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CLASSICAL LIMIT OF THE TRAJECTORY REPRESENTATION OF QUANTUM MECHANICS, LOSS OF INFORMATION AND RESIDUAL INDETERMINACY

Edward R. Floyd

Published Jan 1, 1999• 17 citations
OpenAlex

Abstract

The trajectory representation in the classical limit (¯h → 0) manifests a residual indeterminacy. We show that the trajectory representation in the classical limit goes to neither classical mechanics (Planck’s correspondence principle) nor statistical mechanics. This residual indeterminacy is contrasted to Heisenberg uncertainty. We discuss the relationship between residual indeterminacy and ’t Hooft’s information loss and equivalence classes. PACS Numbers: 3.65.Bz; 3.65.Ca Key words: classical limits, trajectory interpretation, Planck’s correspondence principle, residual indeterminacy, ’t Hooft’s

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The trajectory representation in the classical limit (¯h → 0) manifests a residual indeterminacy. We show that the trajectory representation in the classical limit goes to neither classical mechanics (Planck’s correspondence principle) nor statistical mechanics. This residual indeterminacy is contrasted to Heisenberg uncertainty. We discuss the relationship between residual indeterminacy and ’t Hooft’s information loss and equivalence classes. PACS Numbers: 3.65.Bz; 3.65.Ca Key words: classical limits, trajectory interpretation, Planck’s correspondence principle, residual indeterminacy, ’t Hooft’s
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.
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.
Mechanics can be founded on a principle relating the uncertainty delta-q in the trajectory of an observable particle to its motion relative to the observer. From this principle, p.delta-q=const., p being the q-conjugated momentum, mechanical laws are derived and the meaning of the Lagrangian and Hamiltonian functions are discussed. The connection between the presented principle and Hamilton's Least Action Principle is examined. Wave mechanics and Schrodinger equation appear without additional assumptions by choosing the representation for delta-q in the case the motion is not trajectory describable. The Cramer-Rao inequality serves that purpose. For a particle hidden from direct observation, the position uncertainty determined by the enclosing boundaries leads to thermodynamics in a straightforward extension of the presented formalism. The introduction of uncertainty in classical mechanics formulation enables the translation of mechanical laws into the wide ranging conceptual framework of information theory. The boundaries between classical mechanics, thermodynamics and quantum mechanics are defined in terms of informational changes associated with the system evolution. As a direct application of the proposed formulation upper bounds for the rate of information transfer are derived.
The quantum-classical boundary remains a frontier of research and discovery. One area of attention revolves around Planck’s constant, h, a very small number indeed. In reference to the value of h, Feynman said that if h goes to 0: ‘the classical and quantum results would be the same, and there would be no quantum mechanics to learn!’ This certainly highlights the subtlety of quantum mechanics. Recently, small progress has been made in this field by using Heisenberg’s uncertainty principle to successfully transition from quantum mechanics to classical mechanics and derive Newton’s second law . However, the real question here is… when does quantum become classical? In this chapter we suggest that, regarding conglomerates of atoms, quantum becomes classical when indistinguishability is lost. In other words, when something that was indistinguishable… becomes distinguishable. A heavy atom interferometer experiment is suggested to elucidate this most fascinating unknown.