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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

$$[(2j_3+1)(2j_6+1)(2j_7+1)(2j_8+1)]^\frac{1}{2} \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end{Bmatrix} = \langle ( (j_1j_2)j_3,(j_4j_5)j_6)j_9 | ((j_1 j_4)j_7,(j_2j_5)j_8)j_9\rangle$$.

ecoupling of four angular momentum vectors

Coupling of two angular momenta $$\mathbf{j}_1$$ and$$\mathbf{j}_2 \( is the construction of simultaneous eigenfunctions of \( \mathbf{J}^2\( and J_z, where \( \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 $$\mathbf{j}_1, \mathbf{j}_2, \mathbf{j}_4,$$ and $$\mathbf{j}_5$$ may be written as

$$| ((j_1j_2)j_3, (j_4j_5)j_6)j_9m_9\rangle.$$

Alternatively, one may first couple $$\mathbf{j}_1$$ and $$\mathbf{j}_4$$ to $$\mathbf{j}_7$$ and $$\mathbf{j}_2$$ and $$\mathbf{j}_5$$ to $$\mathbf{j}_8$$, before coupling $$\mathbf{j}_7$$ and $$\mathbf{j}_8$$ to $$\mathbf{j}_9$$:

$$|((j_1j_4)j_7, (j_2j_5)j_8)j_9m_9\rangle.$$

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 ($$m_9$$):

$$|((j_1j_4)j_7, (j_2j_5)j_8)j_9m_9\rangle = \sum_{j_3}\sum_{j6} | ((j_1j_2)j_3, (j_4j_5)j_6)j_9m_9\rangle \langle ( (j_1j_2)j_3,(j_4j_5)j_6)j_9 | ((j_1 j_4)j_7,(j_2j_5)j_8)j_9\rangle.$$

Symmetry relations

A 9-j symbol is invariant under reflection 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 $$(-1)^S$$, where

$$S=\sum_{i=1}^9 j_i.$$

For example:

$$\begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end{Bmatrix} = (-1)^S \begin{Bmatrix} j_4 & j_5 & j_6\\ j_1 & j_2 & j_3\\ j_7 & j_8 & j_9 \end{Bmatrix} = (-1)^S \begin{Bmatrix} j_2 & j_1 & j_3\\ j_5 & j_4 & j_6\\ j_8 & j_7 & j_9 \end{Bmatrix}.$$

Reduction to 6j symbols

The 9j symbols can be calculated as sums over triple-products of 6j symbols where the summation extends over all x admitted by the triangle conditions in the factors:

$$\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} . Special case When \( j_9=0$$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 $$\{j_1j_2j_3\} \$$ is equal to one if the triad $$(j_1j_2j_3)$$satisfies the triangular conditions and zero otherwise.

3n-j symbols

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, and the two different (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.

Clebsch–Gordan coefficients
3-jm symbol
Racah W-coefficient
6-j symbol

References

Biedenharn, L. C.; van Dam, H. (1965). Quantum Theory of Angular Momentum: A collection of Reprints and Original Papers. New York: Academic Press. ISBN 0120960567.
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Condon, Edward U.; Shortley, G. H. (1970). "Chapter 3". The Theory of Atomic Spectra. Cambridge: Cambridge University Press. ISBN 0-521-09209-4.
Maximon, Leonard C. (2010), "3j,6j,9j Symbols", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
Messiah, Albert (1981). Quantum Mechanics (Volume II) (12th ed.). New York: North Holland Publishing. ISBN 0-7204-0045-7.
Brink, D. M.; Satchler, G. R. (1993). "Chapter 2". Angular Momentum (3rd ed.). Oxford: Clarendon Press. ISBN 0-19-851759-9.
Zare, Richard N. (1988). "Chapter 2". Angular Momentum. New York: John Wiley. ISBN 0-471-85892-7.
Biedenharn, L. C.; Louck, J. D. (1981). Angular Momentum in Quantum Physics. Reading, Massachusetts: Addison-Wesley. ISBN 0201135078.
Varshalovich, D. A.; Moskalev, A. N.; Khersonskii, V. K. (1988). Quantum Theory of Angular Momentum. Singapore: World Scientific. ISBN 9971-50-107-4.
Jahn, H. A.; Hope, J. (1954). "Symmetry properties of the Wigner 9j symbol". Physical Review 93 (2): 318. Bibcode:1954PhRv...93..318J. doi:10.1103/PhysRev.93.318.

Stone, Anthony. "Wigner coefficient calculator". (Gives answer in exact fractions)
Plasma Laboratory of Weizmann Institute of Science. "369j-symbol calculator". (Answer as floating point numbers)
Fack, Veerl; van Dyck, Dries. "GYutsis Applet".
Johansson, H.T.; Forssén, C. "(WIGXJPF)". (accurate; C, fortran, python)