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Proceedings of the Royal Society of London Series A Containing Papers of a Mathematical and Physical Character

The adiabatic invariance of the quantum integrals

P. A. M. Dirac

Published Apr 1, 1925• 45 citations
Publisher

Abstract

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.

DOI
10.1098/rspa.1925.0052

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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.
A connection between quantum mechanical and classical adiabatic invariance is shown for the special case of a charged particle moving in a homogeneous, time-dependent magnetic field. The relationship consists of the fact that the transition probability between energy eigenstates corresponding to the magnnetic field at two different times is a function of a certain classical adiabatic invariant alone. From this relation follows the well-known result that during sufficiently slow changes of the magnetic field the particle will remain in one quantum state which passes continuously through the different instantaneous energies. The adiabatic invariance of this classical quantity is shown directly by solving the equations of niotion when the magnetic field changes linearly with time. (auth)
We outline a rather extraordinary series of similarities between classical and quantal behavior in the limit of adiabatic time changes. These include the power laws for the goodness of the respective invariants for isolated eigenstates and invariant tori for integrable systems, the nature of the breakdown of the invariances—level crossing in quantum systems and the role of ever present non-linear resonances is examined in the case of generically non-integrable classical dynamics—and the perhaps surprising relationship for fully chaotic systems where sufficiently slow switching in either classical or quantal systems precisely preserves the number of energy levels up to a given energy. For suitably small values of Planck's constant these similarities yield clear examples of the Bohr correspondence principle linking classical and quantum mechanics; for larger values the details in the classical picture are quenched in the quantum.
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.
Abstract In traditional thermodynamics it is assumed that isentropic, reversible, adiabatic processes can be summoned up on demand and straightforwardly accomplished. By contrast, taking entropy as the maximised uncertainty of a final equilibrium state of a quantised system, it is not obvious that an associated process can always be found that is both rigorously isentropic and reversibly adiabatic. In fact, we find that linear relations between generalized forces X j (such as pressures P j ) and energies E j are necessary and sufficient conditions for a reversible quasi-static and adiabatic change to be truly isentropic. However, such relationships only hold for a few especially simple systems, such as the perfect gas and the idealised paramagnet. They do not generally hold to all orders for more complicated systems. By considering the associated entropy increases up to second order in small changes of the conjugate displacements (such as the volume V j ) we argue that the consequences are nevertheless in practice negligible.
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.
We study quantum adiabatic dynamics, where the slowly moving field is influenced by the system's state (feedback). The feedback is achieved either via mean-field quantum-classical interaction, or, alternatively, via non-disturbing measurements done on an ensemble of identical non-interacting systems. The situation without feedback is governed by the adiabatic theorem: adiabatic energy level populations stay constant, while the adiabatic eigenvectors get a specific phase contribution (Berry phase). However, under feedback the adiabatic theorem does not hold: the adiabatic populations satisfy a closed equation of motion that coincides with the replicator dynamics known by its numerous applications in evolutionary game theory. The feedback generates a new gauge-invariant adiabatic phase, which is free of the constraints on the Berry phase ( e.g. , the new phase is non-zero even for real adiabatic eigenfunctions). In a particular case the adiabatic theorem can still hold, but the new phases are non-trivial.