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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. |
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| == Inverse relation == | | == Inverse relation == |
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| </math> | | </math> |
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| | == Attribution == |
| | {{WPattribution}} |
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| ==References== | | == References ==[[Category:Suggestion Bot Tag]] |
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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* E. P. Wigner, ''On the Matrices Which Reduce the Kronecker Products of Representations of Simply Reducible Groups'', unpublished (1940).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).
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| * A. R. Edmonds, ''Angular Momentum in Quantum Mechanics'', 2nd edition, Princeton University Press, Pinceton, 1960.
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| * D. M. Brink and G. R. Satchler, ''Angular Momentum'', 3rd edition, Clarendon, Oxford, 1993.
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| *L. C. Biedenharn and J. D. Louck, ''Angular Momentum in Quantum Physics'', volume 8 of Encyclopedia of Mathematics, Addison-Wesley, Reading, 1981.
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| * D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, ''Quantum Theory of Angular Momentum'', World Scientific Publishing Co., Singapore, 1988.
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Latest revision as of 11:00, 4 July 2024
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 ==