in that region. So that if
is mainly concentrated in one small stormy area, it is practically certain that the electron is there; we are then able to localise it definitely and conceive of it as a classical particle. But the
-waves of the hydrogen atom are spread about all over the atom; and there is no definite localisation of the electron, though some places are more probable than others.[36]
Attention must be called to one highly important consequence of this theory. A small enough stormy area corresponds very nearly to a particle moving about under the classical laws of motion; it would seem therefore that a particle definitely localised as a moving point is strictly the limit when the stormy area is reduced to a point. But curiously enough by continually reducing the area of the storm we never quite reach the ideal classical particle; we approach it and then recede from it again. We have seen that the wave-group moves like a particle (localised somewhere within the area of the storm) having an energy corresponding to the frequency of the waves; therefore to imitate a particle exactly, not only must the area be reduced to a point but the group must consist of waves of only one frequency. The two conditions are irreconcilable. With one frequency we can only have an infinite succession of waves not terminated by any boundary. A boundary to the group is provided by interference of waves of slightly different length, so that while reinforcing one another at the centre they cancel one another at the boundary. Roughly speaking, if the group has a diameter of 1000 wave-lengths there must be a range of wave-length of 0.1 per cent., so that 1000 of the longest waves and 1001 of the shortest occupy the same distance. If we take a more concentrated stormy area of diameter 10 wave-lengths the range is increased to 10 per cent.; 10 of the longest and 11 of the shortest waves must extend the same distance. In seeking to make the position of the particle more definite by reducing the area we make its energy more vague by dispersing the frequencies of the waves. So our particle can never have simultaneously a perfectly definite position and a perfectly definite energy; it always has a vagueness of one kind or the other unbefitting a classical particle. Hence in delicate experiments we must not under any circumstances expect to find particles behaving exactly as a classical particle was supposed to do—a conclusion which seems to be in accordance with the modern experiments on diffraction of electrons already mentioned.
We remarked that Schrödinger’s picture of the hydrogen atom enabled it to possess something that would be impossible on Bohr’s theory, viz. two energies at once. For a particle or electron this is not merely permissive, but compulsory—otherwise we can put no limits to the region where it may be. You are not asked to imagine the state of a particle with several energies; what is meant is that our current picture of an electron as a particle with single energy has broken down, and we must dive below into the sub-aether if we wish to follow the course of events. The picture of a particle may, however, be retained when we are not seeking high accuracy; if we do not need to know the energy more closely than 1 per cent., a series of energies ranging over 1 per cent, can be treated as one definite energy.
Hitherto I have only considered the waves corresponding to one electron; now suppose that we have a problem involving two electrons. How shall they be represented? “Surely, that is simple enough! We have only to take two stormy areas instead of one.” I am afraid not. Two stormy areas would correspond to a single electron uncertain as to which area it was located in. So long as there is the faintest probability of the first electron being in any region, we cannot make the Schrödinger waves there represent a probability belonging to a second electron. Each electron wants the whole of three-dimensional space for its waves; so Schrödinger generously allows three dimensions for each of them. For two electrons he requires a six-dimensional sub-aether. He then successfully applies his method on the same lines as before. I think you will see now that Schrödinger has given us what seemed to be a comprehensible physical picture only to snatch it away again. His sub-aether does not exist in physical space; it is in a “configuration space” imagined by the mathematician for the purpose of solving his problems, and imagined afresh with different numbers of dimensions according to the problem proposed. It was only an accident that in the earliest problems considered the configuration space had a close correspondence with physical space, suggesting some degree of objective reality of the waves. Schrödinger’s wave-mechanics is not a physical theory but a dodge—and a very good dodge too.
The fact is that the almost universal applicability of this wave-mechanics spoils all chance of our taking it seriously as a physical theory. A delightful illustration of this occurs incidentally in the work of Dirac. In one of the problems, which he solves by Schrödinger waves, the frequency of the waves represents the number of systems of a given kind. The wave-equation is formulated and solved, and (just as in the problem of the hydrogen atom) it is found that solutions only exist for a series of special values of the frequency. Consequently the number of systems of the kind considered must have one of a discrete series of values. In Dirac’s problem the series turns out to be the series of integers. Accordingly we infer that the number of systems must be either 1, 2, 3, 4, ..., but can never be 2¾ for example. It is satisfactory that the theory should give a result so well in accordance with our experience! But we are not likely to be persuaded that the true explanation of why we count in integers is afforded by a system of waves.