  Canonical coordinates for both the Schroedinger and the nonlinear
Schroedinger equations are introduced, making more transparent their
Hamiltonian structures. It is shown that the Schroedinger equation, considered
as a classical field theory, shares with the nonlinear Schroedinger, and more
generally with Liouville completely integrable field theories, the existence of
a "recursion operator" which allows for the construction of infinitely many
conserved functionals pairwise commuting with respect to the corresponding
Poisson bracket. The approach may provide a good starting point to get a clear
interpretation of Quantum Mechanics in the general setting, provided by
Stone-von Neumann theorem, of Symplectic Mechanics. It may give new tools to
solve in the general case the inverse problem of Quantum Mechanics.
