3-jm symbol

Wigner 3"-jm" symbols, also called 3"j" symbols,are related to Clebsch-Gordan coefficientsthrough:

Inverse relation

The inverse relation can be found by noting that "j"₁ - "j"₂ - "m"₃ is an integer number and making the substitution$m\_3\; ightarrow\; -m\_3$:

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::An odd permutation of the columns gives a phase factor::Changing the sign of the $m$ quantum numbers also gives a phase::

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

Relation to integrals of spin-weighted spherical harmonics

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* E. P. Wigner, "On the Matrices Which Reduce the Kronecker Products of Representations of Simply Reducible Groups", unpublished (1940).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)