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 $T^{(k)}$ and $U^{(k)}$ act, respectively, on subsystems 1 and 2 of a system, with subsystem 1 characterized by angular momentum ${\bf j}_1$ and subsystem 2 by the angular momentum ${\bf j}_2$. Then the matrix elements of the scalar product of these two tensor operators in the coupled basis ${\bf J}={\bf j}_1+{\bf j}_2$ are given by

$(\tau_1'j_1'\tau_2'j_2'J'M'\vert T^{(k)}\cdot U^{(k)}\vert\tau_1 j_1\tau_2 j_2 JM)$
$=\delta_{JJ'}\delta_{MM'}(-1)^{j_1+j_2'+J} \left\{\matrix{J & j_2' & j_1'\cr k ... ...\vert\vert\tau_1 j_1)(\tau_2' j_2'\vert\vert U^{(k)}\vert\vert\tau_2 j_2),\quad$	(1)

where $\left\{\matrix{J & j_2' & j_1'\cr k & j_1 & j_2\cr}\right\}$ is the Wigner $6j$-symbol and $\tau_1$ and $\tau_2$ represent additional pertinent quantum numbers characterizing subsystems 1 and 2 (Gordy and Cook 1984).

Edmonds (1968) gives analytic forms of the $6j$-symbol for simple cases, and Shore and Menzel (1968) and Gordy and Cook (1984) give

$\left\{\matrix{a & b & c\cr 0 & c & b\cr}\right\}={(-1)^s\over\sqrt{(2b+1)(2c+1)}}$	(2)
$\left\{\matrix{a & b & c\cr 1 & c & b\cr}\right\}={2(-1)^{s+1}X\over\sqrt{2b(2b+1)(2b+2)2c(2c+1)(2c+2)}}$	(3)
$\left\{\matrix{a & b & c\cr 2 & c & b\cr}\right\}={2(-1)^s[3X(X-1)-4b(b+1)c(c+1)]\over\sqrt{(2b-1)2b(2b+1)(2b+2)(2b+3)(2c-1)2c(2c+1)(2c+2)(2c+3)}},\quad$	(4)

where

$$s \equiv a+b+c$$ (5)

$$X \equiv b(b+1)+c(c+1)-a(a+1).$$ (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 $6j$-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 `$6j$' 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 $3j$ and $6j$ Symbols. Cambridge, MA: MIT Press, 1959.

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