- 3-jm symbol
Wigner 3"-jm" symbols, also called 3"j" symbols,are related to
Clebsch-Gordan coefficients through:egin{pmatrix} j_1 & j_2 & j_3\ m_1 & m_2 & m_3end{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} angle.Inverse relation
The inverse relation can be found by noting that "j"1 - "j"2 - "m"3 is an integer number and making the substitutionm_3 ightarrow -m_3 :langle j_1 m_1 j_2 m_2 | j_3 m_3 angle = (-1)^{j_1-j_2+m_3}sqrt{2j_3+1}egin{pmatrix} j_1 & j_2 & j_3\ m_1 & m_2 & -m_3end{pmatrix}.
Symmetry properties
The symmetry properties of 3"j" symbols are more convenient than those of
Clebsch-Gordan coefficients. A 3"j" symbol is invariant under an evenpermutation of its columns::egin{pmatrix} j_1 & j_2 & j_3\ m_1 & m_2 & m_3end{pmatrix}=egin{pmatrix} j_2 & j_3 & j_1\ m_2 & m_3 & m_1end{pmatrix}=egin{pmatrix} j_3 & j_1 & j_2\ m_3 & m_1 & m_2end{pmatrix}.An odd permutation of the columns gives a phase factor::egin{pmatrix} j_1 & j_2 & j_3\ m_1 & m_2 & m_3end{pmatrix}=(-1)^{j_1+j_2+j_3}egin{pmatrix} j_2 & j_1 & j_3\ m_2 & m_1 & m_3end{pmatrix}=(-1)^{j_1+j_2+j_3}egin{pmatrix} j_1 & j_3 & j_2\ m_1 & m_3 & m_2end{pmatrix}.Changing the sign of the m quantum numbers also gives a phase::egin{pmatrix} j_1 & j_2 & j_3\ -m_1 & -m_2 & -m_3end{pmatrix}=(-1)^{j_1+j_2+j_3}egin{pmatrix} j_1 & j_2 & j_3\ m_1 & m_2 & m_3end{pmatrix}.Selection rules
The Wigner 3j is zero unless all these conditions are satisfied:
:m_1+m_2+m_3=0,
:j_1+j_2 + j_3, is integer
:m_i| le j_i
:j_1-j_2|le j_3 le j_1+j_2.
Scalar invariant
The contraction of the product of three rotational states with a 3"j" symbol,: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 angle |j_2 m_2 angle |j_3 m_3 angleegin{pmatrix} j_1 & j_2 & j_3\ m_1 & m_2 & m_3end{pmatrix},is invariant under rotations.Orthogonality Relations
2j+1)sum_{m_1 m_2}egin{pmatrix} j_1 & j_2 & j\ m_1 & m_2 & mend{pmatrix}egin{pmatrix} j_1 & j_2 & j'\ m_1 & m_2 & m'end{pmatrix}=delta_{j j'}delta_{m m'}.
sum_{j m} (2j+1)egin{pmatrix} j_1 & j_2 & j\ m_1 & m_2 & mend{pmatrix}egin{pmatrix} j_1 & j_2 & j\ m_1' & m_2' & mend{pmatrix}=delta_{m_1 m_1'}delta_{m_2 m_2'}.
Relation to integrals of spin-weighted spherical harmonics
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{hatn)=(-1)^{m_1+s_1} sqrt{frac{(2j_1+1)(2j_2+1)(2j_3+1)}{4piegin{pmatrix} j_1 & j_2 & j_3\ m_1 & m_2 & m_3end{pmatrix}egin{pmatrix} j_1 & j_2 & j_3\ -s_1 & -s_2 & -s_3end{pmatrix}
This should be checked for phase conventions of the harmonics.
ee also
*Clebsch-Gordan coefficients
*Spherical harmonics
*6-j symbol
*9-j symbol
*12-j symbol
*15-j symbol 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).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] (Numerical)
* [http://plasma-gate.weizmann.ac.il/369j.html 369j-symbol calculator at the Plasma Laboratory of Weizmann Institute of Science] (Numerical)
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