= Complete set of commuting observables
A <complete set of commuting observables> consists of mutually commuting <quantum observables> whose joint <eigenvalues> distinguish all vectors of a simultaneous <eigenbasis>, up to phase. For a finite-dimensional <Hilbert space>, commuting <self-adjoint operators> admit a simultaneous <orthonormal basis> of <eigenvectors>; completeness means that every joint <eigenspace> is one-dimensional. An additional <quantum observable> that is already a function of the others supplies no independent <quantum number>.
For the <addition of angular momentum>, the uncoupled labels $(j_1,j_2,m_1,m_2)$ correspond to $\{J_1^2,J_2^2,J_{1z},J_{2z}\}$, whereas the coupled labels $(j_1,j_2,j,m)$ correspond to $\{J_1^2,J_2^2,J^2,J_z\}$. Within a fixed pair of irreducible <angular momentum> spaces each choice gives a natural <eigenbasis>. The <Clebsch-Gordan coefficients> implement the change between them.
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