This discontinuity in the motion of an electron is an instance of a more general fact which has been discovered by the extraordinary minuteness of which physical measurements have become capable. It used always to be supposed that the energy in a body could be diminished or increased continuously, but it now appears that it can only be increased or diminished by jumps of a finite amount. This strange discontinuity would be impossible if the changes in the atom were continuous; it is possible because the atom changes from one state to another by revolution, not by evolution. Evolution in biology and relativity in physics seemed to have established the continuity of natural processes more firmly than ever before; Newton’s action at a distance, which was always considered something of a scandal, was explained away by Einstein’s theory of gravitation. But just when the triumph of continuity seemed complete, and when Bergson’s philosophy had enshrined it in popular thought, this inconvenient discovery about energy came and upset everything. How far it may carry us no one can yet tell. Perhaps we were not speaking correctly a moment ago when we said that an electron passes from one orbit to another “without passing over the intermediate space”; perhaps there is no intermediate space. Perhaps it is merely habit and prejudice that makes us suppose space to be continuous. Poincaré—not the Prime Minister, but his cousin the mathematician, who was a great man—suggested that we should even have to give up thinking of time as continuous, and that we should have to think of a minute, for instance, as a finite number of jerks with nothing between them. This is an uncomfortable idea, and perhaps things are not so bad as that. Such speculations are for the future; as yet we have not the materials for testing them. But the discontinuity in the changes of the atom is much more than a bold speculation; it is a theory borne out by an immense mass of empirical facts.
The relation of the new mechanics to the old is very peculiar. The orbits of electrons, on the new theory, are among those that are possible on the traditional view, but are only an infinitesimal selection from among these. According to the Newtonian theory, an electron ought to be able to move round the nucleus in a circle of any radius, provided it moved with a suitable velocity; but according to the new theory, the only circles in which it can move are those we have already described: a certain minimum circle, and others with radii 4, 9, 16, 25, 36 ... times as large as the radius of the minimum circle. In view of this breach with the old ideas, it is odd that the orbits of electrons, down to the smallest particulars, are such as to be possible on Newtonian principles. Even the minute corrections introduced by Einstein have been utilized by Sommerfeld to explain some of the more delicate characteristics of the hydrogen spectrum. It must be understood that, as regards our present question, Einstein and the theory of relativity are the crown of the old dynamics, not the beginning of the new. Einstein’s work has immense philosophical and theoretical importance, but the changes which it introduces in actual physics are very small indeed until we come to deal with velocities not much less than that of light. The new dynamics of the atom, on the contrary, not merely alters our theories, but alters our view as to what actually occurs, by leading to the conclusion that change is often discontinuous, and that most of the motions which should be possible are in fact impossible. This leaves us quite unable to account for the fact that all the motions that are in fact possible are exactly in accordance with the old principles, showing that the old principles, though incomplete, must be true up to a point. Having discovered that the old principles are not quite true, we are completely in the dark as to why they have as much truth as they evidently have. No doubt the solution of this puzzle will be found in time, but as yet there is not the faintest hint as to how the reconciliation can be effected.
What is known about other elements than hydrogen by means of the spectroscope all goes to show that the same principles apply, and that, when light is emitted, an electron jumps from an outer orbit to an inner one. But when there are many electrons revolving round a single nucleus, the mathematics becomes too difficult for our present powers, and it is impossible to establish such exact and striking coincidences of theory and observation as in the case of hydrogen. Nevertheless, what is known is sufficient to place it beyond reasonable doubt that the explanation of the spectrum of other elements is the same in principle as in the case of hydrogen. There is one case which can be tested to the full, and that is the case of positively electrified helium, which has lost one electron and has only one left. This only differs from hydrogen (as regards the movements of the electron) by the fact that the charge on the nucleus is twice as great as that on the electron, instead of being equal to it, as with hydrogen, and that the mass of the nucleus is four times that of the hydrogen nucleus. The changes which this produces in the spectrum, as compared with hydrogen, are exactly such as theory would predict.
In the present chapter, we have seen what was the conclusion to which Bohr was led as to possible states of the hydrogen atom, but we have not yet seen what was the reasoning by which he was led to this conclusion. In order to understand this reasoning, it is necessary to explain what is called the theory of quanta, of which Bohr’s theory of the atom is a special case. The theory of quanta will be the subject of the next chapter.
VI.
THE THEORY OF QUANTA
THE theory that the energy of a body cannot vary continuously, but only by a certain finite amount, or exact multiples of this amount, was not originally derived from a study of the atom or the spectroscope, but from the study of the radiation of heat. The theory was first suggested by Planck in 1900, thirteen years before Bohr applied it to the atom. Planck showed that it was necessary in order to account for the laws of temperature radiation; roughly speaking, if bodies could part with their warmth continuously, and not by jumps, they ought to grow colder than they do, when they are not exposed to a source of heat. It would take us too far from our subject to go into Planck’s reasoning, which is somewhat abstruse. A good account of it in English will be found in Jean’s Report on Radiation and the Quantum-Theory, published for the Physical Society of London (1914).
Planck’s principle in its original form is as follows. If a body is undergoing any kind of vibration or periodic motion of frequency (i.e. the body goes through its whole period times in a second), then there is a certain fundamental constant
such that the energy of the body owing to this periodic motion is or some exact multiple of
