9-j symbol

Wigner's 9-"j" symbols were introduced by
Eugene Paul Wigner in 1937. They are related to recoupling coefficientsinvolving four angular momenta:

Recoupling of four angular momentum vectors

Coupling of two angular momenta $mathbf\{j\}\_1$ and $mathbf\{j\}\_2$is the construction of simultaneous eigenfunctions of $mathbf\{J\}^2$and $J\_z$, where $mathbf\{J\}=mathbf\{j\}\_1+mathbf\{j\}\_2$,as explained in the article on Clebsch-Gordan coefficients.

Coupling of three angular momenta can be done in several ways, asexplained in the article on Racah W-coefficients. Using the notationand techniques of that article, total angular momentum states that arisefrom coupling the angular momentum vectors $mathbf\{j\}\_1$,$mathbf\{j\}\_2$, $mathbf\{j\}\_4$, and$mathbf\{j\}\_5$ may be written as:$$
((j_1j_2)j_3, (j_4j_5)j_6)j_9m_9 angle.Alternatively, one may first couple $mathbf\{j\}\_1$ and$mathbf\{j\}\_4$ to $mathbf\{j\}\_7$ and$mathbf\{j\}\_2$ and $mathbf\{j\}\_5$ to$mathbf\{j\}\_8$, before coupling $mathbf\{j\}\_7$and $mathbf\{j\}\_8$ to $mathbf\{j\}\_9$::$$
((j_1j_4)j_7, (j_2j_5)j_8)j_9m_9 angle.Both sets of functions provide a complete, orthonormal basis forthe space with dimension $(2j\_1+1)(2j\_2+1)(2j\_4+1)(2j\_5+1)$ spanned by:$$
j_1 m_1 angle |j_2 m_2 angle |j_4 m_4 angle |j_5 m_5 angle, ;; m_1=-j_1,ldots,j_1;;; m_2=-j_2,ldots,j_2;;; m_4=-j_4,ldots,j_4;;;m_5=-j_5,ldots,j_5.Hence, the transformation between the two sets is unitary and the matrix elementsof the transformation are given by the scalar products of the functions.As in the case of the Racah W-coefficients the matrix elements are independentof the total angular momentum projection quantum number ($m\_9$)::$$
((j_1j_4)j_7, (j_2j_5)j_8)j_9m_9 angle = sum_{j_3}sum_{j6}
((j_1j_2)j_3, (j_4j_5)j_6)j_9m_9 angle 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 angle.

ymmetry relations

A $9-j$ symbol is invariant under reflection in either diagonal::

The permutation of any two rows or any two columns yields a phase factor$(-1)^S$, where:$S=sum\_\{i=1\}^9\; j\_i.$For example::

pecial case

When $j\_9=0$ the 9-j symbol is proportional to a 6-j symbol::

Orthogonality relation

The 9-j symbols satisfy this orthogonality relation::The symbol $\{j\_1j\_2j\_3\}$ is equal to one if the triad $(j\_1j\_2j\_3)$satisfies the triangular conditions and zero otherwise.

ee also

* Clebsch-Gordan coefficient
* 3-jm symbol
* Racah W-coefficient
* 6-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

* cite book |last= Varshalovich |first= D. A. |coauthors= Moskalev, A. N.; Khersonskii, V. K.
title= Quantum Theory of Angular Momentum | year= 1988 |publisher= World Scientific |location= Singapore |isbn= 9971-50-107-4

External links

* [http://www-stone.ch.cam.ac.uk/wigner.shtml Anthony Stone’s Wigner coefficient calculator] (Gives exact answer)
* [http://plasma-gate.weizmann.ac.il/369j.html 369j-symbol calculator at the Plasma Laboratory of Weizmann Institute of Science] (Numerical answer)