1.9: Good quantum numbers
( \newcommand{\kernel}{\mathrm{null}\,}\)
It is normal to think of the eigenvalues as labelling states. In that case they are just called quantum numbers. A set of eigenvalues from a complete commuting set of operators are called good quantum numbers. The eigenvalues from a non-commuting operator are a bad quantum numbers, because their values cannot be known simultaneously.
This is not quite as simple as it seems. In real systems the Hamiltonian may contain many small terms (perturbations) which may not commute with the operators which commute with the unperturbed Hamiltonian. Although in principle the quantum numbers are no longer good, in practice they are often used.
An example of this is in spin-orbit coupling of angular momenta in many-electron atoms. Here Lz=∑iliz is a good quantum number in the absence of spin-orbit coupling, but ˆliz does not commute with the spin orbit coupling operator ∑iˆli.ˆsi. Thus for light atoms, where spin-orbit coupling is weak, Lz is often used although it is not strictly a good quantum number.