Symplectic Structures and Quantum Mechanics

TL;DR

This paper introduces a symplectic framework for quantum mechanics, revealing Hamiltonian structures and conserved quantities, which could enhance understanding and solution methods for quantum inverse problems.

Contribution

It presents a symplectic and Hamiltonian formulation of the Schrödinger equation, highlighting integrability and conserved functionals, offering new perspectives for quantum theory interpretation and inverse problems.

Findings

Identifies Hamiltonian structure of Schrödinger equation

Establishes existence of recursion operator and conserved functionals

Suggests new tools for quantum inverse problems

Abstract

Canonical coordinates for the Schr\"odinger equation are introduced, making more transparent its Hamiltonian structure. It is shown that the Schr\"odinger equation, considered as a classical field theory, shares with Liouville completely integrable field theories the existence of a {\sl recursion operator} which allows for the 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 whose solution is given up to now only for one-dimensional systems by the Gel'fand-Levitan-Marchenko formula.