We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple known from classical mechanics. Our approach is genuinely infinite-dimensional and includes a Lagrangian formalism in which self-adjoint operators are viewed as Lagrangian submanifolds associated with the Lagrangian. As a byproduct, we obtain results concerning coadjoint orbits of the unitary group in infinite dimensions, local embeddings of the space of pure states in the unitary group, and self-adjoint extensions of symmetric relations.