How classical is the quantum universe?

We discuss two topics that are usually considered to be exclusively ”quantum”: the Schrödinger equation, and the uncertainty principle. We show (or rather recall) that the Schrödinger equation can be derived from Hamilton’s equations using the metaplectic representation. We also show that the uncertainty principle, stated in the form of the Robertson-Schrödinger-Heisenberg inequalities can be formulated in perfectly classical terms using the topological notion of symplectic capacity.