Vignerov 9-j simbol definisao je Eugen Vigner kao sumu preko 6-j simbola :
{
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9
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=
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{
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{
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{
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{\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}=\sum _{x}(-1)^{2x}(2x+1){\begin{Bmatrix}j_{1}&j_{4}&j_{7}\\j_{8}&j_{9}&x\end{Bmatrix}}{\begin{Bmatrix}j_{2}&j_{5}&j_{8}\\j_{4}&x&j_{6}\end{Bmatrix}}{\begin{Bmatrix}j_{3}&j_{6}&j_{9}\\x&j_{1}&j_{2}\end{Bmatrix}}}
uredi
Vezanjem dva ugaona momenta dobijaju se Klebš-Gordanovi koeficijenti . Tri ugaona momenta možemo da vežemo na nekoliko načina. Četiri ugaona momenta možemo da vežemo isto tako na više načina. Npr.
j
1
{\displaystyle \mathbf {j} _{1}}
j
2
{\displaystyle \mathbf {j} _{2}}
j
4
{\displaystyle \mathbf {j} _{4}}
j
5
{\displaystyle \mathbf {j} _{5}}
j
1
+
j
2
=
j
3
{\displaystyle \mathbf {j} _{1}+\mathbf {j} _{2}=\mathbf {j} _{3}}
j
4
+
j
5
=
j
6
{\displaystyle \mathbf {j} _{4}+\mathbf {j} _{5}=\mathbf {j} _{6}}
a onda:
j
3
+
j
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=
j
9
{\displaystyle \mathbf {j} _{3}+\mathbf {j} _{6}=\mathbf {j} _{9}}
Mi to pišemo u skraćenom obliku kao:
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1
j
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j
3
,
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j
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m
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⟩
.
{\displaystyle |((j_{1}j_{2})j_{3},(j_{4}j_{5})j_{6})j_{9}m_{9}\rangle .}
Drugi način da se vežu 4 ugaona momenta je:
j
1
+
j
4
=
j
7
{\displaystyle \mathbf {j} _{1}+\mathbf {j} _{4}=\mathbf {j} _{7}}
j
2
+
j
5
=
j
8
{\displaystyle \mathbf {j} _{2}+\mathbf {j} _{5}=\mathbf {j} _{8}}
a onda:
j
7
+
j
8
=
j
9
{\displaystyle \mathbf {j} _{7}+\mathbf {j} _{8}=\mathbf {j} _{9}}
odnosno u skraćenom obliku:
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1
j
4
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j
7
,
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j
8
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j
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m
9
⟩
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{\displaystyle |((j_{1}j_{4})j_{7},(j_{2}j_{5})j_{8})j_{9}m_{9}\rangle .}
Transformacija između dva oblika je:
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j
4
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j
7
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j
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j
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m
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⟩
=
{\displaystyle |((j_{1}j_{4})j_{7},(j_{2}j_{5})j_{8})j_{9}m_{9}\rangle =}
∑
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∑
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⟨
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⟩
.
{\displaystyle \sum _{j_{3}}\sum _{j6}|((j_{1}j_{2})j_{3},(j_{4}j_{5})j_{6})j_{9}m_{9}\rangle \langle ((j_{1}j_{2})j_{3},(j_{4}j_{5})j_{6})j_{9}|((j_{1}j_{4})j_{7},(j_{2}j_{5})j_{8})j_{9}\rangle .}
Pri tome 9-j simbol simbol može da se definiše kao:
[
(
2
j
3
+
1
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(
2
j
6
+
1
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(
2
j
7
+
1
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]
1
2
{
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9
}
=
{\displaystyle [(2j_{3}+1)(2j_{6}+1)(2j_{7}+1)(2j_{8}+1)]^{\frac {1}{2}}{\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}=}
⟨
(
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,
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⟩
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{\displaystyle \langle ((j_{1}j_{2})j_{3},(j_{4}j_{5})j_{6})j_{9}|((j_{1}j_{4})j_{7},(j_{2}j_{5})j_{8})j_{9}\rangle .}
9-j simboli zadovoljavaju relaciju ortogonalnosti:
∑
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7
j
8
(
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+
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2
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8
+
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)
{
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j
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}
{
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′
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′
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9
}
=
δ
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3
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′
δ
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6
′
Δ
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Δ
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Δ
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{\displaystyle \sum _{j_{7}j_{8}}(2j_{7}+1)(2j_{8}+1){\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}{\begin{Bmatrix}j_{1}&j_{2}&j_{3}'\\j_{4}&j_{5}&j_{6}'\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}={\frac {\delta _{j_{3}j_{3}'}\delta _{j_{6}j_{6}'}\Delta (j_{1}j_{2}j_{3})\Delta (j_{4}j_{5}j_{6})\Delta (j_{3}j_{6}j_{9})}{(2j_{3}+1)(2j_{6}+1)}}.}
gde je:
Δ
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1
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{\displaystyle \Delta (a,b,c)=[(a+b-c)!(a-b+c)!(-a+b+c)!/(a+b+c+1)!]^{1/2}}
Vignerov 9-j simbol je invarijantan na refleksije oko dijagonale:
{
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=
{
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1
}
.
