_{2}+m_{1}+z)!}{(j_{3}-j_{1}-m_{2}+z)!}}}\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a58798f257cf200edf91ad577a6416e2cdf8043f)
![{\displaystyle {\begin{pmatrix}j&j&0\\m&-m&0\end{pmatrix}}={\frac {(-1)^{j-m}}{(2j+1)^{\frac {1}{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae90a9efa3c434a2c22b2d819ab83661f75e4e1)
![{\displaystyle {\begin{pmatrix}j&j&1\\m&-m&0\end{pmatrix}}=(-1)^{j-m}{\frac {2m}{\left(2j(2j+1)(2j+2)\right)^{\frac {1}{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f21efc045b8db500362feb0640783a08a2321995)
Za
:
![{\displaystyle {\begin{pmatrix}j+{\frac {1}{2}}&j&{\frac {1}{2}}\\m&-m-{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}=(-1)^{j-m-{\frac {1}{2}}}[{\frac {j-m-{\frac {1}{2}}}{(2j+1)(2j+2)}}]^{\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/177df26a9c493c92673af96dc1c8a088fae778cb)
Za
:
![{\displaystyle {\begin{pmatrix}j+{\frac {1}{2}}&j&{\frac {1}{2}}\\m&-m-{\frac {1}{2}}&{\frac {1}{2}}\end{pmatrix}}=(-1)^{j-m-{\frac {1}{2}}}[{\frac {j-m-{\frac {1}{2}}}{(2j+1)(2j+2)}}]^{\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/177df26a9c493c92673af96dc1c8a088fae778cb)
Za
:
![{\displaystyle (-1)^{j-m+{\frac {1}{2}}}{\begin{pmatrix}j_{1}&j&{\frac {3}{2}}\\m&-m-m_{3}&m_{3}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddff9abb912fe64dde659d8cdd986c94af72419a)
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Za
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![{\displaystyle (-1)^{j-m}{\begin{pmatrix}j_{1}&j&2\\m&-m-m_{3}&m_{3}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69d5537b0a4c190099f1aeb664a756fb64772312)
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- 3j, 6j i 9j simboli
- Abramowitz, Milton; Stegun, Irene A., eds. , Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. . New York: Dover. 1965. ISBN 978-0-486-61272-0.
- Edmonds, A. R., Angular Momentum in Quantum Mechanics, Princeton. . New Jersey: Princeton University Press. 1957. ISBN 978-0-691-07912-7.
- Messiah, Albert , Quantum Mechanics (Volume II) (12th ed.). . New York: North Holland Publishing. 1981. ISBN 978-0-7204-0045-8.