When we prepare a physical system for study, we summarize what we know about it (as a result of that preparation) as the “state” of the system. Often (in fact usually) whatever state we have prepared could in principle have been prepared more carefully, so the state we actually have is a statistical mixture of more refined states and we infer the existence of “pure” states for which no further refinement is possible. For example, in the classical mechanics of a system of n particles, the pure states correspond to exact initial values of all the position and momentum coordinates, and the observable quantities that we can measure are then various functions of the later values of those coordinates. And in quantum mechanics we often identify the pure states with configurations of a “wave function” (or more generally and more abstractly with a unit vector in some Hilbert Space), and the observables with self-adjoint operators.

We could just take this formalism as the definition of a quantum theory and get on with it, but it might be helpful to give it some deeper motivation.

One approach is to note that states are really defined in terms of how they are prepared – which involves selecting cases via a preliminary measurement process.

So perhaps we should start with the observables rather than the states.