Revision 772349 of "Simbol 3-jm" on lmowiki

I '''Simbol 3-jm de Wigner''' ciamaa anca 3''j'' simbol, i é relaa ai 
[[coeffizient de Clebsch-Gordan]]
a traves
:<math>
\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.
</math>
== Relazion inversa ==
La relazion inversa la pœul vess trovada notant che ''j''<sub>1</sub> - ''j''<sub>2</sub> - ''m''<sub>3</sub> 
al é un intregh e fent la substituzion
<math> m_3 \rightarrow -m_3 </math>
:<math>
\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}.
</math>
== Propietaa de simmetria ==

Le propietaa de simmetrira di simbol 3''j'' i é mejor di de [[coeffizient de Clebsch-Gordan]]: un simbol 3''j'' al é invariant spta permudazion pera di colomm:
:<math>
\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}.
</math>
una permudazion dispera la da un fator de fasa:
:<math>
\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}.
</math>
cambià el sign de <math>m</math> numer quantigh al da igualament una fasa:
:<math>
\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}.
</math>
== Regul de selzion ==

El Wigner 3j al é zero gavaa che tuti i condizion scia-de-sota i é satisafad:

:<math>m_1+m_2+m_3=0\,</math>

:<math>j_1+j_2 + j_3\,</math> is integer

:<math>|m_i| \le j_i</math> 

:<math>|j_1-j_2|\le j_3 \le j_1+j_2</math>.
== Invariant scalöe ==
La contrazion del produit de 3 stacc rotazional cont un simbol 3''j'' 
:<math>
  \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},
</math>
a l'é invarianta par rotazion
== Relazion d'Ortogonalitaa ==
<math>
(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_{j j'}\delta_{m m'}.
</math>

<math>
\sum_{j m} (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'}.
</math>
==Relazion cont integral d'armònigh sférigh con pes a spin==

<math>
\int d{\mathbf{\hat n}} {}_{s_1} Y_{j_1 m_1}({\mathbf{\hat n}})
{}_{s_2} Y_{j_2m_2}({\mathbf{\hat n}}) {}_{s_3} Y_{j_3m_3}({\mathbf{\hat
n}})=(-1)^{m_1+s_1} \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}
</math>


==Vardii anca==
*[[Coeffizient de Clebsch-Gordan]]
*[[Armònigh sférigh]]
*[[Simbol 6-j]]
*[[Simbol 9-j]]
*[[Simbol 12-j]]
*[[Simbol 15-j]]
==Referenz==
<references />
<div class="references">
*L. C. Biedenharn and J. D. Louck, ''Angular Momentum in Quantum Physics'', volume 8 of Encyclopedia of Mathematics,  Addison-Wesley, Reading, 1981.
* D. M. Brink and G. R. Satchler, ''Angular Momentum'', 3rd edition, Clarendon, Oxford, 1993.  
* A. R. Edmonds,  ''Angular Momentum in Quantum Mechanics'', 2nd edition, Princeton University Press, Princeton, 1960.
*{{dlmf|id=34 |title=3j,6j,9j Symbols|first=Leonard C.|last= Maximon}}
* D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, ''Quantum Theory of Angular Momentum'', World Scientific Publishing Co., Singapore, 1988.
* E. P. Wigner, ''On the Matrices Which Reduce the Kronecker Products of Representations of Simply Reducible Groups'', unpublished (1940). Reprinted in: L. C. Biedenharn and H. van Dam, ''Quantum Theory of Angular Momentum'', Academic Press, New York (1965).
</div>
==Liamm de fœura==
* [http://www-stone.ch.cam.ac.uk/wigner.shtml Anthony Stone’s Wigner coefficient calculator] 
* [http://www.volya.net/vc/vc.php Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator] 
* [http://plasma-gate.weizmann.ac.il/369j.html 369j-symbol calculator at the Plasma Laboratory of Weizmann Institute of Science] 

<!----[[category:Rotational symmetry]]
[[category:Representation theory of Lie groups]]---->
[[category:Mecàniga di quanta]]

[[en:Wigner 3-j symbols]]
[[it:Simboli 3j]]
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[[ru:3j символ]]
[[sq:Simboli 3 - jm]]
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[[uk:3j символи]]