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 $T^{(k_1)}$ and $U^{(k_2)}$ act, respectively, on subsystems 1 and 2. Then the reduced matrix element of the product $T^{(k_1)}\times U^{(k_2)}$ 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

$(\tau'\tau_1'j_1'\tau_2'j_2'J'\vert\vert[T^{(k_1)}\times U^{(k_2)}]^{(k)}\vert\vert\tau \tau_1 j_1\tau_2 j_2 J)$

$=\sqrt{(2J+1)(2J'+1)(2k+1)}\sum_{\tau''} \left\{\matrix{j_1' & j_1 & k_1\cr j_2' & j_2 & k_2\cr J' & J & k\cr}\right\}$

$\times(\tau'\tau_1'j_1'\vert\vert T^{(k_1)}\vert\vert\tau''\tau_1j_1)(\tau''\tau_2'j_2'\vert\vert U^{(k_2)}\vert\vert\tau \tau_2 j_2),\quad$ (1)

where $\left\{\matrix{j_1' & j_1 & k_1\cr j_2' & j_2 & k_2\cr J' & J & k\cr}\right\}$ is a Wigner -symbol (Gordy and Cook 1984).

Shore and Menzel (1968) give the explicit formulas

$\left\{\matrix{a & b & C\cr d & e & F\cr G & H & J\cr}\right\}=\sum_x (-1)^{2x}(2x+1)$

$\times\left\{\matrix{a & b & C\cr F & J & x\cr}\right\}\left\{\matrix{d & e & F\cr b & x & H\cr}\right\}\left\{\matrix{G & H & J\cr x & a & d\cr}\right\}$ (2)

$\left\{\matrix{a & b & J\cr c & d & J\cr K & K & 0\cr}\right\} ={(-1)^{b+c+J+K}\over\sqrt{(2J+1)(2K+1)}}\left\{\matrix{a & b & J\cr d & c & K\cr}\right\}$ (3)

$\left\{\matrix{S & S & 1\cr L & L & 2\cr J & J & 1\cr}\right\}={\left\{\matrix{... ... S & 1\cr}\right\}\over\left\{\matrix{2 & L & L\cr L & 1 & 1\cr}\right\}}.\quad$ (4)

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.

$(\tau'\tau_1'j_1'\tau_2'j_2'J'\vert\vert[T^{(k_1)}\times U^{(k_2)}]^{(k)}\vert\vert\tau \tau_1 j_1\tau_2 j_2 J)$
$=\sqrt{(2J+1)(2J'+1)(2k+1)}\sum_{\tau''} \left\{\matrix{j_1' & j_1 & k_1\cr j_2' & j_2 & k_2\cr J' & J & k\cr}\right\}$
$\times(\tau'\tau_1'j_1'\vert\vert T^{(k_1)}\vert\vert\tau''\tau_1j_1)(\tau''\tau_2'j_2'\vert\vert U^{(k_2)}\vert\vert\tau \tau_2 j_2),\quad$	(1)

$\left\{\matrix{a & b & C\cr d & e & F\cr G & H & J\cr}\right\}=\sum_x (-1)^{2x}(2x+1)$
$\times\left\{\matrix{a & b & C\cr F & J & x\cr}\right\}\left\{\matrix{d & e & F\cr b & x & H\cr}\right\}\left\{\matrix{G & H & J\cr x & a & d\cr}\right\}$	(2)
$\left\{\matrix{a & b & J\cr c & d & J\cr K & K & 0\cr}\right\} ={(-1)^{b+c+J+K}\over\sqrt{(2J+1)(2K+1)}}\left\{\matrix{a & b & J\cr d & c & K\cr}\right\}$	(3)
$\left\{\matrix{S & S & 1\cr L & L & 2\cr J & J & 1\cr}\right\}={\left\{\matrix{... ... S & 1\cr}\right\}\over\left\{\matrix{2 & L & L\cr L & 1 & 1\cr}\right\}}.\quad$	(4)