On Hamiltonian Formulations of the Schrödinger System

Massimo Tessarotto, M. Mond, Davide Batic

Gustavo V. López, Ana Griselda, Carlos R. Martínez-Prieto

We study classical and quantum dynamics of the symmetric harmonic oscillator and the symmetric bouncer defined in 2-D. For these systems we get each of them for two different constants of motion, two Lagrangians and two Hamiltonians which describe the same classical dynamics. However, the quantization of these systems (using Schrödinger equation), using their two equivalents Hamiltonian, describes different quantum dynamics for each of them. This represents an ambiguity on the Hamiltonian formulation of the Quantum Mechanics.

Alexei A. Deriglazov, B. F. Rizzuti

Any classical-mechanics system can be formulated in reparametrization-invariant form. That is, we use the parametric representation for the trajectories, x=x(τ) and t=t(τ) instead of x=x(t). We discuss the quantization rules in this formulation and show that some of the rules become clearer. In particular, both the temporal and the spatial coordinates are subject to quantization, and the canonical Hamiltonian in the reparametrization-invariant formulation is proportional to H̃=pt+H, where H is the usual Hamiltonian and pt is the momentum conjugate to the variable t. Due to reparametrization invariance, H̃ vanishes for any solution, and hence the corresponding quantum-mechanical operator has the property H̃∧Ψ=0, which is the time-dependent Schrödinger equation, iℏ∂tΨ=H∧Ψ. We discuss the quantum mechanics of a relativistic particle as an example.

Leon Bess

Domenico Giulini

Suhaivi Hamdan, Erwin Erwin, Saktioto Saktioto

This article describes the concept of the time-dependent Schrödinger equation (TDSE) as a review to add a detailed understanding of the steps to formulate TDSE which can be seen from the classical mechanic's concept as a mechanical wave function. In this article, a comprehensive approach to the concepts of momentum and energy in particles will be described using operators working on wave functions. The results show that the use of momentum and energy operators can show that TDSE is a Hamiltonian system.