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Physica Scripta

Analogy Between Mechanics and Thermodynamics

A Gaies, A. El Akrmi

Published Jan 1, 2002• 1 citation
Publisher

Abstract

A formal resemblance between thermodynamics and mechanics has been found. In this respect, the fundamental ideas of quantum theory can be obtained from this analogy. We have deduced a correspondence between Caratheodory's theorem and Bohr's stationary states principle. Using the deduced correspondence rules, we obtain the quantification rule of Bohr–Sommerfield. Also, starting from the theory of thermodynamic fluctuations, the Heisenberg uncertainty principle has been deduced without use of any quantum assumption. Our results show then that the Bohr–Sommerfield rule and Heisenberg's principle are the mechanical versions of thermodynamics laws. Exactly as the Schrodinger equation and the quantum mechanics pulse are the mechanical versions of the geometrical optics laws.

DOI
10.1238/physica.regular.065a00300

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A formal resemblance between thermodynamics and mechanics has been found. In this respect, the fundamental ideas of quantum theory can be obtained from this analogy. We have deduced a correspondence between Caratheodory's theorem and Bohr's stationary states principle. Using the deduced correspondence rules, we obtain the quantification rule of Bohr–Sommerfield. Also, starting from the theory of thermodynamic fluctuations, the Heisenberg uncertainty principle has been deduced without use of any quantum assumption. Our results show then that the Bohr–Sommerfield rule and Heisenberg's principle are the mechanical versions of thermodynamics laws. Exactly as the Schrodinger equation and the quantum mechanics pulse are the mechanical versions of the geometrical optics laws.
Abstract It is well known that the experimental facts of atomic physics necessitate a departure from the classical theory of electrodynamics in the description of atomic phenomena. This departure takes the form, in Bohr’s theory, of the special assumptions of the existence of stationary states of an atom, in which it does not radiate, and of certain rules, called quantum conditions, which fix the stationary states and the frequencies of the radiation emitted during transitions between them. These assumptions are quite foreign to the classical theory, but have been very successful in the interpretation of a restricted region of atomic phenomena. The only way in which the classical theory is used is through the assumption that the classical laws hold for the description of the motion in the stationary states, although they fail completely during transitions, and the assumption, called the Correspondence Principle, that the classical theory gives the right results in the limiting case when the action per cycle of the system is large compared to Planck’s constant h, and in certain other special cases. In a recent paper Heisenberg puts forward a new theory, which suggests that it is not the equations of classical mechanics that are in any way at fault, but that the mathematical operations by which physical results are deduced from them require modification. All the information supplied by the classical theory can thus be made use of in the new theory.
In the context of a new analysis of the notebooks of Erwin Schrödinger, the paper deals with the question of the relation between Schrödinger's creation of wave mechanics and the contemporary efforts by Werner Heisenberg and his colleagues to establish a new quantum mechanics. How can one explain, from a broader historical and epistemological perspective, the astonishing simultaneity and complementarity of these discoveries? The paper argues that neither the physical problems with which both approaches deal nor what ultimately turned out to be their common mathematical ground are sufficient to explain their complementarity. Instead, their closeness is explained by analyzing their common roots in classical mechanics and its transformation in the light of the most fundamental new quantum law, the relation between energy and frequency found by Planck. It is shown, in particular, that for both approaches a bridge between quantum and classical aspects involving this relation was crucial. In the case of Heisenberg, this bridge was given by Bohr’s correspondence principle. In the case of Schrödinger it was constituted by Hamilton’s optical-mechanical analogy.
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.
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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.