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{{Short description|Symbol used in quantum mechanics}}
[[File:Jucys diagram for Wigner 9-j symbol.svg|thumb|[[Jucys diagram]] for the Wigner 9-j symbol. The diagram describes a summation over six 3-jm symbols. Plus signs on each nodes indicate an anticlockwise reading of the lines for the 3-jm symbol, whereas minus signs indicate clockwise. Due to its symmetries, there are many ways in which the diagram can be drawn.]]

In physics, Wigner's '''9-''j'' symbols''' were introduced by [[Biography:Eugene Wigner|Eugene Wigner]] in 1937. They are related to [[Physics:Angular momentum coupling|recoupling coefficients]] in [[Physics:Quantum mechanics|quantum mechanics]] involving four angular momenta:

<math>
\sqrt{(2j_3+1)(2j_6+1)(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}
</math>
<math>
=
\langle ( (j_1j_2)j_3,(j_4j_5)j_6)j_9 | ((j_1 j_4)j_7,(j_2j_5)j_8)j_9\rangle.
</math>

==Recoupling of four angular momentum vectors==
Coupling of two angular momenta <math>\mathbf{j}_1</math> and <math>\mathbf{j}_2</math> is the construction of simultaneous eigenfunctions of <math>\mathbf{J}^2</math> and <math>J_z</math>, where <math>\mathbf{J}=\mathbf{j}_1+\mathbf{j}_2</math>, as explained in the article on [[Physics:Clebsch–Gordan coefficients|Clebsch–Gordan coefficients]].

Coupling of three angular momenta can be done in several ways, as explained in the article on [[Racah W-coefficient]]s. Using the notation and techniques of that article, total angular momentum states that arise from coupling the angular momentum vectors <math>\mathbf{j}_1</math>, <math>\mathbf{j}_2</math>, <math>\mathbf{j}_4</math>, and <math>\mathbf{j}_5</math> may be written as
:<math>
| ((j_1j_2)j_3, (j_4j_5)j_6)j_9m_9\rangle.
</math>
Alternatively, one may first couple <math>\mathbf{j}_1</math> and <math>\mathbf{j}_4</math> to <math>\mathbf{j}_7</math> and <math>\mathbf{j}_2</math> and <math>\mathbf{j}_5</math> to <math>\mathbf{j}_8</math>, before coupling <math>\mathbf{j}_7</math> and <math>\mathbf{j}_8</math> to <math>\mathbf{j}_9</math>:
:<math>
|((j_1j_4)j_7, (j_2j_5)j_8)j_9m_9\rangle.
</math>
Both sets of functions provide a complete, [[Orthonormal basis|orthonormal basis]] for the space with dimension <math>(2j_1+1)(2j_2+1)(2j_4+1)(2j_5+1)</math> spanned by
:<math>
|j_1 m_1\rangle |j_2 m_2\rangle |j_4 m_4\rangle |j_5 m_5\rangle, \;\;
m_1=-j_1,\ldots,j_1;\;\; m_2=-j_2,\ldots,j_2;\;\; m_4=-j_4,\ldots,j_4;\;\;m_5=-j_5,\ldots,j_5.
</math>
Hence, the transformation between the two sets is unitary and the matrix elements of the transformation are given by the scalar products of the functions.
As in the case of the [[Racah W-coefficient]]s the matrix elements are independent of the total angular momentum projection [[Physics:Quantum number|quantum number]] (<math>m_9</math>):
:<math>
|((j_1j_4)j_7, (j_2j_5)j_8)j_9m_9\rangle = \sum_{j_3}\sum_{j_6}
| ((j_1j_2)j_3, (j_4j_5)j_6)j_9m_9\rangle
\langle ( (j_1j_2)j_3,(j_4j_5)j_6)j_9 | ((j_1 j_4)j_7,(j_2j_5)j_8)j_9\rangle.
</math>

==Symmetry relations==
A 9-''j'' symbol is invariant under reflection about either diagonal as well as even permutations of its rows or columns:
:<math>
\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}
=
\begin{Bmatrix}
j_7 & j_4 & j_1\\
j_9 & j_6 & j_3\\
j_8 & j_5 & j_2
\end{Bmatrix}.
</math>

An odd permutation of rows or columns yields a phase factor <math>(-1)^S</math>, where
:<math>S=\sum_{i=1}^9 j_i.
</math>
For example:
:<math>
\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}.
</math>

==Reduction to 6j symbols==
The 9-''j'' symbols can be calculated as sums over triple-products of 6-''j'' symbols where the summation extends over all {{math|''x''}} admitted by the triangle conditions in the factors:
:<math>
\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)^{2 x}(2 x + 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}
</math>.

==Special case==
When <math>j_9=0</math> the 9-''j'' symbol is proportional to a [[Physics:6-j symbol|6-j symbol]]:
:<math>
\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}.
</math>

==Orthogonality relation==
The 9-''j'' symbols satisfy this orthogonality relation:
:<math>
\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_3j_3'}\delta_{j_6j_6'} \begin{Bmatrix} j_1 & j_2 & j_3 \end{Bmatrix} \begin{Bmatrix} j_4 & j_5 & j_6\end{Bmatrix} \begin{Bmatrix} j_3 & j_6 & j_9 \end{Bmatrix}}
{(2j_3+1)(2j_6+1)}.
</math>
The ''triangular delta'' {{math|{''j''1  ''j''2  ''j''3}}} is equal to 1 when the triad (''j''1, ''j''2, ''j''3) satisfies the triangle conditions, and zero otherwise.

