Using commuting operators to express states I read that it is natural to use commuting operators to express physical state-vectors in QM. For example, if the energy momentum four vector operators all commute, it is natural to express state-vectors in terms of eigenvectors of the four-momentum.
I know that commuting operators share common eigenstates, but why is it natural to express states in terms of eigenstates of commuting operators? What is the usefulness in doing so?
 A: I disagree a little bit with the way the premise of this question is stated. Let me say how I think of it.
It is natural to express states as a superposition of eigenstates of observables. The reason is that then we can associate a physical meaning to each basis state. In the position representation (assuming one particle), each basis state corresponds to measuring the particle at a particular location. In the momentum representation, each eigenstate corresponds to measuring the particle with a particular momentum. And so on.
In fact I think underlies is a logically crucial step in setting up a quantum mechanical model -- choosing the operators you identify as "observables" connects the Hilbert space to the real world. While you can choose a basis that is not aligned with an observable operator even approximately, it is such an unnatural thing to do that I don't think I've ever seen it done.
However often choosing one observable and using its eigenstates to define a basis is not enough. For one thing, from a physics point of view, we obviously want to know as much as we can about the system, and so using a complete set of commuting observables ensures we can identify all the quantities that can be observed in a given basis state. More subtly, because a given observable can have a degenerate spectrum, using a complete set of commuting observables lets us assign a unique label to each state. Because those labels are the eigenvalues of observables, each eigenstate corresponds to a different experimentally possible outcome.
Tl;dr: What we really want is to express the state in terms of eigenstates of observable operators, as opposed to some random basis. Once we know we want this, it is natural to look for a complete set of observables so that we can fully characterize the states in terms of observable outcomes.
