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Henri Poincaré, Quanta hypothesis: Quanta of action

The new design is seductive by a certain side; for some time the tendency has been to atomism, matter appears to us as formed of indivisible atoms, and the electricity is no longer continuous, it is no longer divisible to infinity, it resolves into electrons all the same charge, all the same between them; we have also for some time magneton, or atom of magnetism. On this account, quanta appear to us as atoms of energy. Unfortunately the comparison does not continue until the end. An atom of hydrogen, for example, is truly invariable; it always preserves the same mass, whatever the compound in which it enters as element; the electrons also preserve their individuality through the most varied vicissitudes; is it the same with the so-called energy atoms? We have for example 3 quanta of energy on a resonator whose wavelength is 3; this energy passes on a second resonator whose wavelength is 5; it then represents not 3, but 5 quanta, since the quantum of the new resonator is smaller and, in the transformation, the number of atoms and the size of each of them has changed.

This is why the theory is not yet satisfactory for the mind; it must also be explained why the quantum of a resonator is in inverse proportion to the wavelength, and this is what decided Mr. Planck to modify the mode of exposition of his ideas; but here I am a little embarrassed, I would not want to betray Mr. Planck by going beyond his thought, by going farther than he wanted to go, or showing where he seems to be leading us. I will first translate his text as accurately as possible, while summarizing it a bit. I recall first that the study of thermodynamic equilibrium has been reduced to a question of statistics and probability.

“The probability of a continuous variable is obtained by considering elementary independent domains of equal probability… In the classic dynamics we use, to find these elementary domains, the theorem that two physical states of which one is the necessary effect of the other are equally probable. In a physical system if we represent by q one of the generalized coordinates and by p the corresponding momentum, according to Liouville’s theorem the domain ∫∫dpdq, considered at a given instant, is invariable with respect to the time if p and q vary according to Hamilton’s equations. On the other hand p and q may, at a given instant take all possible val ues, independent of each other. Whence it follows that the elementary domain is infinitely small, of the magnitude dpdq …. The new hypothesis has for its object to restrict the variability of p and q so that these variables will only change by jumps… Thus the number of elementary domains of probability is reduced and the extent of each is augmented. The hypothesis of quanta of action consists in supposing that these domains are all equal and no longer infinitely small but finite and that for each

∫∫dpdq = h,

h being a constant.”

I think it is necessary to supplement this quotation by some explanations; I cannot explain here what is the action, the generalized coordinates and the moments, or the various integrals that M. Planck brings online; I will confine myself to saying that the element of energy is equal to the product of the frequency by the element of action; and if the quantum of energy is proportional to the frequency, as we have said, it is because the quantum of action is a universal constant, a real atom.

But I must try to clarify what the elementary domains of probability are. These domains are indivisible; that is to say, as soon as we know that we are in one of these domains, everything is thereby determined; otherwise, if the events which are to follow were not therefore entirely known, if they were to differ according to whether we find ourselves in this or that part of this domain, it is because this domain would not be indivisible to the point probability since the probability of certain future events would not be the same in its various parts.

That is to say that all the states of the system which correspond to the same domain cannot be discerned between them, that they constitute one and the same state, and we are thus led to the following statement, more precise than that of Mr. Planck, who is not, I believe, contrary to his thought.

A physical system is susceptible only of a finite number of distinct states; it jumps from one of these states to the other without going through a continuous series of intermediate states.

Suppose for simplicity that the state of the system depends on only three parameters, so that we can represent it geometrically by a point in space. The set of representative points of the various possible states will not then be the whole space, or a region of this space, as is usually supposed; it will be a very large number of isolated points scattering the space. These points, it is true, are very tight, which gives us the illusion of continuity.

All these states should be regarded as equally probable. In fact, if we admit determinism, each of these states must necessarily succeed another state, exactly as well probable, since it is certain that the first leads to the second. We would thus see step by step that if we start from an initial state, all the states to which we will arrive one day or the other are all equally probable; others should not be regarded as possible states.

But our isolated representative points must not be distributed in space in any way; they must be so that, by observing them with our gross senses, we have been able to believe in the common laws of Dynamics and, for example, those of Hamilton. A comparison, which tightens the reality much closer than it seems, may help me to understand. We observe a liquid, and our senses invite us first of all to believe that it is continuous matter; a more precise experiment shows us that this liquid is incompressible, so that the volume of any portion of matter remains constant. Any reason then leads us to think that this liquid is formed of very small and very numerous, but discrete molecules; we will no longer be able to imagine a distribution of these molecules by imposing no obstacle on our imagination; it will be necessary, because of incompressibility, to suppose that two small equal volumes contain the same number of molecules. For the distribution of possible states, Mr. Planck is subjected to a similar restriction, and this is what he expresses by the equations I have quoted above, and which I cannot explain here further.

One could, it is true, imagine mixed hypotheses; let us suppose again that the physical system depends only on three parameters and that its state can be represented by a point of space. The set of representative points of the possible states may be neither a region of space nor an isolated network of points; it may consist of a large number of small surfaces or small curves separated from each other; for example, that one of the material points of the system can only describe certain trajectories; but to describe them in a continuous way except when it jumps from one trajectory to another under the influence of the neighboring points: this may be the case of the resonators of which we have spoken above; or else, the state of the ponderable matter might vary in a discontinuous manner, with a finite number of possible states only, while the state of the ether would vary in a continuous manner. None of this would be inconsistent with Mr. Planck’s thought.

But we will probably prefer the first solution, the frank solution to all these bastard hypotheses; only we must realize the consequences that this entails; what we said should apply to any isolated system and even to the universe. The universe would jump suddenly from one state to another; but in the interval he would remain motionless, the various moments during which he would remain in the same state could no longer be discerned from each other; we would thus arrive at the discontinuous variation of time, at the atom of time.

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