A Librarian's Open Shelf: Essays on Various Subjects

by Arthur E. Bostwick

Still again, the specific heats of solids, instead of being sensibly constant at all temperatures, are found to diminish rapidly in the low temperatures now available in liquid air or hydrogen and apparently tend to disappear at absolute zero. “All takes place,” says Poincaré, “as if these molecules lost some of their degrees of freedom in cooling—as if some of their articulations froze at the limit.”

Planck attempts to explain these facts by introducing the idea of what he calls “quanta” of energy. To quote from Poincaré’s paper:

“How should we picture a radiating body? We know that a Hertz resonator sends into the ether Hertzian waves that are identical with luminous waves; an incandescent body must then be regarded as containing a very great number of tiny resonators. When the body is heated, these resonators acquire energy, start vibrating and consequently radiate.

“Planck’s hypothesis consists in the supposition that each of these resonators can acquire or lose energy only by abrupt jumps, in such a way that the store of energy that it possesses must always be a multiple of a constant quantity, which he calls a ‘quantum’—must be composed of a whole number of quanta. This indivisible unit, this quantum, is not the same for all resonators; it is in inverse ratio to the wave-length, so that resonators of short period can take in energy only in large pieces, while those of long period can absorb or give it out by small bits. What is the result? Great effort is necessary to agitate a short-period resonator, since this requires at least a quantity of energy equal to its quantum, which is great. The chances are, then, that these resonators will keep quiet, especially if the temperature is low, and it is for this reason that there is relatively little short-wave radiation in ‘black radiation’… The diminution of specific-heats is explained similarly: When the temperature falls, a large number of vibrators fall below their quantum and cease to vibrate, so that the total energy diminishes faster than the old theories require.”

Here we have the germs of an atomic theory of energy. As Poincaré now points out, the trouble is that the quanta are not constant. In his study of the matter he notes that the work of Prof. Wilhelm Wien, of Würzburg, leads by theory to precisely the conclusion announced by Planck that if we are to hold to the accepted ideas of statistical equilibrium the energy can vary only by quanta inversely proportional to wave-length. The mechanical property of the resonators imagined by Planck is therefore precisely that which Wien’s theory requires. If we are to suppose atoms of energy, therefore, they must be variable atoms. There are other objections which need not be touched upon here, the whole theory being in a very early stage. To quote Poincaré again:

“The new conception is seductive from a certain standpoint: for some time the tendency has been toward atomism. Matter appears to us as formed of indivisible atoms; electricity is no longer continuous, not infinitely divisible. It resolves itself into equally-charged electrons; we have also now the magneton, or atom of magnetism. From this point of view the quanta appear as atoms of energy. Unfortunately the comparison may not be pushed to the limit; a hydrogen atom is really invariable…. The electrons preserve their individuality amid the most diverse vicissitudes, is it the same with the atoms of energy? We have, for instance, three quanta of energy in a resonator whose wave-length is 3; this passes to a second resonator whose wave-length is 5; it now 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 has changed.”

If, however, we replace the atom of energy by an “atom of action,” these atoms may be considered equal and invariable. The whole study of thermodynamic equilibrium has been reduced by the French mathematical school to a question of 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 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 values, 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 equals h, h being a constant.”

Put a little less mathematically, this simply means that as energy equals action multiplied by frequency, the fact that the quantum of energy is proportional to the frequency (or inversely to the wave-length as stated above) is due simply to the fact that the quantum of action is constant—a real atom. The general effect on our physical conceptions, however, is the same: we have a purely discontinuous universe—discontinuous not only in matter but in energy and the flow of time. M. Poincaré thus puts it: ”A physical system is susceptible only of a finite number of distinct states; it leaps from one of these to the next without passing through any continuous series of intermediate states.”

He notes later:

“The universe, then, leaps suddenly from one state to another; but in the interval it must remain immovable, and the divers instants during which it keeps in the same state can no longer be discriminated from one another; we thus reach a conception of the discontinuous variation of time—the atom of time.”