Wigner 9j-Symbol

A generalization of Clebsch-Gordan Coefficients and Wigner 3j-Symbol and Wigner 6j-Symbol which arises in the coupling of four angular momenta and can be written in terms of the Wigner 3j-Symbol and Wigner 6j-Symbol. Let tensor operators and act, respectively, on subsystems 1 and 2. Then the reduced matrix element of the product of these two irreducible operators in the coupled representation is given in terms of the reduced matrix elements of the individual operators in the uncoupled representation by
 (1)
where is a Wigner -symbol (Gordy and Cook 1984).

Shore and Menzel (1968) give the explicit formulas

 (2) (3) (4)

See also Clebsch-Gordan Coefficient, Racah V-Coefficient, Racah W-Coefficient, Wigner 3j-Symbol, Wigner 6j-Symbol

References

Gordy, W. and Cook, R. L. Microwave Molecular Spectra, 3rd ed. New York: Wiley, pp. 807-809, 1984.

Messiah, A. `' Symbols.'' Appendix C.III in Quantum Mechanics, Vol. 2. Amsterdam, Netherlands: North-Holland, pp. 567-569 and 1066-1068, 1962.

Shore, B. W. and Menzel, D. H. Principles of Atomic Spectra. New York: Wiley, pp. 279-284, 1968.

© 1996-9 Eric W. Weisstein
1999-05-26