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.
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.
[1] From the formal definition of angular momentum, recursion relations for the Clebsch–Gordan coefficients can be found.
It is diagonal and its eigenvalue characterizes the particular irreducible representation of the angular momentum algebra
This is physically interpreted as the square of the total angular momentum of the states on which the representation acts.
They comprise the spherical basis, are complete, and satisfy the following eigenvalue equations,
where the ladder coefficient is given by: In principle, one may also introduce a (possibly complex) phase factor in the definition of
The angular momentum states are orthogonal (because their eigenvalues with respect to a Hermitian operator are distinct) and are assumed to be normalized,
Here the italicized j and m denote integer or half-integer angular momentum quantum numbers of a particle or of a system.
On the other hand, the roman jx, jy, jz, j+, j−, and j2 denote operators.
We are then going to define a family of "total angular momentum" operators acting on the tensor product space
The action of the total angular momentum operator on this space constitutes a representation of the SU(2) Lie algebra, but a reducible one.
The total[nb 1] angular momentum operators are defined by the coproduct (or tensor product) of the two representations acting on V1⊗V2,
The total angular momentum operators can be shown to satisfy the very same commutation relations,
Hence, a set of coupled eigenstates exist for the total angular momentum operator as well,
The total angular momentum quantum number J must satisfy the triangular condition that
These rules may be iterated to, e.g., combine n doublets (s=1/2) to obtain the Clebsch-Gordan decomposition series, (Catalan's triangle),
The coupled states can be expanded via the completeness relation (resolution of identity) in the uncoupled basis The expansion coefficients
The recursion relations were discovered by physicist Giulio Racah from the Hebrew University of Jerusalem in 1941.
Taking the upper sign with the condition that M = J gives initial recursion relation:
A convenient way to derive these relations is by converting the Clebsch–Gordan coefficients to Wigner 3-j symbols using 3.
It is useful to observe that any phase factor for a given (ji, mi) pair can be reduced to the canonical form:
(Note that this form is only locally canonical: it fails to take into account the rules that govern combinations of (ji, mi) pairs such as the one described in the next paragraph.)
An additional rule holds for combinations of j1, j2, and j3 that are related by a Clebsch-Gordan coefficient or Wigner 3-j symbol:
Factorials of negative numbers are conventionally taken equal to zero, so that the values of the 3j symbol at, for example,
In the case where integers are involved, the coefficients can be related to integrals of spherical harmonics:
[8][9] 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.