9-j symbol
Encyclopedia
Wigner's 9-j symbols were introduced by
Eugene Paul Wigner in 1937. They are related to recoupling coefficients
involving four angular momenta
and 
is the construction of simultaneous eigenfunctions of
and
, where 
,
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-coefficient
s. Using the notation
and techniques of that article, total angular momentum states that arise
from coupling the angular momentum vectors
,
, 
, and
may be written as
Alternatively, one may first couple
and
to 
and
and 
to
, before coupling 
and
to 
:
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-coefficient
s the matrix elements are independent
of the total angular momentum projection quantum number (
):
symbol is invariant under reflection in either diagonal:
The permutation of any two rows or any two columns yields a phase factor
, where
For example:
the summation extends over all x admitted by the triangle conditions in the factors:
.
the 9-j symbol is proportional to a 6-j symbol:
The symbol
is equal to one if the triad 
satisfies the triangular conditions and zero otherwise.
that are defined as sums of products of of Wigner's 3-jm
coefficients. The sums are over all combinations of that
the 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
vertices and 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, and the
two different (non-isomorphic) 12-j symbols
with the Q 3 and Wagner graph
s on 8 vertices.
Symmetry relations are generally representative of the
automorphism group of these graphs.
Eugene Paul Wigner in 1937. They are related to recoupling coefficients
involving four angular momenta
Recoupling of four angular momentum vectors
Coupling of two angular momenta and is the construction of simultaneous eigenfunctions of
and
, where ,as explained in the article on Clebsch-Gordan coefficients
Clebsch-Gordan coefficients
In physics, the Clebsch–Gordan coefficients are sets of numbers that arise in angular momentum coupling under the laws of quantum mechanics.In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum...
.
Coupling of three angular momenta can be done in several ways, as
explained in the article on Racah W-coefficient
Racah W-coefficient
Racah's W-coefficients were introduced by Giulio Racah in 1942. These coefficients have a purely mathematical definition. In physics they are used in calculations involving the quantum mechanical description of angular momentum, for example in atomic theory....
s. Using the notation
and techniques of that article, total angular momentum states that arise
from coupling the angular momentum vectors
,, , and may be written asAlternatively, one may first couple
and to and and to, before coupling and
to :Both sets of functions provide a complete, orthonormal basis for
the space with dimension
spanned byHence, 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-coefficient
Racah W-coefficient
Racah's W-coefficients were introduced by Giulio Racah in 1942. These coefficients have a purely mathematical definition. In physics they are used in calculations involving the quantum mechanical description of angular momentum, for example in atomic theory....
s the matrix elements are independent
of the total angular momentum projection quantum number (
):Symmetry relations
A symbol is invariant under reflection in either diagonal:The permutation of any two rows or any two columns yields a phase factor
, whereFor example:
Reduction to 6j symbols
The 9j symbols can be calculated as sums over triple-products of 6j symbols wherethe summation extends over all x admitted by the triangle conditions in the factors:
.Special case
When the 9-j symbol is proportional to a 6-j symbol:Orthogonality relation
The 9-j symbols satisfy this orthogonality relation:The symbol
is equal to one if the triad satisfies the triangular conditions and zero otherwise.
3n-j symbols
The 6-j symbol is the first representative, , of 3n-j symbolsthat are defined as sums of products of of Wigner's 3-jm
coefficients. The sums are over all combinations of that
the 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
Table of simple cubic graphs
The connected 3-regular simple graphs are listed for small vertex numbers.-Connectivity:The count of simple cubic graphs is 1, 2, 5, 19,... for 2, 4, 6, 8,... vertices . A classification according to edge connectivity is made as follows: the 1-connected and 2-connected graphs are defined as usual...
with
vertices and nodes. The 6-j symbol is associated with
the K4
Complete graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.-Properties:...
graph on 4 vertices,
the 9-j symbol with the utility graph on
6 vertices, and the
two different (non-isomorphic) 12-j symbols
with the Q 3 and Wagner graph
Wagner graph
In the mathematical field of graph theory, the Wagner graph is a 3-regular graph with 8 vertices and 12 edges. It is the 8-vertex Möbius ladder graph.-Properties:...
s on 8 vertices.
Symmetry relations are generally representative of the
automorphism group of these graphs.
See also
- Clebsch-Gordan coefficient
- 3-jm symbol
- Racah W-coefficientRacah W-coefficientRacah's W-coefficients were introduced by Giulio Racah in 1942. These coefficients have a purely mathematical definition. In physics they are used in calculations involving the quantum mechanical description of angular momentum, for example in atomic theory....
- 6-j symbol