Clebsch-Gordan (CG) coefficients ⟨j₁,m₁; j₂,m₂ | J,M⟩ are the expansion coefficients when combining two angular momenta j₁ and j₂. The total angular momentum J ranges from |j₁−j₂| to j₁+j₂, and M = m₁+m₂ always. The CG coefficients are real and satisfy orthogonality and completeness, forming a unitary transformation between product and coupled bases. They appear throughout quantum mechanics: atomic spectroscopy (selection rules), nuclear shell model, particle physics (quark model multiplets), and as a special case of Wigner 3j-symbols which have nicer symmetry properties.