6-j symbol

Wigner's 6-j symbols were introduced by Eugene Paul Wigner in 1940, and published in 1965. They are defined by a sum over products of four 3jm symbols,

$\begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} = \sum_{m_i} (-1)^S \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & -m_3 \end{pmatrix} \begin{pmatrix} j_1 & j_5 & j_6\\ -m_1 & m_5 & m_6 \end{pmatrix} \begin{pmatrix} j_4 & j_5 & j_3\\ m_4 & -m_5 & m_3 \end{pmatrix} \begin{pmatrix} j_4 & j_2 & j_6\\ -m_4 & -m_2 & -m_6 \end{pmatrix} .$

with phase $S=\sum_{k=1}^6 (j_k-m_k)$. The summation is over all six m_i, effectively confined by the selection rules of the 3jm symbols. They are related to Racah's W-coefficients by

$\begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} = (-1)^{j_1+j_2+j_4+j_5}W(j_1j_2j_5j_4;j_3j_6).$

They have higher symmetry than Racah's W-coefficients.

Symmetry relations

The 6-j symbol is invariant under the permutation of any two columns:

$\begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} = \begin{Bmatrix} j_2 & j_1 & j_3\\ j_5 & j_4 & j_6 \end{Bmatrix} = \begin{Bmatrix} j_1 & j_3 & j_2\\ j_4 & j_6 & j_5 \end{Bmatrix} = \begin{Bmatrix} j_3 & j_2 & j_1\\ j_6 & j_5 & j_4 \end{Bmatrix}.$

The 6-j symbol is also invariant if upper and lower arguments are interchanged in any two columns:

$\begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} = \begin{Bmatrix} j_4 & j_5 & j_3\\ j_1 & j_2 & j_6 \end{Bmatrix} = \begin{Bmatrix} j_1 & j_5 & j_6\\ j_4 & j_2 & j_3 \end{Bmatrix} = \begin{Bmatrix} j_4 & j_2 & j_6\\ j_1 & j_5 & j_3 \end{Bmatrix}.$

These equations reflect the 24 symmetry operations of the automorphism group that leave the associated tetrahedral Yutsis graph with 6 edges invariant: mirror operations that exchange two vertices and a swap an adjacent pair of edges.

The 6-j symbol

$\begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix}$

is zero unless j₁, j₂, and j₃ satisfy triangle conditions, i.e.,

$j_1 = |j_2-j_3|, \ldots, j_2+j_3.$

In combination with the symmetry relation for interchanging upper and lower arguments this shows that triangle conditions must also be satisfied for (j₁,j₅,j₆), (j₄,j₂,j₆), and (j₄,j₅,j₃).

Special case

When j₆ = 0 the expression for the 6-j symbol is:

$\begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & 0 \end{Bmatrix} = \frac{\delta_{j_2,j_4}\delta_{j_1,j_5}}{\sqrt{(2j_1+1)(2j_2+1)}} (-1)^{j_1+j_2+j_3}\{j_1,j_2,j_3\}.$

The function {j₁,j₂,j₃} is equal to 1 when the triad (j₁,j₂,j₃) satisfies the triangle conditions, and zero otherwise. The symmetry relations can be used to find the expression when another j is equal to zero.

Orthogonality relation

The 6-j symbols satisfy this orthogonality relation:

$\sum_{j_3} (2j_3+1) \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6' \end{Bmatrix} = \frac{\delta_{j_6^{}j_6'}}{2j_6+1} \{j_1,j_5,j_6\} \{j_4,j_2,j_6\}.$

See also

• Clebsch–Gordan coefficients
• 3-jm symbol
• Racah W-coefficient
• 9-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 0-12-096056-7. 

• Edmonds, A. R. (1957). Angular Momentum in Quantum Mechanics. Princeton, New Jersey: Princeton University Press. ISBN 0-691-07912-9. 

• 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. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248, http://dlmf.nist.gov/34 
• 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 0-201-13507-8. 

External links

• Regge, T. (1959). "Simmetry Properties of Racah's Coefficients". Nuovo Cimento 11 (1): 116–117. doi:10.1007/BF02724914. 
• Stone, Anthony. "Wigner coefficient calculator". http://www-stone.ch.cam.ac.uk/wigner.shtml.  (Gives exact answer)
• Simons, Frederik J.. "Matlab software archive, the code SIXJ.M". http://geoweb.princeton.edu/people/simons/software.html. 
• Volya, A. "Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator". http://www.volya.net/vc. 
• Plasma Laboratory of Weizmann Institute of Science. "369j-symbol calculator". http://plasma-gate.weizmann.ac.il/369j.html. 
• GNU scientific library. "Coupling coefficients". http://www.gnu.org/software/gsl/manual/html_node/Coupling-Coefficients.html. 
• Mathar, Richard J.. "6j vector coupling coefficients". http://www.strw.leidenuniv.nl/~mathar/progs/6jSymb.  Static table for j₁ <= 11/2. Python3 implementation