compatible observables

Two observables are called compatible when their operators commute — when the order in which you measure them makes no difference. For such pairs there exists a common set of eigenstates, states that are simultaneously eigenstates of both operators, so both quantities can have sharp, definite values at the same time. Measuring one does not disturb your knowledge of the other.

Compatibility is what allows physicists to label quantum states with several quantum numbers at once. The energy, the total angular momentum, and one component of angular momentum of a hydrogen atom are mutually compatible, so a state can carry a definite value of each, and that trio of labels pins the state down. A maximal set of compatible observables — a complete set of commuting observables — gives every state a unique address.

Compatibility is the friendly opposite of the uncertainty principle. Where incompatible observables force an unavoidable trade-off, compatible ones live together peacefully: you can measure them repeatedly, in any order, and keep getting the same consistent answers. The dividing line between the two cases is drawn entirely by whether the commutator of their operators is zero.