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- Wigner's 9-j symbols were introduced by Eugene Paul Wigner in 1937. They are related to recoupling coefficients involving four angular momenta [(2j_3+1)(2j_6+1)(2j_7+1)(2j_8+1)]^\frac{1}{2} \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end{Bmatrix} \langle (j_3,j_6)j_9 | (j_7,j_8)j_9\rangle. Recoupling of four angular momentum vectors Coupling of two angular momenta <math>\mathbf{j}_1 and <math>\mathbf{j}_2 is the construction of simultaneous eigenfunctions of <math>\mathbf{J}^2 and <math>J_z, where <math>\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 <math>\mathbf{j}_1, <math>\mathbf{j}_2, <math>\mathbf{j}_4, and <math>\mathbf{j}_5 may be written as | (j_3, j_6)j_9m_9\rangle. Alternatively, one may first couple <math>\mathbf{j}_1 and <math>\mathbf{j}_4 to <math>\mathbf{j}_7 and <math>\mathbf{j}_2 and <math>\mathbf{j}_5 to <math>\mathbf{j}_8, before coupling <math>\mathbf{j}_7 and <math>\mathbf{j}_8 to <math>\mathbf{j}_9: |(j_7, j_8)j_9m_9\rangle. Both sets of functions provide a complete, orthonormal basis for the space with dimension <math>(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 (<math>m_9): |(j_7, j_8)j_9m_9\rangle \sum_{j_3}\sum_{j6} | (j_3, j_6)j_9m_9\rangle \langle (j_3,j_6)j_9 | (j_7,j_8)j_9\rangle. Symmetry relations A <math>9-j symbol is invariant under reflection in either diagonal: \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}. The permutation of any two rows or any two columns yields a phase factor <math>(-1)^S, where S\sum_{i1}^9 j_i. For example: \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}. Special case When <math>j_90 the 9-j symbol is proportional to a 6-j symbol: \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}. Orthogonality relation The 9-j symbols satisfy this orthogonality relation: \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_3j_3'}\delta_{j_6j_6'} \{j_1j_2j_3\} \{j_4j_5j_6\} \{j_3j_6j_9\}} {(2j_3+1)(2j_6+1)}. The symbol <math>\{j_1j_2j_3\} is equal to one if the triad <math>(j_1j_2j_3) satisfies the triangular conditions and zero otherwise. See also Clebsch-Gordan coefficient 3-jm symbol Racah W-coefficient 6-j symbol References Boisvert, Ronald F. ; Clark, Charles W. ; Lozier, Daniel M. et al. , eds. (2008), "9-j symbol", Digital Library of Mathematical Functions, N.I.S.T. External links Anthony Stone’s Wigner coefficient calculator (Gives exact answer) 369j-symbol calculator at the Plasma Laboratory of Weizmann Institute of Science (Numerical answer)
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