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Journal of Physics Conference Series

Understanding the relation between classical and quantum mechanics: prospects for undergraduate teaching

Marco Di Mauro, A. Drago, Adele Naddeo

Published Mar 1, 2024• 0 citations
Publisher

Abstract

Abstract Classical and quantum mechanics are two very different theories, each describing the world within its own range of validity. It is often stated that classical mechanics emerges from quantum mechanics in a certain limit. This is known as the correspondence principle. According to Planck’s version of the correspondence principle, classical mechanics is recovered when the limit in which a dimensionless parameter containing Planck’s constant h goes to zero is taken, while Bohr’s version entails taking the limit of large quantum numbers. However, despite what is usually stated in textbooks, the relation between the two theories is much more complex to state and understand. Here we deal with this issue by analysing some key examples, in some of which also the analogously subtle relation between wave and geometric optics is considered. Implications for quantum mechanics teaching at undergraduate level are carefully discussed.

DOI
10.1088/1742-6596/2727/1/012013

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Abstract Classical and quantum mechanics are two very different theories, each describing the world within its own range of validity. It is often stated that classical mechanics emerges from quantum mechanics in a certain limit. This is known as the correspondence principle. According to Planck’s version of the correspondence principle, classical mechanics is recovered when the limit in which a dimensionless parameter containing Planck’s constant h goes to zero is taken, while Bohr’s version entails taking the limit of large quantum numbers. However, despite what is usually stated in textbooks, the relation between the two theories is much more complex to state and understand. Here we deal with this issue by analysing some key examples, in some of which also the analogously subtle relation between wave and geometric optics is considered. Implications for quantum mechanics teaching at undergraduate level are carefully discussed.
Planck formulated the correspondence principle between quantum and classical mechanics as the limit in which the Planck constant h goes to zero. Bohr formulated the correspondence principle to be the limit of large quantum numbers. For three common quantum mechanical systems it is shown that in order for eigenvalues of quantum mechanical observables to have meaningful classical limits, it is necessary to take the double limit as both the Planck constant goes to zero and the quantum number goes to infinity, subject to the constraint that their product is equal to an appropriate classical action. This synthesis of the Bohr and Planck formulations of the correspondence principle is also used to show that the quantum mechanical transition frequency between adjacent levels approaches the corresponding classical frequency. The features these systems have in common in their classical limit are explained by general considerations of the classical limit of the Schrödinger equation.
Conceptual and formal problems Many of the problems surrounding the interpretation of quantum mechanics stem from the absence of a sufficiently general and physically transparent analysis of the connection with the domain in which classical concepts are assumed to apply. It is the purpose of this chapter to describe with the aid of a few examples how the quantum potential approach provides a context within which the conceptual and formal issues raised by this problem may be clarified. A key question is to explain how the state or context dependence which is so characteristic of the quantum domain turns into state independence at a certain level, characterized as ‘classical’. To illustrate one aspect of the conceptual problem we recall that in Bohr's interpretation the validity of classical concepts is already presupposed since, it is suggested, it is only in terms of these that one can unambiguously communicate the results of experiments in the quantum domain. Thus, classical physics must be considered as prior to quantum mechanics and the latter is a generalization of the former in that it provides a new set of laws governing the application of classical concepts. According to this view, any procedure by which classical mechanics is recovered from quantum mechanics as a mathematical limit can only be a demonstration of consistency with the already postulated epistemological relation between the two theories and not as a ‘derivation’ of classical mechanics from quantum mechanics (George, Prigogine & Rosenfeld, 1972).
Quantum mechanics and classical mechanics are distinctly different theories, but the correspondence principle states that quantum particles behave classically in the limit of high quantum number. In recent years much research has been done on extending both quantum and classical mechanics into the complex domain. These complex extensions continue to exhibit a correspondence, and this correspondence becomes more pronounced in the complex domain. The association between complex quantum mechanics and complex classical mechanics is subtle and demonstrating this relationship requires the use of asymptotics beyond all orders.
The correspondence principle addresses the connection between classical and quantum physics. The simple statement that quantum mechanics reduces to classical mechanics in the limit where the principal quantum number n approaches infinity, while found in many textbooks, is not true in general. In this article we will give special attention to the notion that the quantum frequency spectrum of a periodic system reduces to the classical spectrum in this limit. Two simple counter-examples—a particle in a cubical box, and a rigid rotator—will show us that the classical result is not always recovered in the limit of large quantum numbers.
The similarities between quantum mechanics and paraxial optics were already well-known to the founding fathers of quantum mechanics; indeed knowledge of paraxial optics partly informed quantum mechanics as a wave theory. Likewise quantum mechanical methods have been employed to better understand optics, for example, to determine which optical transformations are in principle realisable and which not. Recently the notion of classical entanglement has been mooted, ushering in a new avenue to explore, perhaps bridging the classical-quantum divide. These developments have raised questions as to which quantum tasks could be implemented with classical light, taking advantage of the wealth of four centuries of experience in optics. In this article we review the similarities as well as differences between optics and quantum mechanics, providing a quantum notation for classical light. We review the evidence for cross-fertilisation between quantum mechanics and classical optics, in particular considering the issue of classical entanglement and its exploitation for quantum tasks. Our work provides a concise theoretical framework punctuated with relevant examples, and critically assesses the current state of the field and its limits.