THE NEWER 'MATTER'
would be one associated with wave-optics, which would give the classical results in all cases where the wave-length was negligible in comparison with the dimensions of the path. Schrodinger suggests that a correct extension of the analogy would be to regard the wave system as sine-waves. In this connection it should be recollected that Fourier has shown that any given form of waves can be represented by the superposition of sine-waves, and that therefore a sine-wave (see Chapter XXXII) may be considered as a general formulation.
The ray methods in physics worked only to some extent, that is to say, in the cases where the radii of curvature of such rays and the dimensions of the spreads were large in comparison with the wave-length. When this is not the case we have to consider waves and not rays. Naturally, dealing with atomic dimensions, which are very small, instead of using the paths of the particles or the ^-rays, we have to use the ^-waves. It appears that this difference was the main, rather puzzling distinction between the classical mechanics and the quantum mechanics, between the macroscopic theories and the sub-microscopic ones.
The above realization, and its formulation into a mathematical theory, seems to be an important and extremely fruitful generalization, which probably will be retained as a method.
One of the puzzling features of the quantum theory was the structural appearance of the whole-number laws of the 'orbits'. That some such whole-number relation is justified seems to be well established, yet it contradicted the older 'continuous' mechanics. A new theory, to be at all satisfactory, should be able to fit these whole-number empirical data. The first test of the new wave mechanics, and also its first success, was precisely in this field.
If a ray of the ^-wave was supposed to run around in a circle for a stationary state, the circumference must be a whole multiple of the wave-length, or
where n is an integer. We see that the quantum condition of the Bohr theory, that the angular momentum must be a whole multiple of k/2ir, is only the result of the requirement that the wave-function ^ shall be single-valued, which is another way of saying that the circumference (2vr) must contain a whole number of wave-lengths. It may be compared perhaps with waves travelling around a circular loop of string. If they travelled both ways we would have stationary waves.
At this point a very important structural feature of the new wave mechanics makes its appearance. In the above interpretation the 'velocity of the electron' has lost its physical meaning, it becomes simply the wave length of the ^-waves. In the wave mechanics, as well as in the matrix mechanics, there is no meaning to the older 'position of an electron on its orbit'. So the wave mechanics again embodies the advantages of the matrix mechanics by not postulating entities which can never be observed. The whole numbers, as Schrodinger remarks, 'appear as naturally as do "integers" in the theory of vibrating strings'. In the theory of string vibrations these whole numbers are determined by certain