Hamiltonian Formulation of Quantum Mechanics

Abstract

The Schrödinger equation is a deterministic equation for the quantum mechanical state. It can be cast into the form of a Hamiltonian system (familiar from analytical mechanics) on the space of pure states (rays) . Quantum states (pure or mixed) then formally become distribution functions This gives, to some extent, an analogy to classical statistical mechanics, where mixed stated are also given by distributions over phase space (the space of pure states) . This analogy in itself is quite interesting and has, moreover, been successfully employed to compute the dynamics of open quantum systems when viewed as stochastic process on the space of rays. However, we know that there are essential differences between quantum mechanics and a classical statistical system, which should also show up in this approach and hence render the envisaged analogy incomplete. We think that it is of interest to point out where this analogy fails, or, if you like, at what price the analogy can be uphold. To avoid technical complications, which are not our concern here, we restrict our discussion to finite-dimensional Hilbert spaces. More details can be found in (Cirelli, Manià, and Pizzocchero 1990) and (Gibbons 1992).