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. Review of the angular momentum operators Angular momentum operators are self-adjoint operators , , and that satisfy the commutation relations where is the Levi-Civita symbol. Together the three operators define a vector operator, a rank one Cartesian tensor operator, It is also known as a spherical vector, since it is also a spherical tensor operator. It is only for rank one that spherical tensor operators coincide with the Cartesian tensor operators. By developing this concept further, one can define another operator as the inner product of with itself: This is an example of a Casimir operator. 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. One can also define raising () and lowering () operators, the so-called ladder operators, Spherical basis for angular momentum eigenstates It can be shown from the above definitions that commutes with , , and : When two Hermitian operators commute, a common set of eigenstates exists. Conventionally, and are chosen. From