From C-Numbers to Q-Numbers: The Classical Analogy in the History of Quantum Theory: 8 - Rilegato

Darrigol, Olivier

Rilegato

ISBN 10: 0520078225 ISBN 13: 9780520078222

Casa editrice: Univ of California Pr, 1993

Riassunto

The history of quantum theory is a maze of conceptual problems, through which Olivier Darrigol provides a lucid and learned guide, tracking the role of formal analogies between classical and quantum theory. From Planck's first introduction of the quantum of action to Dirac's formulation of quantum mechanics, Darrigol illuminates not only the history of quantum theory but also the role of analogies in scientific thinking and theory change.

Unlike previous works, which have tended to focus on qualitative, global arguments, Darrigol's study follows the lines of mathematical reasoning and symbolizing and so is able to show the motivations of early quantum theorists more precisely—and provocatively—than ever before. Erudite and original, From c-Numbers to q-Numbers sets a new standard as a philosophically perceptive and mathematically precise history of quantum mechanics. For years to come it will influence historical and philosophical discussions of twentieth-century physics.

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Informazioni sull?autore

A physicist and historian of physics, Olivier Darrigol is a chargé de recherches at the Centre National de la Recherche Scientifique in Paris and a lecturer in history of physics at the Ecole Normale Supérieure.

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From c-Numbers to q-Numbers: The Classical Analogy in the History of Quantum Theory

By Olivier Darrigol

University of California Press

Copyright © 1993 Olivier Darrigol
All right reserved.
ISBN: 0520078225
Introduction

Most of Planck's early work was carried out with the principal goal of proving that the second law of thermodynamics was strictly valid and that the entropy of a closed system always increases. Accordingly, he first rejected kinetic gas theory, for it considered the second law to be only statistically valid—until he came to develop his radiation theory on the basis of an analogy with Boltzmann's gas theory. One may wonder how any reasonable theoretician could draw his inspiration from the theory which he is trying to disprove. A plausible answer could be that Planck was converted to Boltzmann's ideas. But he believed too much in the absolute validity of the entropy law to do so. The key that gave him access to the formal apparatus of Boltzmann's theory was in fact a reinterpretation of this theory in nonstatistical terms.1

According to Planck, the central concept of both radiation and kinetic gas theory had to be that of "elementary disorder." In his opinion the main difficulty encountered in these theories was that in the derivation of equations for the evolution of directly observable quantities from fundamental electrodynamic or mechanical processes, there were terms depending on uncontrollable details of the state of the system, for instance the position of individual molecules or the electromagnetic field at a precise point of space. According to Boltzmann, these terms really existed, but they could be neglected when considering the statistical behavior of a large

Allan Needell was the first to point out the central importance of the idea of absolute irreversibility m Planck's work before 1914 (Needell 1980, 1988).

number of exemplars of the system. The resulting evolution of the directly observable quantities (the one given by the Boltzmann equation) was irreversible, but only in a statistical manner. Instead, according to Planck's notion of elementary disorder the unknown structural details of the system, for instance the structure of the walls of the container or the internal structure of electric resonators, had to be adjusted in such a way that the unwanted terms completely disappeared. This warranted a strictly deterministic (and irreversible) evolution of "directly observable quantities."2

The relation between Planck's and Boltzmann's work in thermodynamics, then, is a subtle and intricate one, and an elucidation of it will be one of the principal goals of this chapter. Planck's appropriation of some of Boltzmann's computational methods has often misled his modern readers, who generally understand these problems from the point of view of statistical thermodynamics, which is essentially Boltzmann's. To a reader aware of the pitfall of incommensurability, Planck's approach will appear far more coherent and conservative than usually assumed.

The analogies used by Planck in his radiation theory were drawn from a reinterpreted version of Boltzmann's theory. Yet, in any analogy there is a risk of overestimating the similarities between the systems compared. Planck certainly did. In Boltzmann's irreversibility theorem, not only was irreversible behavior derived but the final equilibrium state of the system was shown to be unique. Planck initially believed that such uniqueness also held for the electrodynamic system which he considered in his radiation theory. More specifically, he thought he could show that Wien's law was the only possible distribution for thermal radiation. Under the pressure of new empirical data, however, he came to realize that any thermal radiation law was compatible with his irreversibility theorem.

At this stage, Planck thought of adapting the analogy between his and Boltzmann's theory to another method of determining the equilibrium state of a system, through Boltzmann's quantitative relation between entropy and "probability." Naturally, he did this within the context of his reinterpretation of Boltzmann's theory: he freed Boltzmann's "probability" from its original ties with the statistical conception of the entropy law

"Elementary disorder" was the generic expression used by Planck to characterize molecular chaos and natural radiation in his lectures on radiation theory: see Planck 1906, e.g., on 134: "The assertion that in nature every state and process involving a great number of uncontrollable elements is elementarily disordered provides the condition and also the strict guarantee for the unequivocal determination of measurable processes in both mechanics and electrodynamics and for the validity of the second principle of thermodynamics."

and interpreted it instead as a quantitative measure of the elementary disorder that warranted strict irreversible behavior.

Here the reinterpretation had considerable effects. Most important, we shall see that it permitted finite energy-elements to appear in the final expression of entropy (whereas they disappeared in Boltzmann's original method); at the same time it allowed maintaining the continuous equations for the evolution of the electrodynamic system, without apparent inconsistency. More generally, Planck's quantum hypothesis was meant to complete the existing electrodynamic theories, not to contradict them. Closely connected to elementary disorder, this hypothesis found its logical place in the uncontrollable details of electrodynamic systems and left untouched the laws ruling directly observable quantities.3

Altogether, tight connections existed between the central concepts of Planck's radiation theory, namely: absolute irreversibility, disorder, entropy, and energy quanta. By 1905, however, Einstein perceived an inconsistency in the corresponding reinterpretation of Boltzmann's theory. In his opinion the separation between directly observable quantities and internal structure necessary to Planck's idea of disorder could not be maintained. One then had to return to orthodox statistical thermodynamics, and this led, in the case of Planck's electrodynamic system, to absurd results. The observed properties of thermal radiation, Einstein concluded, could not be explained without a sharp break from ordinary electrodynamics. Nevertheless, the formal skeleton of Planck's derivation of his blackbody law remained valid. This should be seen as a virtue of the symbolic part of Planck's analogies, the resulting equations being "more clever than their inventor," as Born once put it.4

That Planck did not introduce a "quantum discontinuity" in 1900 was first asserted in Kuhn 1978.

Born to Bohr, undated (early 1926), AHQP: "For the time being our mathematical formulae tend to be more clever than we are. The formulae come to us quite naturally, but the interpretation is often difficult."

Continues...

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Dettagli dell'edizione

• Editore: Univ of California Pr
• Data di pubblicazione: 1993
• Lingua: Inglese
• ISBN 10: 0520078225
• ISBN 13: 9780520078222
• Rilegatura: Copertina rigida
• Numero di pagine: 388
• Contatto del produttore: non disponibile
• Persona responsabile: non disponibile