I have a question. Suppose we have a quantum mechanical system, of which we can measure 3 different observables. Each of the observables has an eigenvector spectrum and we can express a state as a linear combination of the eigenvectors (we assume that they form a basis). My question is the following.
The system is in particular state $|\psi\rangle$.
If we want to express it as a linear combination of basis vectors, which basis vectors do we use (we have 3 different types since we have 3 observable)?
Or maybe, if we assume that the observables commute and form a complete set, is the basis made of eigenvectors that belong to all 3 observables?
Now let's say we have an observable that has a discrete basis and another that has a continuous one. Assuming that they commute, will the basis with eigenvectors of both of them be a mix of a discrete and a continuous eigenvector spectrum?