9-j symbol

Coupling of two angular momenta ${\displaystyle \mathbf {j} _{1}}$ and ${\displaystyle \mathbf {j} _{2}}$ is the construction of simultaneous eigenfunctions of ${\displaystyle \mathbf {J} ^{2}}$ and ${\displaystyle J_{z}}$, where ${\displaystyle \mathbf {J} =\mathbf {j} _{1}+\mathbf {j} _{2}}$, 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 ${\displaystyle \mathbf {j} _{1}}$, ${\displaystyle \mathbf {j} _{2}}$, ${\displaystyle \mathbf {j} _{4}}$, and ${\displaystyle \mathbf {j} _{5}}$ may be written as

${\displaystyle |((j_{1}j_{2})j_{3},(j_{4}j_{5})j_{6})j_{9}m_{9}\rangle .}$

Alternatively, one may first couple ${\displaystyle \mathbf {j} _{1}}$ and ${\displaystyle \mathbf {j} _{4}}$ to ${\displaystyle \mathbf {j} _{7}}$ and ${\displaystyle \mathbf {j} _{2}}$ and ${\displaystyle \mathbf {j} _{5}}$ to ${\displaystyle \mathbf {j} _{8}}$, before coupling ${\displaystyle \mathbf {j} _{7}}$ and ${\displaystyle \mathbf {j} _{8}}$ to ${\displaystyle \mathbf {j} _{9}}$:

${\displaystyle |((j_{1}j_{4})j_{7},(j_{2}j_{5})j_{8})j_{9}m_{9}\rangle .}$

Both sets of functions provide a complete, orthonormal basis for the space with dimension ${\displaystyle (2j_{1}+1)(2j_{2}+1)(2j_{4}+1)(2j_{5}+1)}$ spanned by

${\displaystyle |j_{1}m_{1}\rangle |j_{2}m_{2}\rangle |j_{4}m_{4}\rangle |j_{5}m_{5}\rangle ,\;\;m_{1}=-j_{1},\ldots ,j_{1};\;\;m_{2}=-j_{2},\ldots ,j_{2};\;\;m_{4}=-j_{4},\ldots ,j_{4};\;\;m_{5}=-j_{5},\ldots ,j_{5}.}$

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 (${\displaystyle m_{9}}$):

${\displaystyle |((j_{1}j_{4})j_{7},(j_{2}j_{5})j_{8})j_{9}m_{9}\rangle =\sum _{j_{3}}\sum _{j_{6}}|((j_{1}j_{2})j_{3},(j_{4}j_{5})j_{6})j_{9}m_{9}\rangle \langle ((j_{1}j_{2})j_{3},(j_{4}j_{5})j_{6})j_{9}|((j_{1}j_{4})j_{7},(j_{2}j_{5})j_{8})j_{9}\rangle .}$

A 9-j symbol is invariant under reflection about either diagonal as well as even permutations of its rows or columns:

${\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}={\begin{Bmatrix}j_{1}&j_{4}&j_{7}\\j_{2}&j_{5}&j_{8}\\j_{3}&j_{6}&j_{9}\end{Bmatrix}}={\begin{Bmatrix}j_{9}&j_{6}&j_{3}\\j_{8}&j_{5}&j_{2}\\j_{7}&j_{4}&j_{1}\end{Bmatrix}}={\begin{Bmatrix}j_{7}&j_{4}&j_{1}\\j_{9}&j_{6}&j_{3}\\j_{8}&j_{5}&j_{2}\end{Bmatrix}}.}$

An odd permutation of rows or columns yields a phase factor ${\displaystyle (-1)^{S}}$, where

${\displaystyle S=\sum _{i=1}^{9}j_{i}.}$

For example:

${\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}=(-1)^{S}{\begin{Bmatrix}j_{4}&j_{5}&j_{6}\\j_{1}&j_{2}&j_{3}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}=(-1)^{S}{\begin{Bmatrix}j_{2}&j_{1}&j_{3}\\j_{5}&j_{4}&j_{6}\\j_{8}&j_{7}&j_{9}\end{Bmatrix}}.}$

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:

${\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}=\sum _{x}(-1)^{2x}(2x+1){\begin{Bmatrix}j_{1}&j_{4}&j_{7}\\j_{8}&j_{9}&x\end{Bmatrix}}{\begin{Bmatrix}j_{2}&j_{5}&j_{8}\\j_{4}&x&j_{6}\end{Bmatrix}}{\begin{Bmatrix}j_{3}&j_{6}&j_{9}\\x&j_{1}&j_{2}\end{Bmatrix}}}$.

When ${\displaystyle j_{9}=0}$ the 9-j symbol is proportional to a 6-j symbol:

${\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&0\end{Bmatrix}}={\frac {\delta _{j_{3},j_{6}}\delta _{j_{7},j_{8}}}{\sqrt {(2j_{3}+1)(2j_{7}+1)}}}(-1)^{j_{2}+j_{3}+j_{4}+j_{7}}{\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{5}&j_{4}&j_{7}\end{Bmatrix}}.}$

The 9-j symbols satisfy this orthogonality relation:

${\displaystyle \sum _{j_{7}j_{8}}(2j_{7}+1)(2j_{8}+1){\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}{\begin{Bmatrix}j_{1}&j_{2}&j_{3}'\\j_{4}&j_{5}&j_{6}'\\j_{7}&j_{8}&j_{9}\end{Bmatrix}}={\frac {\delta _{j_{3}j_{3}'}\delta _{j_{6}j_{6}'}{\begin{Bmatrix}j_{1}&j_{2}&j_{3}\end{Bmatrix}}{\begin{Bmatrix}j_{4}&j_{5}&j_{6}\end{Bmatrix}}{\begin{Bmatrix}j_{3}&j_{6}&j_{9}\end{Bmatrix}}}{(2j_{3}+1)(2j_{6}+1)}}.}$

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 vertices 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 (K_3,3), 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.

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

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