A collision problem in the wave mechanics
C. G. DARWIN
Abstract In the present work the wave mechanics are applied to the discussion of a collision problem of a more complicated type than has, I think, hitherto been done. Though there is nothing very new in the process it is rather interesting to see how a wave function behaves which involves two bodies, and is therefore outside our ordinary space intuitions. But the work had a much more general aim, as an attempt to see whether a sharper line of demarcation could be drawn between the particle-like and the wave-like properties of matter. Put briefly, I take a problem which would be regarded at first sight as irreconcilable with a pure wave theory, but thoroughly typical of the behaviour of particles, and show how in fact the correct result arises naturally from the consideration of wave alone. I have been led in this way to certain not very original speculation on the basis of the quantum theory and these are given at the end of the paper. These are not in the least intended to be a general formulation of the theory, but rather as an examination of a particular type of attempted formulation. I fear that the views expressed will be found partly obvious and perhaps partly inconsistent or even erroneous. My excuse for giving them is that the quantum theory seem already sufficiently developed for us to attempt something more than a mathematical formulation of it, and that a generally accepted physical understanding will be more quickly reached by agreement or disagreement with expressed opinions, than by waiting for the arrival of a full-grown theory. It has been an extraordinarily difficult task to express the ideas I intended, and I wish to thank Prof. D. R. Hartree for many valuable criticisms of my first attempts to do so, though he must be exonerated from all blame if the final result is unintelligible or erroneous. 1. The quantum mechanics have been expressed in a variety of ways, which place the emphasis differently, but it has been shown that they are all exactly equivalent, so what may choose which we will. I shall use the following description, limiting myself to the simpler cases which do not involve considerations of relativity. We take a partial differential equation of the wave type, set down according to certain rules depending on the particular system considered. We solve this subject to certain boundary conditions, expressive of all the limitations imposed by silts, mirrors, diffraction, gratings, moving shutters, etc., that may occur in the proposed system, and obtain as result the wave function Ψ, a function of all the co-ordinates of the system and of the time. From this we from a set of functions, of which the simplest is |Ψ|2; this process may be called taking the intensities. We next have to prepare for the particle aspect, by applying certain rules so as to normalise the intensities, a process entirely foreign to the idea of waves. Finally, we make the interpretation, in which we say that the normalised intensities measure the probability of the particles having such and such positions, momenta, energies, etc., and these are the results verified by observation.
Proceedings of the Royal Society of London Series A Containing Papers of a Mathematical and Physical Character•Jun 4, 1929•49 citations