722 X. ON THE STRUCTURE OF 'MATTER'
where a set of ^-waves, which vary continuously in direction and in frequency in a small range, reinforce each other to give a wave-group travelling with what we usually call 'the velocity of the particle'.
As the waves have different frequencies, they travel with different velocities and so we have to face the problem of the dispersing medium, where, according to classical theories, the region of reinforcement has a velocity different from that of the phase. The ordinary expression for 'group velocity' gives, on the wave interpretation, the magnitude of the 'velocity of the particle'.12
That which, in the classical theories, we called the 'motion of the particle' is represented by the motion of the region of reinforcement of ^-waves. The direction of motion is represented by the direction of a ray-, or wave-normal. The particular ray selected as the position of the 'path of the particle' is represented by the ray cut in coinciding phase by a set of ^-waves with slightly varying directions. The 'position of the particle' is given by the small region occupied by the group of waves of slightly different frequencies and velocities.
In this connection we should notice an important point which is necessitated by the methods of generalization and of translation from macroscopic to sub-microscopic events and vice versa. Schrodinger, by formulating the differential equations for the ^-waves, has brought out clearly the important structural point that wave mechanics bears a similar relationship to classical mechanics of particles as wave-optics bears to ray-optics.
Here again we have the macroscopic phenomena capable of being treated by mathematical methods different from those used for the small-scale phenomena.
In classical mechanics the state of a system whose co-ordinates were qi . . . q, and whose momenta were pi . . . p was represented by a point in a 2s-dimensionaI ^-spread and the changes in the system were represented by the passage of the point along some curve, a 'ray', so to say.
Schrodinger regards the classical mechanics as only an approximation, while rigorous treatment must be made by the aid of wave mechanics.
The large-scale, or macroscopic, mechanical processes correspond to a wave signal in the g-spread and can be regarded as a point in comparison with the geometrical structure of the path. In small-scale phenomena, such as the atomic processes, a rigorous wave formulation must be used.
This analysis can be carried further and Hamilton knew well and used the analogy between mechanics and geometrical optics. The Hamilton variation
Sj'Ldt^O, is Fermat's principle for a wave motion; the Hamilton-
Jacobi equation expresses Huygens' principle for wave motion; and the new wave mechanics expresses the Kirchhoff analysis of physical optics. As Huygens' principle could deal with the problems of physical optics up to a certain point, so the Hamilton-Jacobi equations could deal with atomic problems up to a certain point. At the exact wave analysis of Kirchhoff was needed to clear up the finer points of physical optics, so the new wave mechanics is required for the exact solution of the atomic problems.13
A detailed analysis shows that classical mechanics was associated with geometrical optics (ray-optics). Obviously, a more exact system of mechanics