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Coefficients in angular momentum eigenstates of quantum systems

In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum decomposition of the tensor product of two irreducible representations (i. e., a reducible representation into irreducible representations, in cases where the numbers and types of irreducible components are already known abstractly). The name derives from the German mathematicians Alfred Clebsch and Paul Gordan, who encountered an equivalent problem in invariant theory. From a vector calculus perspective, the CG coefficients associated with the SO(3) group can be defined simply in terms of integrals of products of spherical harmonics and their complex conjugates. The addition of spins in quantum mechanical terms can be read directly from this approach as spherical harmonics are eigenfunctions of total angular momentum and projection thereof onto an axis, and the integrals correspond to the Hilbert space inner product. From the formal definition of angular momentum, recursion relations for the Clebsch–Gordan coefficients can be found. There also exist complicated explicit formulas for their direct calculation. The formulas below use Dirac's bra–ket notation and the Condon–Shortley phase convention is adopted.

A convenient way to derive these relations is by converting the Clebsch–Gordan coefficients to Wigner 3 j symbols using The symmetry properties of Wigner 3 j symbols are much simpler.

Clebsch–Gordan coefficients are related to Wigner 3 j symbols which have more convenient symmetry relations. The factor (−1)^(2 j2) is due to the Condon–Shortley constraint that , while (–1)^(J − M) is due to the time reversed nature of .

For arbitrary groups and their representations, Clebsch–Gordan coefficients are not known in general. However, algorithms to produce Clebsch–Gordan coefficients for the special unitary group SU(n) are known. In particular, SU(3) Clebsch Gordan coefficients have been computed and tabulated because of their utility in characterizing hadronic decays, where a flavor SU(3) symmetry exists that relates the up, down, and strange quarks. A web interface for tabulating SU(N) Clebsch–Gordan coefficients is readily available. Clebsch–Gordan coefficients for symmetric group are also known as Kronecker coefficients.