3j-symbol: Difference between revisions

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Wigner '''3''-jm'' symbols''', also called 3''j'' symbols,
{{subpages}}
are related to [[Clebsch-Gordan coefficients]]
In [[physics]] and [[mathematics]],  Wigner '''3''-jm'' symbols''', also called 3''j'' symbols,
through
are related to the [[Clebsch-Gordan coefficients]] of the [[group]]s [[SU(2)]] and [[SO(3)]] through
:<math>
:<math>
\begin{pmatrix}
\begin{pmatrix}
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\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} \rangle.
\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} \rangle.
</math>
</math>
The 3''j'' symbols show more symmetry in permutation of the labels than  the corresponding Clebsch-Gordan coefficients.


== Inverse relation ==
== Inverse relation ==
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</math>
</math>


== Attribution ==
{{WPattribution}}


==References==
== References ==[[Category:Suggestion Bot Tag]]
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  See http://en.wikipedia.org/wiki/Wikipedia:Footnotes for a
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* E. P. Wigner, ''On the Matrices Which Reduce the Kronecker Products of Representations of Simply Reducible Groups'', unpublished (1940).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).
* A. R. Edmonds,  ''Angular Momentum in Quantum Mechanics'', 2nd edition, Princeton University Press, Pinceton, 1960.
* D. M. Brink and G. R. Satchler, ''Angular Momentum'', 3rd edition, Clarendon, Oxford, 1993. 
*L. C. Biedenharn and J. D. Louck, ''Angular Momentum in Quantum Physics'', volume 8 of Encyclopedia of Mathematics,  Addison-Wesley, Reading, 1981.
* D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, ''Quantum Theory of Angular Momentum'', World Scientific Publishing Co., Singapore, 1988.
</div>

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In physics and mathematics, Wigner 3-jm symbols, also called 3j symbols, are related to the Clebsch-Gordan coefficients of the groups SU(2) and SO(3) through

The 3j symbols show more symmetry in permutation of the labels than the corresponding Clebsch-Gordan coefficients.

Inverse relation

The inverse relation can be found by noting that j1 - j2 - m3 is an integral number and making the substitution

Symmetry properties

The symmetry properties of 3j symbols are more convenient than those of Clebsch-Gordan coefficients. A 3j symbol is invariant under an even permutation of its columns:

An odd permutation of the columns gives a phase factor:

Changing the sign of the quantum numbers also gives a phase:

Selection rules

The Wigner 3j is zero unless , is integer, and .

Scalar invariant

The contraction of the product of three rotational states with a 3j symbol,

is invariant under rotations.

Orthogonality Relations

Attribution

Template:WPattribution

== References ==