Hamiltonian Formulation of Quantum Mechanics

Tal Hendel

Abstract The equation of motion in the standard formulation of non-relativistic quantum mechanics, the Schrödinger equation, is based on the Hamiltonian. In contrast, in Feynman’s path-integral formulation of quantum mechanics, the equation of motion is the propagation equation, which is based on the Lagrangian. That these two different equations of motion are equivalent was shown by Feynman, who provided a derivation of the Schrödinger equation from the propagation equation. Surprisingly, however, while in classical mechanics there exists a simple relationship between the Hamiltonian and the Lagrangian, there is nothing in Feynman’s derivation that gives any indication of this relationship. Here I show that the equations of motion in the Hamiltonian and Lagrangian formulations of quantum mechanics are, in fact, simply related to each other.

D. Arsenović, Nikola Burić, Duska B Popovic +2 more

In the Hilbert space formulation of quantum mechanics, ideal measurements of physical variables are discussed using the spectral theory of Hermitian operators and the corresponding projector-valued measures (PVMs). However, more general types of measurements require the treatment in terms of positive-operator-valued measures (POVMs). In the Hamiltonian formulation of quantum mechanics, canonical coordinates are related to PVM. In this paper the results of an analysis of various aspects of applications of POVMs in the Hamiltonian formulation are reported. Several properties of state parameters and quantum observables given by POVMs or represented in an overcomplete basis, including the general Hamiltonian treatment of the Neumark extension, are presented. An analysis of the phase operator, given by the corresponding POVMs, in the Hilbert space and the Hamiltonian frameworks is also given.

Stanley Gudder

R. McWeeny

Maurice A. de Gosson

In their simplest formulations, classical dynamics is the study of Hamiltonian flows and quantum mechanics of propagators. Both are linked, and emerge from the datum of a single classical concept, the Hamiltonian function. We study and emphasize the analogies between Hamiltonian flows and quantum propagators; this allows us to verify G. Mackey's observation that quantum mechanics (in its Weyl formulation) is a refinement of Hamiltonian mechanics. We discuss in detail the metaplectic representation, which very explicitly shows the close relationship between classical mechanics and quantum mechanics; the latter emerging from the first by lifting Hamiltonian flows to the double covering of the symplectic group. We also give explicit formulas for the factorization of Hamiltonian flows into simpler flows, and prove a quantum counterpart of these results.