{\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}={\begin{Bmatrix}j_{1}&j_{4}&j_{7}\\j_{2}&j_{5}&j_{8}\\j_{3}&j_{6}&j_{9}\end{Bmatrix}}={\begin{Bmatrix}j_{9}&j_{6}&j_{3}\\j_{8}&j_{5}&j_{2}\\j_{7}&j_{4}&j_{1}\end{Bmatrix}}.}
Ako se permutiraju bilo koja dva reda ili dve kolone :
{
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=
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S
{
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=
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S
{
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}
.
{\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}=(-1)^{S}{\begin{Bmatrix}j_{4}&j_{5}&j_{6}\\j_{1}&j_{2}&j_{3}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}=(-1)^{S}{\begin{Bmatrix}j_{2}&j_{1}&j_{3}\\j_{5}&j_{4}&j_{6}\\j_{8}&j_{7}&j_{9}\end{Bmatrix}}.}
tada se množi faznim faktorom
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−
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S
{\displaystyle (-1)^{S}}
S
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∑
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=
1
9
j
i
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{\displaystyle S=\sum _{i=1}^{9}j_{i}.}
Za
j
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=
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{\displaystyle j_{9}=0}
{
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0
}
=
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δ
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{
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7
}
.
{\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&0\end{Bmatrix}}={\frac {\delta _{j_{3},j_{6}}\delta _{j_{7},j_{8}}}{\sqrt {(2j_{3}+1)(2j_{7}+1)}}}(-1)^{j_{2}+j_{3}+j_{4}+j_{7}}{\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{5}&j_{4}&j_{7}\end{Bmatrix}}.}
∑
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j
24
(
−
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)
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+
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+
j
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−
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34
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+
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)
(
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24
+
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)
{
j
1
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12
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4
j
34
j
13
j
24
j
}
{
j
1
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13
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2
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24
j
14
j
23
j
}
=
{
j
1
j
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12
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3
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34
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14
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23
j
}
.
{\displaystyle \sum _{j_{13}\,j_{24}}(-1)^{2j_{2}+j_{24}+j_{23}-j_{34}}(2j_{13}+1)(2j_{24}+1){\begin{Bmatrix}j_{1}&j_{2}&j_{12}\\j_{3}&j_{4}&j_{34}\\j_{13}&j_{24}&j\end{Bmatrix}}{\begin{Bmatrix}j_{1}&j_{3}&j_{13}\\j_{4}&j_{2}&j_{24}\\j_{14}&j_{23}&j\end{Bmatrix}}={\begin{Bmatrix}j_{1}&j_{2}&j_{12}\\j_{4}&j_{3}&j_{34}\\j_{14}&j_{23}&j\end{Bmatrix}}.}
Abramowitz, Milton; Stegun, Irene A., ur. (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . New York: Dover. ISBN 978-0-486-61272-0 . Edmonds, A. R. (1957). Angular Momentum in Quantum Mechanics . Princeton, New Jersey: Princeton University Press. ISBN 978-0-691-07912-7 . Messiah, Albert (1981). Quantum Mechanics . II (12th izd.). New York: North Holland Publishing. ISBN 978-0-7204-0045-8 . 
![{\displaystyle [(2j_{3}+1)(2j_{6}+1)(2j_{7}+1)(2j_{8}+1)]^{\frac {1}{2}}{\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6571875aea26d9955e41faf0210878f988cdca01)
![{\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)
