3j-symbol

From Citizendium
Jump to: navigation, search
This article is basically copied from an external source and has not been approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.
The content on this page originated on Wikipedia and is yet to be significantly improved. Contributors are invited to replace and add material to make this an original article.

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