9-j symbol

$\sqrt{(2 j_{3} + 1) (2 j_{6} + 1) (2 j_{7} + 1) (2 j_{8} + 1)} {\begin{matrix} j_{1} & j_{2} & j_{3} \\ j_{4} & j_{5} & j_{6} \\ j_{7} & j_{8} & j_{9} \end{matrix}}$ $= ⟨ ((j_{1} j_{2}) j_{3}, (j_{4} j_{5}) j_{6}) j_{9} | ((j_{1} j_{4}) j_{7}, (j_{2} j_{5}) j_{8}) j_{9} ⟩ .$

Recoupling of four angular momentum vectors

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

| ((j_{1} j_{2}) j_{3}, (j_{4} j_{5}) j_{6}) j_{9} m_{9} ⟩ .

Alternatively, one may first couple $j_{1}$ and $j_{4}$ to $j_{7}$ and $j_{2}$ and $j_{5}$ to $j_{8}$ , before coupling $j_{7}$ and $j_{8}$ to $j_{9}$ :

| ((j_{1} j_{4}) j_{7}, (j_{2} j_{5}) j_{8}) j_{9} m_{9} ⟩ .

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

| j_{1} m_{1} ⟩ | j_{2} m_{2} ⟩ | j_{4} m_{4} ⟩ | j_{5} m_{5} ⟩, m_{1} = - j_{1}, \dots, j_{1}; m_{2} = - j_{2}, \dots, j_{2}; m_{4} = - j_{4}, \dots, j_{4}; m_{5} = - j_{5}, \dots, 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_{1} j_{4}) j_{7}, (j_{2} j_{5}) j_{8}) j_{9} m_{9} ⟩ = \sum_{j_{3}} \sum_{j_{6}} | ((j_{1} j_{2}) j_{3}, (j_{4} j_{5}) j_{6}) j_{9} m_{9} ⟩ ⟨ ((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} ⟩ .

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Recoupling of four angular momentum vectors

Symmetry relations

Reduction to 6j symbols

Special case

Orthogonality relation

3n-j symbols

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