9-j symbol
In physics, Wigner's 9-j symbols were introduced by Eugene Paul Wigner in 1937.
They are related to recoupling coefficients in quantum mechanics involving four angular momenta:
( 2
j
3
+ 1 ) ( 2
j
+ 1 ) ( 2
+ 1 ) ( 2
j
1
{\displaystyle {\sqrt {(2j_{3}+1)(2j_{6}+1)(2j_{7}+1)(2j_{8}+1)}}{\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}}
Coupling of two angular momenta
is the construction of simultaneous eigenfunctions of
, as explained in the article on Clebsch–Gordan coefficients.
Coupling of three angular momenta can be done in several ways, as explained in the article on Racah W-coefficients.
Using the notation and techniques of that article, total angular momentum states that arise from coupling the angular momentum vectors
may be written as Alternatively, one may first couple
, before coupling
: Both sets of functions provide a complete, orthonormal basis for the space with dimension
spanned by Hence, the transformation between the two sets is unitary and the matrix elements of the transformation are given by the scalar products of the functions.
As in the case of the Racah W-coefficients the matrix elements are independent of the total angular momentum projection quantum number (
): A 9-j symbol is invariant under reflection about either diagonal as well as even permutations of its rows or columns: An odd permutation of rows or columns yields a phase factor
, where For example: The 9-j symbols can be calculated as sums over triple-products of 6-j symbols where the summation extends over all x admitted by the triangle conditions in the factors: When
the 9-j symbol is proportional to a 6-j symbol: The 9-j symbols satisfy this orthogonality relation: The triangular delta {j1 j2 j3} is equal to 1 when the triad (j1, j2, j3) satisfies the triangle conditions, and zero otherwise.
The 6-j symbol is the first representative, n = 2, of 3n-j symbols that are defined as sums of products of n of Wigner's 3-jm coefficients.
The sums are over all combinations of m that the 3n-j coefficients admit, i.e., which lead to non-vanishing contributions.
If each 3-jm factor is represented by a vertex and each j by an edge, these 3n-j symbols can be mapped on certain 3-regular graphs with 3n edges and 2n nodes.
The 6-j symbol is associated with the K4 graph on 4 vertices, the 9-j symbol with the utility graph on 6 vertices (K3,3), and the two distinct (non-isomorphic) 12-j symbols with the Q3 and Wagner graphs on 8 vertices.
Symmetry relations are generally representative of the automorphism group of these graphs.
