Clebsch-Gordan coefficients

In physics, more precisely in quantum mechanics, the Clebsch-Gordan coefficients are the numerical constants that express the probability amplitude for the spins j₁,j₂ with z-projections m₁,m₂ to add to j with projection

m = m₁ + m₂.

In more mathematical terms, the CG coefficients are used in the representation theory, particularly of compact Lie groups, to perform the explicit direct sum decomposition of the tensor product of two irreducible representations, into irreducible representations, in cases where the numbers and types of irreducible components are already known abstractly. The naming derives from the German mathematicians Alfred Clebsch (1833-1872) and Paul Gordan (1837-1912) of the nineteenth century, who in invariant theory encountered an equivalent problem.

In fact in terms of classical mathematics CG coefficients, or at least those associated to the group SO(3), may be defined much more directly, by means of formulae for the multiplication of spherical harmonics. The addition of spins in quantum-mechanical terms can be read directly from this approach.

See also

• Clebsch-Gordan series
• Wigner 3-j symbol
• 6-j symbol
• Spherical harmonics
• Associated Legendre polynomials

References

• A.R. Edmonds, Angular Momentum in Quantum Mechanics, (1957) Princeton University Press, ISBN 0-691-07912-9.
• E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, (1970) Cambridge at the University Press, ISBN 521-09209-4 See chapter 3.
• Albert Messiah, Quantum Mechanics (Volume II), (1966) North Holland Publishing, ISBN ????