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European Journal of Science Education

Classical mechanics and quantum mechanics: an elementary approach to the comparison of the two viewpoints

Leonello Paoloni

Published Jul 1, 1982• 7 citations
Publisher

Abstract

Summaries English The relationship between classical mechanics and quantum mechanics is illustrated by considering how we form the images of moving objects. Brownian motion is an helpful example because it can be analyzed in two related, but different frames which show how the images we construct depend on the conditions of our observation and on some intrinsic characters of the observed objects.

DOI
10.1080/0140528820040303

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Summaries English The relationship between classical mechanics and quantum mechanics is illustrated by considering how we form the images of moving objects. Brownian motion is an helpful example because it can be analyzed in two related, but different frames which show how the images we construct depend on the conditions of our observation and on some intrinsic characters of the observed objects.
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.
Replace the potential energy in the Hamiltonian operator from electromagnetic interaction potential energy to gravitational interaction potential energy. Define that the kinetic energy of a moving object is equal to its de Broglie wave's hν. Successfully obtained the Schrödinger equation describing planetary motion. The Schrödinger equation itself is a combination of classical mechanics and quantum mechanics. Quantum mechanics based on it cannot completely exclude classical mechanics. Many examples have been introduced where classical mechanics and quantum theory can be combined to describe the same object (whether it is macroscopic or microscopic). Local realism and determinism still have a market in modern physics. Quantum mechanics and classical mechanics have a compatible and complementary relationship.
Classical objectivity as a property of quantum states —a view proposed to explain the observer-independent character of our world from quantum theory, is an important step in bridging the quantum-classical gap. It was recently derived in terms of spectrum broadcast structures for small objects embedded in noisy photon-like environments. However, two fundamental problems have arisen: a description of objective motion and applicability to other types of environments. Here we derive an example of objective states of motion in quantum mechanics by showing the formation of dynamical spectrum broadcast structures in the celebrated, realistic model of decoherence —Quantum Brownian Motion. We do it for realistic, thermal environments and show their noise-robustness. This opens a potentially new method of studying the quantum-to-classical transition.
Recent advances in our understanding of foundations of quantum mechanics have shown that information can be made objective through quantum states. Such objectification processes, predicted e.g. in a variety of quantum open systems, must accompany any realistic quantum-to-classical transition mechanism in order to reproduce the objective character of the classical limit of our world. However, so far only examples of simple, unstructured information, such as a value of an observable, have been studied. In this work we show that a more complicated form of information, given by a Cartesian reference frame, can also be made (at least partially) objective in quantum mechanics. The non-trivial internal gauge structure of reference frames, given by the transformation group, leads to a more general form of objectivity, where all observers see the same but modulo their relative orientations, like it happens in modern understanding of geometry. This opens a way to extend quantum objectivity beyond simple scenarios and possibly link it to the foundations of modern geometry.
By using the formulation of quantum mechanics with classical pictures, the motion of a quantum is analysed for linear harmonic oscillator. On this occasion it is postulated that the quantum mechanical momentum of the quantum is approximately equal to the corresponding classical mechanical momentum in the region of validity of the coordinates. According to this analysis it is shown that the quantum mechanical motion well corresponds to the classical motion, and that the wave equation determines the associated wave along the whole orbit of the quantum in the configuration space and the uniqueness of the wave throughout the orbit plays an important role in quantum mechanics. In addition to the above, an attempt is made to make clear the well-known fact that classical mechanics with the quantization gives approximately the same results as quantum mechanics.