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arXiv:1510.05701 · Progress of Theoretical and Experimental Physics

On the Hamilton–Jacobi method in classical and quantum nonconservative systems

A. de Souza Dutra, R. A. C. Correa, P. H. R. S. Moraes

Published Aug 1, 2016• 1 citation
arXivHTMLPDF

Abstract

In this work we show how to complete some Hamilton–Jacobi solutions of linear, nonconservative classical oscillatory systems which appeared in the literature, and we extend these complete solutions to the quantum mechanical case. In addition, we obtain the solution of the quantum Hamilton–Jacobi equation for an electric charge in an oscillating pulsing magnetic field. We also argue that for the case where a charged particle is under the action of an oscillating magnetic field, one can apply nuclear magnetic resonance techniques in order to find experimental results regarding this problem. We obtain all results analytically, showing that the quantum Hamilton–Jacobi formalism is a powerful tool to describe quantum mechanics.

DOI
10.1093/ptep/ptw105

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In this work we show how to complete some Hamilton–Jacobi solutions of linear, nonconservative classical oscillatory systems which appeared in the literature, and we extend these complete solutions to the quantum mechanical case. In addition, we obtain the solution of the quantum Hamilton–Jacobi equation for an electric charge in an oscillating pulsing magnetic field. We also argue that for the case where a charged particle is under the action of an oscillating magnetic field, one can apply nuclear magnetic resonance techniques in order to find experimental results regarding this problem. We obtain all results analytically, showing that the quantum Hamilton–Jacobi formalism is a powerful tool to describe quantum mechanics.
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Abstract The Hamilton-Jacobi theory is the apotheosis of Lagrangian and Hamiltonian mechanics: action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria. These action functions are the solutions of a nonlinear, first-order partial differential equation, called the Hamilton-Jacobi equation. The characteristic equations of this differential equation are the extended Hamilton equations. Solution of a class of mechanics problems is thus reduced to the solution of a single partial differential equation. Aside from its use as a problem-solving tool, the Hamilton-Jacobi theory has particular importance because of its close relation to the Schroedinger formulation of quantum mechanics. This chapter discusses the connection between the Hamilton-Jacobi theory and the Schroedinger formulation, the Bohm hidden variable model and Feynman path integral method that are derived from it, Hamilton’s characteristic equations, complete integrals, separation of variables, canonical transformations, general integrals, mono-energetic integrals, relativistic Hamilton-Jacobi equation, quantum Cauchy problem, quantum mechanics, and classical mechanics.
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In this chapter, the authors introduce the Hamilton-Jacobi equation showing that classical mechanics can be described with a wave equation for the classical action. In order to find the equation showing that classical mechanics already contains a wave behavior in matter particles, the people need to go back to the Lagrange/Hamilton formalism. Davisson-Germer experiment has shown without any doubt that electrons behave as massive particles whose dynamics is governed by some sort of wave. A free particle is apparently the conceptually simplest system to study because the TISE will reduce to its simplest form. The simple (linear) harmonic oscillator is of paramount importance in classical as well as quantum mechanics. A proper solution to the Harmonic oscillator problem can be found by solving Schrodinger equation analytically. Charles Hermite was a French mathematician who gave important contributions in many fields of mathematics used in physics like, for example, Hermitian operators and Hermite polynomials.