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:They have higher symmetry than Racah's W-coefficients.

ymmetry relations

The $6-j$ symbol is invariant under the permutation of any two columns::The $6-j$ symbol is also invariant if upper and lower argumentsare interchanged in any two columns::The $6-j$ symbol: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::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::

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= 0201135078

External 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]