Vignerov 6-j simbol definisao je 1940. Eugen Paul Vigner. Definišu se preko sume produkata 3-j simbola:
![{\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}=\sum _{m_{i}}(-1)^{S}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&-m_{3}\end{pmatrix}}\times }](https://wikimedia.org/api/rest_v1/media/math/render/svg/52fec32aa027d9a417d2d080d5e31bf4f9ea0e50)
![{\displaystyle \quad \times {\begin{pmatrix}j_{1}&j_{5}&j_{6}\\-m_{1}&m_{5}&m_{6}\end{pmatrix}}{\begin{pmatrix}j_{4}&j_{5}&j_{3}\\m_{4}&-m_{5}&m_{3}\end{pmatrix}}{\begin{pmatrix}j_{4}&j_{2}&j_{6}\\-m_{4}&-m_{2}&-m_{6}\end{pmatrix}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76144c53681db6e9aa0489a2d239588c881cb1cb)
sa fazom
. Sumira se preko svih šest mi, a selekciona pravila 3jm ograničavaju sumu. Povezani su sa Rakovim koeficijentima:
![{\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}=(-1)^{j_{1}+j_{2}+j_{4}+j_{5}}W(j_{1}j_{2}j_{5}j_{4};j_{3}j_{6}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8e0f68de73e6fe8fd65da8280d5df765f99d8aa)
Vignerov 6-j simbol može da se prikaže preko konačne sume:
![{\displaystyle {\begin{Bmatrix}a&b&c\\d&e&f\end{Bmatrix}}=(-1)^{a+c+d+f}{\frac {\Delta (abc)\Delta (bdf)}{\Delta (aef)\Delta (cde)}}\times }](https://wikimedia.org/api/rest_v1/media/math/render/svg/78e1735bc0945863b5120d5713e5b894fd1dbe0a)
![{\displaystyle \quad \times \sum _{n}(-1)^{n}{\frac {(a-b+d+e-n)!(-b+c+e+f-n)!(a+c+d+f+1-n)!}{n!(a-b+c-n)!(-b+d+f-n)!(a+e+f+1-n)!(c+d+e+1-n)!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c884052c3f9c420f633a150fe660050f9862babe)
a tu se sumacija odvija po svim n sve dok faktorijeli ne postanu negativni.
Pri tome funkcija
je jednaka 1 ako je zadovoljena relacija triangularnosti za
, a 0 ako nije definisana je sledećim izrazom:
![{\displaystyle \Delta (a,b,c)=[(a+b-c)!(a-b+c)!(-a+b+c)!/(a+b+c+1)!]^{1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/608c37a1b2ad80b357f61e667e5fd2c8b97bca03)
Vignerovi simboli zadovoljavaju relacije ortogonalnosti:
![{\displaystyle \sum _{j_{3}}(2j_{3}+1){\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}{\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}'\end{Bmatrix}}={\frac {\delta _{j_{6}^{}j_{6}'}}{2j_{6}+1}}\Delta (j_{1},j_{5},j_{6})\Delta (j_{4},j_{2},j_{6}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c46ce8418ac7899f9e28c1e97ffa239d25b40f71)
U slučaju da je
dobija se:
![{\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&0\end{Bmatrix}}={\frac {\delta _{j_{2},j_{4}}\delta _{j_{1},j_{5}}}{\sqrt {(2j_{1}+1)(2j_{2}+1)}}}(-1)^{j_{1}+j_{2}+j_{3}}\Delta (j_{1},j_{2},j_{3}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52b5c708b51dc81019a5e0dbae78c6628052f7b9)
Pri tome funkcija
je jednaka 1 ako je zadovoljena relacija triangularnosti za
, a 0 ako nije.
Vignerov 6-j simbol invarijantan je na permutaciju dve kolone, tako da vredi:
![{\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}={\begin{Bmatrix}j_{2}&j_{1}&j_{3}\\j_{5}&j_{4}&j_{6}\end{Bmatrix}}={\begin{Bmatrix}j_{1}&j_{3}&j_{2}\\j_{4}&j_{6}&j_{5}\end{Bmatrix}}={\begin{Bmatrix}j_{3}&j_{2}&j_{1}\\j_{6}&j_{5}&j_{4}\end{Bmatrix}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1c2b640518187af11cc7957f4f1037be51394b0)
Vignerov 6-j simbol invarijantan je i na zamenu dva argumenta u gornjim kolonama sa dva argumenta u donjim kolonama:
![{\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}={\begin{Bmatrix}j_{4}&j_{5}&j_{3}\\j_{1}&j_{2}&j_{6}\end{Bmatrix}}={\begin{Bmatrix}j_{1}&j_{5}&j_{6}\\j_{4}&j_{2}&j_{3}\end{Bmatrix}}={\begin{Bmatrix}j_{4}&j_{2}&j_{6}\\j_{1}&j_{5}&j_{3}\end{Bmatrix}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ef51ee016f847891f5755c6f6511550a96b1d17)
Vignerov 6-j simbol
![{\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44d296c2586f084defaa47a6226afe252dfccafc)
je nula sem ako j1, j2 i j3 ne zadovoljavaju triangularne uslove:
![{\displaystyle j_{1}=|j_{2}-j_{3}|,\ldots ,j_{2}+j_{3}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a82ed50699e6d41dcc4cf94c0bd2170458c7584)
Asimptotska formula je razvijena za slučaj kada svih šest kvantnih brojeva j1, ..., j6 teži velikim brojevima. Asimptotska formula je dana sa:
![{\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}\sim {\frac {1}{\sqrt {12\pi |V|}}}\cos {\left(\sum _{i=1}^{6}J_{i}\theta _{i}+{\frac {\pi }{4}}\right)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2de21a4c381832111302fc4411a727ed0dc1592d)
- Abramowitz, Milton; Stegun, Irene A., ur. (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. ISBN 9780486612720.
- Edmonds, A. R. (1957). Angular Momentum in Quantum Mechanics. Princeton, New Jersey: Princeton University Press. ISBN 9780-691-07912-7.
- Messiah, Albert (1981). Quantum Mechanics. II (12th izd.). New York: North Holland Publishing. ISBN 9780-7204-0045-8.