Consider the Schrödinger equation with confining potential V in the Euclidean space. The question of observability in an open set U in time T can be stated as follows: given a square-integrable initial datum v, is it true that the mass of the solution v(t) that one observes in U over the time interval [0, T] is bounded from below by a fixed fraction (say 1/10) of the mass of the initial datum? We will see that we can characterize the open sets U and times T for which such a statement is true (up to slightly enlarging the observation set U). The observability condition that we find states that any trajectory of a certain flow must enter the open set U in time T. This so-called Hamiltonian flow describes the trajectory of a point mass confined in the potential V, evolving according to Newton's second law. This condition is the result of some form of quantum-classical correspondence, a concept that one can study using semiclassical analysis. In two dimensions, we shall take a closer look at the example of harmonic oscillators, where the potential V is quadratic. For rotation-invariant observation sets U, we will see that the optimal observation time can be estimated from Diophantine properties of the characteristic frequencies of the oscillator.