Wigner 3j-Symbol

The Wigner 3j-symbols (j_1 j_2 j_3; m_1 m_2 m_3), also known as "3j symbols" (Messiah 1962, p. 1056) or Wigner coefficients (Shore and Menzel 1968, p. 275) are quantities that arise in considering coupled angular momenta in two quantum systems.

They are returned by the Wolfram Language function ThreeJSymbol[{j1, m1}, {j2, m2}, {j3, m3}].

The parameters of the 3j symbol (j_1 j_2 J; m_1 m_2 -M) (where m_3 has been written as -M) are either integers or half-integers. Additionally, they satisfy the follow selection rules (Messiah 1962, pp. 1054-1056; Shore and Menzel 1968, p. 272).

1. m_1 in {-|j_1|,...,|j_1|}, m_2 in {-|j_2|,...,|j_2|}, and M in {-|J|,...,|J|}.

2. m_1+m_2=M.

3. The triangular inequalities |j_1-j_2|<=J<=j_1+j_2.

4. Integer perimeter rule: j_1+j_2+J is an integer.

Note that not all these rules are independent, since rule (4) is implied by the other three. If these conditions are not satisfied, (j_1 j_2 J; m_1 m_2 -M)=0.

The Wigner 3j-symbols have the symmetries

(j_1 j_2 j; m_1 m_2 m)=(j_2 j j_1; m_2 m m_1)
=(j j_1 j_2; m m_1 m_2)
=(-1)^(j_1+j_2+j)(j_2 j_1 j; m_2 m_1 m)
=(-1)^(j_1+j_2+j)(j_1 j j_2; m_1 m m_2)
=(-1)^(j_1+j_2+j)(j j_2 j_1; m m_2 m_1)
=(-1)^(j_1+j_2+j)(j_1 j_2 j; -m_1 -m_2 -m)

(Messiah 1962, p. 1056).

The 3j-symbols can be computed using the Racah formula

 (a b c; alpha beta gamma)=(-1)^(a-b-gamma) 

where Delta(abc) is a triangle coefficient,


and the sum is over all integers t for which the factorials in f(t) all have nonnegative arguments (Messiah 1962, p. 1058; Shore and Menzel 1968, p. 273). In particular, the number of terms is equal to nu+1, where nu is the smallest of the nine numbers

 a+/-alpha b+/-beta c+/-gamma; a+b-c b+c-a c+a-b

(Messiah 1962, p. 1058).

The symbols obey the orthogonality relations

 sum_(j,m)(2j+1)(j_1 j_2 j; m_1 m_2 m)(j_1 j_2 j; m_1^' m_2^' m)=delta_(m_1m_1^')delta_(m_2m_2^')
 sum_(m_1,m_2)(2j+1)(j_1 j_2 j; m_1 m_2 m)(j_1 j_2 j^'; m_1 m_2 m^')=delta_(jj^')delta_(mm^'),

where delta_(ij) is the Kronecker delta.

General formulas are very complicated, but some specific cases are

(l l 0; m -m 0)=((-1)^(l-m))/(sqrt(2l+1))
(j_1 j_2 j_1+j_2; m_1 m_2 -M)=(-1)^(j_1-j_2+M)[((2j_1)!(2j_2)!)/((2j_1+2j_2+1)!)((j_1+j_2+M)!(j_1+j_2-M)!)/((j_1+m_1)!(j_1-m_1)!(j_2+m_2)!(j_2-m_2)!)]^(1/2)
(j_1 j_2 j; j_1 -j_1 -m)=(-1)^(-j_1+j_2+m)[((2j_1)!(-j_1+j_2+j)!)/((j_1+j_2+j+1)!(j_1-j_2+j)!)((j_1+j_2+m)!(j-m)!)/((j_1+j_2-j)!(-j_1+j_2-m)!(j+m)!)]^(1/2)
(j_1 j_2 j; 0 0 0)={(-1)^gsqrt(((2g-2j_1)!(2g-2j_2)!(2g-2j)!)/((2g+1)!))(g!)/((g-j_1)!(g-j_2)!(g-j)!) if J=2g; 0 if J=2g+1,

for J=j_1+j_2+j (Condon and Shortley 1951, pp. 76-77; Messiah 1962, pp. 1058-1060; Shore and Menzel 1968, p. 275; Abramowitz and Stegun 1972, pp. 1006-1010).

For spherical harmonics Y_l^m(theta,phi),

 Y_(l_1)^(m_1)(theta,phi)Y_(l_2)^(m_2)(theta,phi)=sum_(l,m)sqrt(((2l_1+1)(2l_2+1)(2l+1))/(4pi))(l_1 l_2 l; m_1 m_2 m)Y^__l^m(theta,phi)(l_1 l_2 l; 0 0 0).

For values of l_3 obeying the triangle condition Delta(l_1l_2l_3),

=sqrt(((2l_1+1)(2l_2+1)(2l_3+1))/(4pi))(l_1 l_2 l_3; 0 0 0)(l_1 l_2 l_3; m_1 m_2 m_3)


 1/2intP_(l_1)(costheta)P_(l_2)(costheta)P_(l_3)(costheta)sinthetadtheta=(l_1 l_2 l_3; 0 0 0)^2.

They can be expressed using the related Clebsch-Gordan coefficients C_(m_1m_2)^j=(j_1j_2m_1m_2|j_1j_2jm) (Condon and Shortley 1951, pp. 74-75; Wigner 1959, p. 206), or Racah V-coefficients V(j_1j_2j;m_1m_2m).

Connections among the Wigner 3j-, Clebsch-Gordan, and Racah V-symbols are given by

 (j_1j_2m_1m_2|j_1j_2jm)=(-1)^(m+j_1-j_2)sqrt(2j+1)(j_1 j_2 j; m_1 m_2 -m)
 V(j_1j_2j;m_1m_2m)=(-1)^(-j_1+j_2+j)(j_1 j_2 j_1; m_2 m_1 m_2).

See also

Clebsch-Gordan Coefficient, Racah V-Coefficient, Racah W-Coefficient, Wigner 6j-Symbol, Wigner 9j-Symbol

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Abramowitz, M. and Stegun, I. A. (Eds.). "Vector-Addition Coefficients." §27.9 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1006-1010, 1972.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.Condon, E. U. and Shortley, G. The Theory of Atomic Spectra. Cambridge, England: Cambridge University Press, Shalit, A. and Talmi, I. Nuclear Shell Theory. New York: Academic Press, 1963.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. 804-811, 1984.Messiah, A. "Clebsch-Gordan (C.-G.) Coefficients and '3j' Symbols." Appendix C.I in Quantum Mechanics, Vol. 2. Amsterdam, Netherlands: North-Holland, pp. 1054-1060, 1962.Racah, G. "Theory of Complex Spectra. II." Phys. Rev. 62, 438-462, 1942.Rose, M. E. Elementary Theory of Angular Momentum. New York: Dover, 1995.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. 275-276, 1968.Sobel'man, I. I. "Angular Momenta." Ch. 4 in Atomic Spectra and Radiative Transitions, 2nd ed. Berlin: Springer-Verlag, 1992.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 3j-Symbol

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Weisstein, Eric W. "Wigner 3j-Symbol." From MathWorld--A Wolfram Web Resource.

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