Wigner 6j-Symbol

The Wigner -symbols (Messiah 1962, p. 1062), commonly simply called the -symbols, are a generalization of Clebsch-Gordan coefficients and Wigner 3j-symbol that arise in the coupling of three angular momenta. They are variously called the " symbols" (Messiah 1962, p. 1062) or 6- symbols (Shore and Menzel 1968, p. 279).

The Wigner -symbols are returned by the Wolfram Language function SixJSymbol[j1, j2, j3, j4, j5, j6].

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

$(tau_1^'j_1^'tau_2^'j_2^'J^'M^'|T^((k))·U^((k))|tau_1j_1tau_2j_2JM) =delta_(JJ^')delta_(MM^')(-1)^(j_1+j_2^'+J){J j_2^' j_1^'; k j_1 j_2}(tau_1^'j_1^'||T^((k))||tau_1j_1)(tau_2^'j_2^'||U^((k))||tau_2j_2),$

(1)

where ${J j_2^' j_1^'; k j_1 j_2}$ is the Wigner -symbol and and represent additional pertinent quantum numbers characterizing subsystems 1 and 2 (Gordy and Cook 1984).

The symbols are denoted ${j_1 j_2 j_3; J_1 J_2 J_3}$ and are defined for integers and half-integers , , , , , whose triads , , , and satisfy the following conditions (Messiah 1962, p. 1063).

1. Each triad satisfies the triangular inequalities.

2. The sum of the elements of each triad is an integer. Therefore, the members of each triad are either all integers or contain two half-integers and one integer.

If these conditions are not satisfied, ${j_1 j_2 j_3; J_1 J_2 J_3}=0$ .

The -symbols are invariant under permutation of their columns, e.g.,

${j_1 j_2 j_3; J_1 J_2 J_3}={j_2 j_1 j_3; J_2 J_1 J_3}$

(2)

and under exchange of two corresponding elements between rows, e.g.,

${j_1 j_2 j_3; J_1 J_2 J_3}={J_1 J_2 j_3; j_1 j_2 J_3}$

(3)

(Messiah 1962, pp. 1063-1064).

The -symbols can be computed using the Racah formula

${j_1 j_2 j_3; J_1 J_2 J_3} =sqrt(Delta(j_1j_2j_3)Delta(j_1J_2J_3)Delta(J_1j_2J_3)Delta(J_1J_2j_3))×sum_(t)((-1)^t(t+1)!)/(f(t)),$

(4)

where is a triangle coefficient,

f(t)=(t-j_1-j_2-j_3)!(t-j_1-J_2-J_3)!(t-J_1-j_2-J_3)!(t-J_1-J_2-j_3)!(j_1+j_2+J_1+J_2-t)!(j_2+j_3+J_2+J_3-t)!(j_3+j_1+J_3+J_1-t)!,

(5)

and the sum is over all integers for which the factorials in all have nonnegative arguments (Wigner 1959; Messiah 1962, p. 1065; Shore and Menzel 1968, p. 279). In particular, the number of terms is equal to , where is the smallest of the twelve numbers

j_1+j_2-j_3 j_1+J_2-J_3 J_1+j_2-J_3 J_1+J_2-j_3; j_2+j_3-j_1 J_2+J_3-j_1 j_2+J_3-J_1 J_2+j_3-J_1; j_3+j_1-j_2 J_3+j_1-J_2 J_3+J_1-j_2 j_3+J_1-J_2

(6)

(Messiah 1962, p. 1064).

The symbols satisfy the so-called Racah-Elliot and orthogonality relations,

$sum_(x)(-1)^(2x)(2x+1){a b x; a b f}=1$	(7)
$sum_(x)(-1)^(a+b+x)(2x+1){a b x; b a f}$	(8)
	(9)
$sum_(x)(2x+1){a b x; c d f}{c d x; a b g}=(delta_(fg))/(2f+1)$	(10)
$sum_(x)(-1)^(f+g+x)(2x+1){a b x; c d f}{c d x; b a g}$	(11)
	(12)
$sum_(x)(-1)^(a+b+c+d+e+f+g+h+x+j)(2x+1){a b x; c d g}×{c d x; e f h}{e f x; b a j}$	(13)
	(14)

(Messiah 1962, p. 1065).

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

			(15)
			(16)
			(17)

where

			(18)
			(19)

(Edmonds 1968; Shore and Menzel 1968, p. 281; Gordy and Cook 1984, p. 809). Note that since must be an integer, , so replacing the definition of with its negative above gives an equivalent result.

Messiah (1962, p. 1066) gives the additional special cases

			(20)
			(21)

for .

The Wigner -symbols are related to the Racah W-coefficients by

$(-1)^(a+b+c+d)W(abcd;ef)={a b c; d e f}$

(22)

(Messiah 1962, p. 1062; Shore and Menzel 1968, p. 279).

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References

Biedenharn, L. C. and Louck, J. D. The Racah-Wigner Algebra in Quantum Theory. Reading, MA: Addison-Wesley, 1981.Biedenharn, L. C. and Louck, J. D. Angular Momentum in Quantum Physics: Theory and Applications. Reading, MA: Addison-Wesley, 1981.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 '

' Symbols." Appendix C.II in Quantum Mechanics, Vol. 2. Amsterdam, Netherlands: North-Holland, pp. 567-569 and 1061-1066, 1962.Racah, G. "Theory of Complex Spectra. II." Phys. Rev. 62, 438-462, 1942.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.Wigner, E. P. Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, expanded and improved ed. New York: Academic Press, 1959.

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Wigner 6j-Symbol

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Weisstein, Eric W. "Wigner 6j-Symbol." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Wigner6j-Symbol.html

Wigner 6j-Symbol

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