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Journal of Anyang Normal University

On the Classical Limit of the Quantum Mechanics Based on the Hamilton-Jacobi Equation

Li-Feng Wang

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

Based on the Hamilton-Jacobi equation and the Bohm's quantum potential,it is pointed out that the classical limit of the quantum mechanics is the classical statistical mechanics.Through analyzing the Gaussian packet from two approaches,the perspective of the classical limit via the H-J equation is shown to be more profound than that via the Ehrenfest theorem.

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Based on the Hamilton-Jacobi equation and the Bohm's quantum potential,it is pointed out that the classical limit of the quantum mechanics is the classical statistical mechanics.Through analyzing the Gaussian packet from two approaches,the perspective of the classical limit via the H-J equation is shown to be more profound than that via the Ehrenfest theorem.
The Feynman path integral formulation of quantum mechanics is a path integral representation for a propagator or probability amplitude in going between two points in space-time. The wave function is expressed in terms of an integral equation from which the Schrodinger equation can be derived. On taking the limit h — 0, the method of stationary phase can be applied and Newton's second law of motion is obtained. Also, the condition the phase vanishes leads to the Hamilton - Jacobi equation. The secondary objective of this paper is to study ways of relating quantum mechanics and classical mechanics. The Ehrenfest theorem is applied to a particle in an electromagnetic field. Expressions are found which are the hermitian Lorentz force operator, the hermitian torque operator, and the hermitian power operator.
The problem of the reduction of quantum mechanics to classical mechanics is considered. It is demonstrated that the customary derivation of classical mechanics from the Schrödinger equation via the Hamilton-Jacobi action equation is incomplete. An improved derivation, involving the expectation value of the action, rather than the action itself, is given. Under this formulation the Hamilton-Jacobi description of classical mechanics is obtained from quantum mechanics, for suitable wave packets as h goes to zero.
dga te Q L D 4017, Au stra l i a ( Recei ved Jul y 3, 2002) A possibl e way fro m qua ntum me chanics to classical mechanics can b e achie ved w ith an exp onential substitutio n used in the Schr odin ger equation , and then considerin g the classical limit .T his gives a picture of classical Ûuid and an ensemble of classical tra jectories.In di ˜erence from this approach to the classical limit, w hile utilisi ng the same substitution , w e assume a minimum uncertainty w ave packet.It is show n that this approach to the classical limit of quantum mechanics yields a single traj ectory traced by the centroid of the minimum uncertainty w ave packet.T he momentum and the centroid of such packet satisf y the classical H amilton { J acobi equation.
We study how the classical Hamilton's principal and characteristic functions are generated from the solutions of the quantum Hamilton–Jacobi equation. While in the classically forbidden regions these quantum quantities directly tend to the classical ones, this is not the case in the allowed regions. There, the limit is reached only if the quantum fluctuations are eliminated by means of coarse-graining averages. Analogously, the classical Hamilton–Jacobi scheme bringing to the motion's equations arises from a similar formal quantum procedure. Quanta 2022; 11: 42–52.
It is pointed out that, if classical mechanics is to be regarded as a limiting case of quantum mechanics, then it must admit superpositions of states, which are generally regarded as unacceptable. To avoid this it is proposed that classical mechanics be described by a nonlinear equation which is equivalent to the Hamilton-Jacobi equation and is not always the limit of the Schrödinger equation. It is conjectured that the transition from the wave-mechanical equation to the classical equation is characterized by a mass m0 = ħc/γ12 ≈ 10−5g. This nonlinear equation may help one to understand the process of measurement of a quantum-mechanical system by means of a classical measuring instrument.