Wigner 6j-Symbol

A generalization of Clebsch-Gordan Coefficients and Wigner 3j-Symbol which arises in the coupling of three angular momenta. Let tensor operators and act, respectively, on subsystems 1 and 2 of a system, with subsystem 1 characterized by angular momentum and subsystem 2 by the angular momentum . Then the matrix elements of the scalar product of these two tensor operators in the coupled basis are given by

 (1)
where is the Wigner -symbol and and represent additional pertinent quantum numbers characterizing subsystems 1 and 2 (Gordy and Cook 1984).

Edmonds (1968) gives analytic forms of the -symbol for simple cases, and Shore and Menzel (1968) and Gordy and Cook (1984) give

 (2) (3) (4)
where

 (5) (6)

See also Clebsch-Gordan Coefficient, Racah V-Coefficient, Racah W-Coefficient, Wigner 3j-Symbol, Wigner 9j-Symbol

References

Carter, J. S.; Flath, D. E.; and Saito, M. The Classical and Quantum -Symbols. Princeton, NJ: Princeton University Press, 1995.

Edmonds, A. R. Angular Momentum in Quantum Mechanics, 2nd ed., rev. printing. Princeton, NJ: Princeton University Press, 1968.

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

Messiah, A. Racah Coefficients and `' Symbols.'' Appendix C.II in Quantum Mechanics, Vol. 2. Amsterdam, Netherlands: North-Holland, pp. 567-569 and 1061-1066, 1962.

Rotenberg, M.; Bivens, R.; Metropolis, N.; and Wooten, J. K. The and Symbols. Cambridge, MA: MIT Press, 1959.

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