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

$\sqrt \begin j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end = \langle ( (j_1j_2)j_3,(j_4j_5)j_6)j_9 , ((j_1 j_4)j_7,(j_2j_5)j_8)j_9\rangle.$

Recoupling of four angular momentum vectors

Coupling of two angular momenta $\mathbf_1$ and $\mathbf_2$ is the construction of simultaneous eigenfunctions of $\mathbf^2$ and $J_z$, where $\mathbf=\mathbf_1+\mathbf_2$, 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 $\mathbf_1$, $\mathbf_2$, $\mathbf_4$, and $\mathbf_5$ may be written as
:$, ((j_1j_2)j_3, (j_4j_5)j_6)j_9m_9\rangle.$
Alternatively, one may first couple $\mathbf_1$ and $\mathbf_4$ to $\mathbf_7$ and $\mathbf_2$ and $\mathbf_5$ to $\mathbf_8$, before coupling $\mathbf_7$ and $\mathbf_8$ to $\mathbf_9$:
:$, ((j_1j_4)j_7, (j_2j_5)j_8)j_9m_9\rangle.$
Both sets of functions provide a complete, orthonormal basis for the space with dimension $(2j_1+1)(2j_2+1)(2j_4+1)(2j_5+1)$ spanned by
:$, j_1 m_1\rangle , j_2 m_2\rangle , j_4 m_4\rangle , j_5 m_5\rangle, \;\; m_1=-j_1,\ldots,j_1;\;\; m_2=-j_2,\ldots,j_2;\;\; m_4=-j_4,\ldots,j_4;\;\;m_5=-j_5,\ldots,j_5.$
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 ($m_9$):
:$, ((j_1j_4)j_7, (j_2j_5)j_8)j_9m_9\rangle = \sum_\sum_ , ((j_1j_2)j_3, (j_4j_5)j_6)j_9m_9\rangle \langle ( (j_1j_2)j_3,(j_4j_5)j_6)j_9 , ((j_1 j_4)j_7,(j_2j_5)j_8)j_9\rangle.$

Symmetry relations

A 9-''j'' symbol is invariant under reflection about either diagonal as well as even permutations of its rows or columns:
:$\begin j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end = \begin j_1 & j_4 & j_7\\ j_2 & j_5 & j_8\\ j_3 & j_6 & j_9 \end = \begin j_9 & j_6 & j_3\\ j_8 & j_5 & j_2\\ j_7 & j_4 & j_1 \end = \begin j_7 & j_4 & j_1\\ j_9 & j_6 & j_3\\ j_8 & j_5 & j_2 \end.$

An odd permutation of rows or columns yields a phase factor $(-1)^S$, where
:$S=\sum_^9 j_i.$
For example:
:$\begin j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end = (-1)^S \begin j_4 & j_5 & j_6\\ j_1 & j_2 & j_3\\ j_7 & j_8 & j_9 \end = (-1)^S \begin j_2 & j_1 & j_3\\ j_5 & j_4 & j_6\\ j_8 & j_7 & j_9 \end.$

Reduction to 6j symbols

The 9-''j'' symbols can be calculated as sums over triple-products of 6-''j'' symbols where the summation extends over all admitted by the triangle conditions in the factors:
:$\begin j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6 \\ j_7 & j_8 & j_9 \end = \sum_x (-1)^(2 x + 1) \begin j_1 & j_4 & j_7 \\ j_8 & j_9 & x \end \begin j_2 & j_5 & j_8 \\ j_4 & x & j_6 \end \begin j_3 & j_6 & j_9 \\ x & j_1 & j_2 \end$.

Special case

When $j_9=0$ the 9-''j'' symbol is proportional to a 6-j symbol:
:$\begin j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & 0 \end = \frac (-1)^ \begin j_1 & j_2 & j_3\\ j_5 & j_4 & j_7 \end.$

Orthogonality relation

The 9-''j'' symbols satisfy this orthogonality relation:
:$\sum_ (2j_7+1)(2j_8+1) \begin j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end \begin j_1 & j_2 & j_3'\\ j_4 & j_5 & j_6'\\ j_7 & j_8 & j_9 \end = \frac .$
The ''triangular delta'' is equal to 1 when the triad (''j''₁, ''j''₂, ''j''₃) satisfies the triangle conditions, and zero otherwise.

3''n''-j symbols

The 6-j symbol is the first representative, , of -''j'' symbols 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 -''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 (''K''_3,3), and the two distinct (non-isomorphic) 12-''j'' symbols with the ''Q''₃ and Wagner graphs on 8 vertices.
Symmetry relations are generally representative of the automorphism group of these graphs.

See also

* Clebsch–Gordan coefficients
* 3-j symbol, also called 3-jm symbol
* Racah W-coefficient
* 6-j symbol

References

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External links

* (Gives answer in exact fractions)
* (Answer as floating point numbers)
*
* (accurate; C, fortran, python)
* {{cite web
, first1=H.T.
, last1=Johansson
, title=(FASTWIGXJ)
, url=http://fy.chalmers.se/subatom/fastwigxj/
(fast lookup, accurate; C, fortran)

Rotational symmetry
Representation theory of Lie groups
Quantum mechanics