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Journal of Baoji University of Arts and Sciences

Hamilton-Jacobi theory and the establishment of Schrdinger equation

Wang Run-xuan

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

The highly profound connection,between classical mechanics Hamilton-Jacobi state and Schr o¨dinger equation in mathematical form and physical meaning,is studied logically and systematically.Base on Hamilton equation of classical mechanics,the Schr o¨dinger equation of nonrelativity and relativity are established.

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American Journal of Physics•May 13, 2011•8 citations
Quantum mechanics and classical mechanics are closely related to each other.The Hamilton-Jacobi Equation of classical mechanics plays an important role in the extension of Schrdinger equation,and it is also a way to extend Schrdinger equation in the teaching of quantum mechanics.Besides,the Hamilton-Jacobi Equation under the theory of relativity can extend Klein-Gordon equation of quantum mechanics of special relativity.Considering and analyzing the differences betweeen Klein-Gordon equation and Dirac Equation further,this paper linearized the Hamilton-Jacobi Equation under the theory of relativity,and extended Dirac Equation which was more important than quantum mechanics of special relativity.It made the route by which Hamilton-Jacobi Equation,as the classical mechanics,goes to quantum mechanics more extensive,and this study further explored the corresponding relationship between classical mechanics and quantum mechanics.
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.
In his extremely important paper [1] E. Schro¨dinger proposed a wave equation which is the basic equation of the “undulatorical” mechanics and can be considered as a substitution to the Hamilton-Jacobi partial differential equations (HJ) of the ordinary (classical) mechanics. The wave equation was established on condition that the Lagrange function contains no terms linear in velocities. Schro¨dinger writes (footnote on p. 514 l.c.): “In relativistic mechanics and in the case of a magnetic field the expression of the (HJ) is more complicated. In the case of a single electron this equation means the constancy of the four-dimensional gradient diminished by a given vector (the four-dimensional potential). The wave mechanical translation of this Ansatz meets some difficulties.”
The Hamilton-Jacobi equation of classical mechanics is approached as a model reduction of conservative particle mechanics where the velocity degrees-of-freedom are eliminated. This viewpoint allows an extension of the association of the Hamilton-Jacobi equation from conservative systems to general Newtonian particle systems involving non-conservative forces, including dissipative ones. A geometric optics approximation leads to a dissipative Schrödinger equation, with the expected limiting form when the associated classical force system involves conservative forces.
The Hamilton–Jacobi equation in classical mechanics and the related eikonal equation in geometrical optics are often described as the “point of closest approach” between classical and quantum mechanics. Most textbook treatments of Hamilton–Jacobi theory are aimed at graduate students and derive the equation only after a long introduction to canonical transformations. Most treatments of the eikonal equation only emphasize its use in geometrical optics. We show that both the Hamilton–Jacobi equation and the eikonal equation can be derived by a common procedure using only elementary aspects of the Lagrangian and Hamiltonian formalisms introduced in undergraduate classical mechanics courses. Through this common approach, we hope to highlight to undergraduates the deep connections between classical mechanics, classical wave theory, and Schrödinger’s wave mechanics.