proton spin Two components of the angular momentum operator do not commute, but satisfy the commutation relations: So, any CSCO cannot involve more than one component of . It can be shown that the square of the angular momentum operator, , commutes with . Also, the Hamiltonian is a function of only and has rotational invariance, where is the reduced mass of the system. Since the components of are generators of rotation, it can be shown that Therefore, a commuting set consists of , one component of (which is taken to be ) and . The solution of the problem tells us that disregarding spin of the electrons, the set forms a CSCO. Let be any basis state in the Hilbert space of the hydrogenic atom. Then That is, the set of eigenvalues or more simply, completely specifies a unique eigenstate of the Hydrogenic atom. The free particle For a free particle, the Hamiltonian is invariant under translations. Translation commutes with the Hamiltonian: . However, if we express the Hamiltonian in the basis of the translation operator, we will find that has doubly degenerate eigenvalues. It can be shown that to make the CSCO in this case, we need another operator called the parity operator , such that . forms a CSCO. Again, let and be the degenerate eigenstates of corresponding the eigenvalue , i.e. The degeneracy in is removed by the momentum operator . So, forms a CSCO. Addition of angular momenta We consider the case of two systems, 1 and 2, with respective angular momentum operators and . We can write the eigenstates of and as and of and as . Then the basis states of the complete system are given by Therefore, for the complete system, the set of eigenvalues completely specifies a unique basis state, and forms a CSCO. Equivalently, there exists another set of basis states for the system, in terms of the total angular momentum operator . The eigenvalues of are where takes on the values , and those of are where . The basis states of the operators and are . Thus we may also specify a unique basis state in the Hilbert space of the complete system by the set of eigenvalues , and the corresponding CSCO is . See also Quantum number Good quantum number Degenerate energy levels Mathematical structure of quantum mechanics Operators in Quantum Mechanics Canonical commutation relation Measurement