From my studying of QM, I have been learning about state functions that are supposed to describe a particular system. Now for most purposes, it seems that I have been computing the position wave function (or, through Fourier transforms, the momentum) which evolves with time. But I am just wondering: can any (Hermitian) operator act on this same wave function and yield sensible output? From my studying, I have learned that non-commuting operators act on different eigenvectors to yield direct/immutable measurements (eigenvalues), but that they can still act on different functions and just yield different results. I have attached a small piece from Griffiths that says when there is coupling between different observables, that they have to be included in the state function. But if they are not coupled, do we just separate these different observables at all times? Why exactly do we call it a state function if only certain operators can act on it? Is there a way to generalize this wave function so that any operator (e.g. spin, energy, angular momentum, position) can act on it and it would yield the expected results from a measurement of this system we are trying to describe? What's the most general way to describe a system, or do we have to use different functions to characterize different aspects of a system (e.g. spin, position, energy)?