- 6-j symbol
Wigner's 6-"j" symbols were introduced by
Eugene Paul Wigner in 1940, and published in 1965.They are related to Racah's W-coefficientsby:egin{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_1j_2j_5j_4;j_3j_6).They have higher symmetry than Racah's W-coefficients.ymmetry relations
The 6-j symbol is invariant under the permutation of any two columns::egin{Bmatrix} j_1 & j_2 & j_3\ j_4 & j_5 & j_6 end{Bmatrix} = egin{Bmatrix} j_2 & j_1 & j_3\ j_5 & j_4 & j_6 end{Bmatrix}= egin{Bmatrix} j_1 & j_3 & j_2\ j_4 & j_6 & j_5 end{Bmatrix}= egin{Bmatrix} j_3 & j_2 & j_1\ j_6 & j_5 & j_4 end{Bmatrix}.The 6-j symbol is also invariant if upper and lower argumentsare interchanged in any two columns::egin{Bmatrix} j_1 & j_2 & j_3\ j_4 & j_5 & j_6 end{Bmatrix} = egin{Bmatrix} j_4 & j_5 & j_3\ j_1 & j_2 & j_6 end{Bmatrix} = egin{Bmatrix} j_1 & j_5 & j_6\ j_4 & j_2 & j_3 end{Bmatrix} = egin{Bmatrix} j_4 & j_2 & j_6\ j_1 & j_5 & j_3 end{Bmatrix}.The 6-j symbol:egin{Bmatrix} j_1 & j_2 & j_3\ j_4 & j_5 & j_6 end{Bmatrix}is zero unless j_1, j_2, and j_3 satisfy triangle conditions,i.e.,:j_1 = |j_2-j_3|, ldots, j_2+j_3.In combination with the symmetry relation for interchanging upper and lower arguments thisshows that triangle conditions must also be satisfied for j_1,j_5,j_6),j_4,j_2,j_6), and j_4,j_5,j_3).
pecial case
When j_6=0 the expression for the 6-j symbol is::egin{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).The function Delta(j_1,j_2,j_3) is equal to 1 when j_1,j_2,j_3) satisfy the triangle conditions,and zero otherwise. The symmetry relations can be used to find the expression when another j is equalto zero.
Orthogonality relation
The 6-j symbols satisfy this orthogonality relation::sum_{j_3} (2j_3+1) egin{Bmatrix} j_1 & j_2 & j_3\ j_4 & j_5 & j_6 end{Bmatrix} egin{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).
ee also
*
Clebsch-Gordan coefficient
*3-jm symbol
*Racah W-coefficient
*9-j symbol References
* cite book |last= Biedenharn |first= L. C. |coauthors= van Dam, H.
title= Quantum Theory of Angular Momentum: A collection of Reprints and Original Papers
year= 1965 |publisher=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* cite book |last= Condon |first= Edward U. |coauthors= Shortley, G. H. |title= The Theory of Atomic Spectra |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 edition
publisher= North Holland Publishing |location= New York |isbn= 0-7204-0045-7* cite book |last= Brink |first= D. M. |coauthors= Satchler, G. R. |title= Angular Momentum
year= 1993 |edition= 3rd edition |publisher=Clarendon Press |location= Oxford |isbn= 0-19-851759-9 |chapter= Chapter 2* cite book |last= Zare |first= Richard N. |title= Angular Momentum |year=1988
publisher= John Wiley |location= New York |isbn= 0-471-85892-7 |chapter= Chapter 2* cite book |last= Biedenharn |first= L. C. |coauthors= Louck, J. D. |title= Angular Momentum in Quantum Physics
year= 1981 |publisher=Addison-Wesley |location= Reading, Massachusetts |isbn= 0201135078External links
* [http://www-stone.ch.cam.ac.uk/wigner.shtml Anthony Stone’s Wigner coefficient calculator] (Gives exact answer)
* [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]
Wikimedia Foundation. 2010.