New Quantum Theory

NEW QUANTUM THEORY So far we have been describing a theory which, broadly speaking, consists of two not very coherent parts, namely, Ein stein's light-quantum theory and the theory based on stationary states and Bohr's postulate. Although as we have seen, it has been very successful, it does suffer from certain serious defects— quite apart from its logical weaknesses and obscurities. The chief of these are : I. It fails to meet the dilemma of the corpuscular and undula tory behaviour of light.

2. It fails to render any adequate account of those magnetic phenomena which are usually termed "anomalous," such as the Zeeman effect and the gyromagnetic effect.

3. The presence in the formulae of the Zeeman effect and band spectra of half quanta, i.e., of multiples of h which are half integers, finds no explanation in the "classical" quantum theory. This difficulty would obviously still remain, even if a new quantum equal to 11/2 were introduced.

There is a certain parallelism between the advance from the older or classical quantum theory to the newer theory and that from the old elastic solid theory of light to the later and much more adequate electromagnetic theory of Clerk Maxwell, and, no doubt, we shall long continue to use the language of the older quantum theory, and the fairly simple pictures it provides of atomic phenomena, just as we still use that of the elastic solid theory of light. The new theory has developed along two different lines, which, however, lead invariably to the same mathematical consequences, so that they are almost certainly just different aspects of one and the same theory. One of these aspects, known as wave mechanics, and historically the earlier and more interesting, was initiated by Louis de Broglie and further de veloped by E. Schroedinger. L. de Broglie's immediate objective was the solution of the dilemma (I) the other line of development is due to W. Heisenberg. He was inspired by a scientific outlook very prevalent among German physicists since the time of the Austrian philosopher, Ernst Mach, who insisted on the rigorous observance of the Newtonian principle of the avoidance of all hypotheses—"hypotheses seu metaphysicae, seu physicae, seu qualitatum occultarum, seu mechanicae, in philosophic experimen tali locum non liabent" (Newton, Principia Liber tertius p. 53o,

editio tertia).

Wave Mechanics.

The similarity of the laws of classical mechanics and geometrical optics was known long ago and for gotten or overlooked in modern times. The laws of reflection and refraction, it will be remembered, were subsumed by Fermat in the statement that the path selected by a ray of light in travel ling from a point A to a second point B is that which takes a minimum time. (See LIGHT.) To be quite accurate we ought to say that the path of a ray of light is always one which occupies a minimum or a maximum time, with, of course, the intermediate possibility exemplified by the behaviour of curved mirrors and lenses, that all paths within wide limits may occupy the same time. The time taken by any point on a wave front to travel a short distance dq in a direction normal to the wave front is obvi ously dq/u, where u is the velocity of the wave (the phase velocity). The whole time occupied between two points A and B is made up of the sum of such elements dq/u and is written Fermat's principle of minimum (or maximum) time may there fore be expressed by The symbol 6 is used to indicate that the expression in (49) represents the difference between the value of the integral (the time) measured along the actual path of the ray and that meas ured along a neighbouring path. It is important to note that the frequency v (or the period r) is taken to be the same for both paths. Remembering this and that the velocity u where X is the length of a wave, we see at once that an alternative statement of Fermat's principle is or the path of the ray of light is the one which contains the minimum (or maximum) number of waves.