TOPICS
Search

Racah W-Coefficient

The Racah W-coefficients, sometimes simply called the Racah coefficients (Shore and Menzel 1968, p. 279), are quantities introduced by Racah (1942) that are related to the Clebsch-Gordan coefficients by

(J_1J_2[J^']J_3|J_1,J_2J_3[J^('')])=sqrt((2J^'+1)(2J^('')+1))W(J_1J_2JJ_3;J^'J^(''))
(1)

and

(J_1J_2[J^']J_3|J_1J_3[J^('')]J_2)=sqrt((2J^'+1)(2J^('')+1))W(J_1^'J_3J_2J^('');JJ_1).
(2)

The Racah W-coefficients are related to the Wigner 6j-symbols by

(-1)^(a+b+c+d)W(abcd;ef)={a b c; d e f}
(3)

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

Regge (1959) discovered additional symmetries of the Racah W-coefficients beyond their standard permutation symmetries. Together, these symmetries form the order-144 Regge group and have the geometric interpretation described by the Regge symmetry.

See also

Clebsch-Gordan Coefficient, Racah V-Coefficient, Regge Group, Regge Symmetry, Wigner 3j-Symbol, Wigner 6j-Symbol, Wigner 9j-Symbol

Explore with Wolfram|Alpha

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.Messiah, A. "Racah Coefficients and '6j' Symbols." Appendix C.II in Quantum Mechanics, Vol. 2. Amsterdam, Netherlands: North-Holland, pp. 1061-1066, 1962.Racah, G. "Theory of Complex Spectra. II." Phys. Rev. 62, 438-462, 1942.Regge, T. "Symmetry properties of Racah's coefficients." Nuovo Cim. 11, 116-117, 1959. https://doi.org/10.1007/BF02724914.Shore, B. W. and Menzel, D. H. Principles of Atomic Spectra. New York: Wiley, 1968.Sobel'man, I. I. "Angular Momenta." Ch. 4 in Atomic Spectra and Radiative Transitions, 2nd ed. Berlin: Springer-Verlag, 1992.

Referenced on Wolfram|Alpha

Racah W-Coefficient

Cite this as:

Weisstein, Eric W. "Racah W-Coefficient." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RacahW-Coefficient.html

Subject classifications