==3''n''-j symbols==
The [[Physics:6-j symbol|6-j symbol]] is the first representative, {{math|''n'' {{=}} 2}}, of {{math|3''n''}}-''j'' symbols that are defined as sums of products of {{math|''n''}} of Wigner's 3-''jm'' coefficients. The sums are over all combinations of {{math|m}} that the {{math|3''n''}}-''j'' coefficients admit, i.e., which lead to non-vanishing contributions.

If each 3-''jm'' factor is represented by a vertex and each j by an edge, these {{math|3''n''}}-''j'' symbols can be mapped on certain [[Table of simple cubic graphs|3-regular graphs]] with {{math|3''n''}} edges and {{math|2''n''}} nodes. The 6-''j'' symbol is associated with the [[Complete graph|K4]] graph on 4 vertices, the 9-''j'' symbol with the [[Utility graph|utility graph]] on 6 vertices (''K''3,3), and the two distinct (non-isomorphic) 12-''j'' symbols with the [[Hypercube graph|''Q''3]] and [[Wagner graph]]s on 8 vertices.
Symmetry relations are generally representative of the automorphism group of these graphs.

==See also==

* [[Physics:Clebsch–Gordan coefficients|Clebsch–Gordan coefficients]]
* [[Physics:3-j symbol|3-j symbol]], also called 3-jm symbol
* [[Racah W-coefficient]]
* [[Physics:6-j symbol|6-j symbol]]

==References==
{{no footnotes|date=September 2010}}
* {{cite book |last1= Biedenharn |first1= L. C. |last2= van Dam |first2= H.
|title= Quantum Theory of Angular Momentum: A collection of Reprints and Original Papers
|year= 1965 |publisher= [[Company:Academic Press|Academic Press]] |location= New York |isbn= 0120960567}}
* {{cite book |last= Edmonds |first= A. R. |title= Angular Momentum in Quantum Mechanics |year= 1957 |publisher= [[Princeton University Press]] |location= Princeton, New Jersey |isbn= 0-691-07912-9 |url-access= registration |url= https://archive.org/details/angularmomentumi0000edmo }}
* {{cite book |last= Condon |first= Edward U. |author2= Shortley, G. H. |title= The Theory of Atomic Spectra |url= https://archive.org/details/in.ernet.dli.2015.212979 |year= 1970
|publisher= Cambridge University Press |location= Cambridge |isbn= 0-521-09209-4 |chapter= Chapter 3}}
*{{dlmf|id=34 |title=3j,6j,9j Symbols|first=Leonard C.|last= Maximon}}
* {{cite book |last= Messiah |first= Albert |title= Quantum Mechanics (Volume II) |year= 1981 | edition= 12th
|publisher= [[Company:Elsevier|North Holland Publishing]] |location= New York |isbn= 0-7204-0045-7}}
* {{cite book |last= Brink |first= D. M. |author2= Satchler, G. R. |title= Angular Momentum |year= 1993 |edition= 3rd |publisher= Clarendon Press |location= Oxford |isbn= 0-19-851759-9 |chapter= Chapter 2 |url-access= registration |url= https://archive.org/details/angularmomentum0000brin }}
* {{cite book |last= Zare |first= Richard N. |title= Angular Momentum |year=1988
|publisher= [[Company:John Wiley & Sons|John Wiley]] |location= New York |isbn= 0-471-85892-7 |chapter= Chapter 2}}
* {{cite book |last= Biedenharn |first= L. C. |author2= Louck, J. D. |title= Angular Momentum in Quantum Physics
|year= 1981 |publisher= Addison-Wesley |location= Reading, Massachusetts |isbn= 0201135078 }}
* {{cite book |last= Varshalovich |first= D. A. |author2= Moskalev, A. N.|author3= Khersonskii, V. K.
|title= Quantum Theory of Angular Momentum | year= 1988 |publisher= [[Company:World Scientific|World Scientific]] |location= Singapore |isbn= 9971-50-107-4 }}
*{{cite journal
|first1=H. A.
|last1=Jahn
|first2=J.
|last2=Hope
|title=Symmetry properties of the Wigner 9j symbol
|journal=Physical Review
|year=1954
|volume=93
|issue=2
|page=318
|doi=10.1103/PhysRev.93.318
|bibcode=1954PhRv...93..318J
}}

==External links==
* {{cite web
|first1=Anthony
|last1=Stone
|url=http://www-stone.ch.cam.ac.uk/wigner.shtml
|title=Wigner coefficient calculator
}} (Gives answer in exact fractions)
* {{ cite web
|url=http://plasma-gate.weizmann.ac.il/369j.html
|title=369j-symbol calculator
|author=Plasma Laboratory of Weizmann Institute of Science
}} (Answer as floating point numbers)
* {{cite web
|url=http://caagt.ugent.be/yutsis/GYutsisApplet.caagt
|title=GYutsis Applet
|first1=Veerl
|last1=Fack
|first2=Dries
|last2=van Dyck
}}
* {{cite web
|first1=H.T.
|last1=Johansson
|first2=C.
|last2=Forssén
|title=(WIGXJPF)
|url=http://fy.chalmers.se/subatom/wigxjpf/
}} (accurate; C, fortran, python)
* {{cite web
|first1=H.T.
|last1=Johansson
|title=(FASTWIGXJ)
|url=http://fy.chalmers.se/subatom/fastwigxj/
}} (fast lookup, accurate; C, fortran)

[[Category:Rotational symmetry]]
[[Category:Representation theory of Lie groups]]
[[Category:Quantum mechanics]]

{{Sourceattribution|9-j symbol}}

Physics:9-j symbol - Revision history

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