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arXiv:1201.0150 · American Journal of Physics

What is the limit ℏ → of quantum theory?

U. Klein

Published Oct 17, 2012• 31 citations
arXivHTMLPDF

Abstract

An analysis is made of the relation between quantum theory and classical mechanics, in the context of the limit ℏ→0. Several ways in which this limit may be performed are considered. It is shown that Schrödinger's equation for a single particle moving in an external potential V does not, except in special cases, lead, in this limit, to Newton's equation of motion for the particle. This shows that classical mechanics cannot be regarded as emerging from quantum mechanics—at least in this sense—upon straightforward application of the limit ℏ→0.

DOI
10.1119/1.4751274

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An analysis is made of the relation between quantum theory and classical mechanics, in the context of the limit ℏ→0. Several ways in which this limit may be performed are considered. It is shown that Schrödinger's equation for a single particle moving in an external potential V does not, except in special cases, lead, in this limit, to Newton's equation of motion for the particle. This shows that classical mechanics cannot be regarded as emerging from quantum mechanics—at least in this sense—upon straightforward application of the limit ℏ→0.
Classical and quantum physics represent two distinct theories; however, quantum physics is regarded as the more fundamental of the two. It is posited that classical mechanics should arise from quantum mechanics under certain limiting conditions. Nevertheless, this remains a challenging objective. In this work, we explore the potential for unifying the dynamics of classical and quantum physics. This discussion does not suggest that classical behavior emerges from quantum mechanics; rather, it demonstrates the exact equivalence between the dynamics of quantum observables and their classical counterparts. It is shown that the Heisenberg equations of motion can be cast in a form that is structurally equivalent to Newton's equations of motion, with ℏ\hbarℏ dropping out from the equations. In a generalized analysis, the Heisenberg equations are cast in a form that is identical to the classical Hamilton's equations of motion. This implies that both quantum and classical dynamics are governed by the same form of equations, with the Heisenberg operators substituting the classical observables.
The laws of quantum mechanics differ significantly from the laws of classical mechanics. The mathematical apparatus of quantum mechanics is based on the amazing Schrödinger equation. This is a specific partial differential equation with respect to a complex-valued function called the wave function. This characteristic allows, in particular, estimating the probability of finding a quantum mechanical particle in a given volume. We establish the Schrödinger equation for a free particle and for a particle moving in an external field. Then the problem of passing a potential barrier is investigated. It is shown that, in contrast to the classical case, even a relatively slow quantum mechanical particle is capable of overcoming a sufficiently high potential barrier. The Appendix considers the concept of a normalized wave function and analyzes the movement of a particle in a potential well with infinite walls.
Newton's second law of motion, F = ma, has long been considered a cornerstone of classical mechanics, providing a fundamental framework for understanding the behavior of macroscopic objects. However, as physics delves into the quantum realm, the applicability of Newton's second law becomes limited, and the principles of quantum mechanics emerge as the dominant description of particle behavior. Beginning with an overview of the historical development of classical mechanics by Newton and the emergence of quantum mechanics in the early 20th century, the article delves into the foundational principles of framework. The classical description of particles as point masses with deterministic trajectories is contrasted with the probabilistic nature of quantum particles described by wave functions and superposition principles. Practical applications and implications of these theories are also discussed, shedding light on their significance in various scientific disciplines. This research article presents a comparative analysis of Newton's second law of motion and quantum physics, examining their fundamental principles and implications through the lens of classical and quantum mechanics. Beginning with an overview of Newton's second law and its significance in classical mechanics, the article delves into the foundational postulates of quantum mechanics and their departure from classical concepts. This explores the inherent limitations of Newton's second law when applied to quantum phenomena, comparing and contrasting its principles with those of quantum physics. Through a comprehensive analysis of evidence and theoretical frameworks, this article elucidates the boundaries of classical mechanics and highlights the need for quantum mechanics to describe phenomena at the atomic and subatomic levels. Through an exploration of key differences and similarities, this aims to provide insights into the behavior of particles at both macroscopic and microscopic scales
We develop a dynamical theory, based on a system of ordinary differential equations describing the motion of particles which reproduces the results of quantum mechanics. The system generalizes the Hamilton equations of classical mechanics to the quantum domain, and turns into them in the classical limit ℏ→0\hbar\rightarrow 0ℏ→0. The particles' motions are completely determined by the initial conditions. In this theory, the wave function ψψψ of quantum mechanics is equal to the exponent of an action function, obtained by integrating some Lagrangian function along particle trajectories, described by equations of motion. Consequently, the equation for the logarithm of a wave function is related to the equations of motion in the same way as the Hamilton-Jacobi equation is related to the Hamilton equations in classical mechanics. We demonstrate that the probability density of particles, moving according to these equations, should be given by a standard quantum-mechanical relation, ρ=∣ψ∣2ρ=|ψ|^2ρ=∣ψ∣2. The theory of quantum measurements is presented, and the mechanism of nonlocal correlations between results of distant measurements with entangled particles is revealed. In the last section, we extend the theory to particles with nonzero spin.
Quantum mechanics can only provide the probability of finding moving particles at the atomic level. The Schrodinger equation provides the information of a wavefunction, which gives only the quantitative prediction of probability of finding a moving particle at a particular location. Quantum mechanics appears as a strange theory, it can provide a powerful tool to explain many observed phenomena at the atomic level. Modeling of quantum mechanics using the Schrodinger equation is only one of the choices. Quantum mechanics can only provide information about moving particles in a statistical sense. For the case of “light waves,” defines the wavelength of a photon of light. This is also known as the quantum theory of light. In quantum mechanics, the wave function in the Schrodinger equation is associated with the likelihood of finding the particle at a particular time and at a particular place.