6-j symbol

Wigner's 6-j symbols were introduced by Eugene Paul Wigner in 1940, and published in 1965. They are related to Racah's W-coefficients by

$\begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} = (-1)^{j_1%2Bj_2%2Bj_4%2Bj_5}W(j_1j_2j_5j_4;j_3j_6).$

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

1 Symmetry relations
2 Special case
3 Orthogonality relation
4 See also
5 References
6 External links

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:

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

The 6-j symbol

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

is zero unless $j_1$ , $j_2$ , and $j_3$ satisfy triangle conditions, i.e.,

$j_1 = |j_2-j_3|, \ldots, j_2%2Bj_3.$

In combination with the symmetry relation for interchanging upper and lower arguments this shows that triangle conditions must also be satisfied for $(j_1,j_5,j_6)$ , $(j_4,j_2,j_6)$ , and $(j_4,j_5,j_3)$ .

Special case

When $j_6=0$ the expression for the 6-j symbol is:

$\begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & 0 \end{Bmatrix} = \frac{\delta_{j_2,j_4}\delta_{j_1,j_5}}{\sqrt{(2j_1%2B1)(2j_2%2B1)}} (-1)^{j_1%2Bj_2%2Bj_3}\Delta(j_1,j_2,j_3).$

The function $\Delta(j_1,j_2,j_3)$ is equal to 1 when $(j_1,j_2,j_3)$ satisfy 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:

$\sum_{j_3} (2j_3%2B1) \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6 \end{Bmatrix} \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6' \end{Bmatrix} = \frac{\delta_{j_6^{}j_6'}}{2j_6%2B1} \Delta(j_1,j_5,j_6) \Delta(j_4,j_2,j_6).$

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, MR 2723248, 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)
Frederik J Simons: Matlab software archive, the code SIXJ.M
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.