Compton effect – the change of frequency and wave vector of X-ray radiation in process of scattering on free electron is usually considered as convincing demonstration of corpuscular aspect of electromagnetic field.
However Schrödinger has shown, that this effect can be completely explained by wave theory, without any necessity to enter the idea of corpuscular photon.
This explanation is based on Schrödinger's conception of real electronic field which is described by equation of his name.
Under the influence of X-ray radiation the resting electronic wave gradually comes to movement, and the motionless part of the electronic field together with moving part forms the standing wave. On this periodic structure X-ray radiation scatters just as light on the standing wave of ultrasound.
At the beginning of his paper Schrödinger (1927) wrote: “According to the wave theory of light all changes in frequency and wave normal can he predicted, as is well known, on the basis of very simple and general considerations on the phase, without going into any detail of the phenomenon... If the as*sumption is correct that with the de Broglie waves we have at hand a tool, equivalent to wave optics, for the treatment of those phenomena that were formerly understood exclusively as motions of particles, we should expect and demand that on the basis of simple phase considerations of the stated kind one should understand the changes in direction and frequency of the aether wave in the Compton effect connected with the velocity change of the electron”.
Detailed calculations on the basis of Schrödinger's was made by J. Strnad in 1986. The electronic field is described by quantum mechanics taking into account relativistic effects, and radiation – by classic electrodynamics.
The account of Bragg conditions and Doppler's frequency shift leads to the equations which in accuracy describe the experimental results received by observation of Compton effect.
Über den Comptoneffekt; von E. Schrödinger. Annale der Physik. 1927, 28, 257-64.
http://www.apocalyptism.ru/Compton-Schrodinger.htm
The Compton effect - Schrödinger's treatment. J. Strnad. Eur. J. Phys. 7 (1986), 217-221.
http://www.apocalyptism.ru/Comptoneffect-treatment.htm