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CRC Press eBooks

1 On Rayleigh’s Pendulum

L.D. Faddeev, L.A. Khalfin, I.V. Komarov

Published May 21, 2004• 0 citations
Publisher

Abstract

The importance of Ehrenfest’s “Adiabatic Hypothesis” for the present and future of the quantum theory makes very desirable an exact examination of its purely mechanical meaning. Some years ago one of the authors1 found a general method to look for the adiabatic invariants, whereas the other author2 investigated the case of a degenerated, conditionally periodic system which had not been considered in the first paper (see also the paper by Burgers3). An objection which can be attributed to this theory is that in the course of calculations at some point a simplifying assumption was made in the integrated differential equations, namely that its right-hand side is subject to an averaging process; to explain this approximation, arguments connected with the slowness of changes of the system parameters were used. This shortcoming makes it difficult to use the ordinary methods and to check the adiabatic invariance of several quantities. Therefore it seems reasonable to consider a very simple example which we can integrate without any additional assumptions and only then use the slowness of parameter changes.

DOI
10.1201/9780203643204-6

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The importance of Ehrenfest’s “Adiabatic Hypothesis” for the present and future of the quantum theory makes very desirable an exact examination of its purely mechanical meaning. Some years ago one of the authors1 found a general method to look for the adiabatic invariants, whereas the other author2 investigated the case of a degenerated, conditionally periodic system which had not been considered in the first paper (see also the paper by Burgers3). An objection which can be attributed to this theory is that in the course of calculations at some point a simplifying assumption was made in the integrated differential equations, namely that its right-hand side is subject to an averaging process; to explain this approximation, arguments connected with the slowness of changes of the system parameters were used. This shortcoming makes it difficult to use the ordinary methods and to check the adiabatic invariance of several quantities. Therefore it seems reasonable to consider a very simple example which we can integrate without any additional assumptions and only then use the slowness of parameter changes.
Abstract The postulate of the existence of stationary states in multiply periodic dynamical systems requires that if the condition of such a system, when quantised, is changed in any way by the application of an external field or by the alteration of one of the internal constraints, the new state of the system must also be correctly quantised. It follows that the laws of classical mechanics cannot in general be true, even approximately, during the transition. There is one kind, of change, however, during which one may expect the classical laws to hold, namely, the so-called adiabatic change, which takes place infinitely slowly and regularly, so that the system practically remains multiply periodic all the time. In this case the quantum numbers cannot change, and it should be possible to deduce from the classical laws that the quantum integrals remain invariant. This was attempted by Burgers, who showed that they are invariant provided there are no linear relations of the type Σrmrωr = 0 (1) between the frequencies ωr, of the system, where the mr are integers. In general, however, the frequencies will alter during the adiabatic change, and in so doing will pass through an infinity of values for which relations such as (1) hold. A closer investigation is therefore necessary, as was pointed out by Burgers himself. In the following work, conditions which are rigorously sufficient to ensure the invariance of the quantum integrals, are obtained in such a form that it is possible for one to see whether they are satisfied or not without having to integrate the equations of adiabatic motion.
In this paper we find the quantities that are adiabatic invariants of any desired order for a general slowly time-dependent Hamiltonian. In a preceding paper, we chose a quantity that was initially an adiabatic invariant to first order, and sought the conditions to be imposed upon the Hamiltonian so that the quantum mechanical adiabatic theorem would be valid to mth order. [We found that this occurs when the first (m − 1) time derivatives of the Hamiltonian at the initial and final time instants are equal to zero.] Here we look for a quantity that is an adiabatic invariant to mth order for any Hamiltonian that changes slowly in time, and that does not fulfill any special condition (its first time derivatives are not zero initially and finally).
We examine the development of the concept of parametric invariance in classical mechanics, quantum mechanics, statistical mechanics, and thermodynamics, and particularly its relation to entropy. The parametric invariance was used by Ehrenfest as a principle related to the quantization rules of the old quantum mechanics. It was also considered by Rayleigh in the determination of pressure caused by vibration, and the general approach we follow here is based on his. Specific calculation of invariants in classical and quantum mechanics are determined. The Hertz invariant, which is a volume in phase space, is extended to the case of a variable number of particles. We show that the slow parametric change leads to the adiabatic process, allowing the definition of entropy as a parametric invariance.
When a theory is framed trying to explain a discrepant system of facts, it is a necessary process of thought to take some branch of the theory as more completely true than the rest, and to adjust the remaining parts in such a way that they will fit in with this base, though they may still conflict with one another. This has certainly been true of the quantum theory; the speculations connected with it have as their base the law of the conservation of energy. Now a critical examination of fundamentals does not by any means justify this faith. It is of course a fact of observation that in the gross energy is conserved, but this only means an averaged energy; and as pure dynamics has failed to explain many atomic phenomena there seems no reason to maintain the exact conservation of energy, which is only one of the consequences of the dynamical equations. Indeed it is hardly too much to say that had the photoelectric effect been discovered a century ago, it is probable that no one would ever have suggested that the status of the first law of thermodynamics was in any way different from that of the second. On the other hand, Bohr's theory, and especially Sommerfeld's extension of it, have given great encouragement to the belief that in dynamics lay the way to the complete truth, so that in consequence of the triumphs of that theory there has been little thought in other directions. Another impediment is that our whole ideas are saturated with the principles of energy, so that denying it leaves hardly any foundation from which to start.