In two previous papers, the authors introduced the concept of binormal differential equations. It was shown that invariants of the Courant-Snyder type are associated with the scalar products of the column vectors associated to an ordinary differential equation and to its binormal. This paper shows the equivalence of the above invariant and the Lewis form. It also introduces a density matrix for a second-order differential equation and clarifies the geometrical meaning of Twiss parameters. Within the framework of accelerator physics, the importance of the above results in the analysis of quantum problems such as the evolution of squeezed states is stressed.