- 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:2j_3+1)(2j_6+1)(2j_7+1)(2j_8+1)] ^frac{1}{2} egin{Bmatrix} j_1 & j_2 & j_3\ j_4 & j_5 & j_6\ j_7 & j_8 & j_9 end{Bmatrix} = 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.Recoupling of four angular momentum vectors
Coupling of two angular momenta mathbf{j}_1 and mathbf{j}_2is the construction of simultaneous eigenfunctions of mathbf{J}^2and 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-coefficient s. 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, andmathbf{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 andmathbf{j}_4 to mathbf{j}_7 andmathbf{j}_2 and mathbf{j}_5 tomathbf{j}_8, before coupling mathbf{j}_7and 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 theRacah W-coefficient s 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::egin{Bmatrix} j_1 & j_2 & j_3\ j_4 & j_5 & j_6\ j_7 & j_8 & j_9 end{Bmatrix} = egin{Bmatrix} j_1 & j_4 & j_7\ j_2 & j_5 & j_8\ j_3 & j_6 & j_9 end{Bmatrix} = egin{Bmatrix} j_9 & j_6 & j_3\ j_8 & j_5 & j_2\ j_7 & j_4 & j_1 end{Bmatrix}.
The permutation of any two rows or any two columns yields a phase factor1)^S, where:S=sum_{i=1}^9 j_i.For example::egin{Bmatrix} j_1 & j_2 & j_3\ j_4 & j_5 & j_6\ j_7 & j_8 & j_9 end{Bmatrix} = (-1)^S egin{Bmatrix} j_4 & j_5 & j_6\ j_1 & j_2 & j_3\ j_7 & j_8 & j_9 end{Bmatrix} = (-1)^S egin{Bmatrix} j_2 & j_1 & j_3\ j_5 & j_4 & j_6\ j_8 & j_7 & j_9 end{Bmatrix}.
pecial case
When j_9=0 the 9-j symbol is proportional to a
6-j symbol ::egin{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} egin{Bmatrix} j_1 & j_2 & j_3\ j_5 & j_4 & j_7 end{Bmatrix}.Orthogonality relation
The 9-j symbols satisfy this orthogonality relation::sum_{j_7 j_8} (2j_7+1)(2j_8+1) egin{Bmatrix} j_1 & j_2 & j_3\ j_4 & j_5 & j_6\ j_7 & j_8 & j_9 end{Bmatrix} egin{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'} {j_1j_2j_3} {j_4j_5j_6} {j_3j_6j_9 {(2j_3+1)(2j_6+1)}.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= 9971-50-107-4External 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)
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