Thomson Scattering when low-intensity light meets an orbital electron

Can you explain to me the reason why Thomson Scattering can not explain what happens when light meets an electron at low intensity, and what does that have to do with light being a wave or particle or relativistic/QM effects?

https://en.wikipedia.org/wiki/Compton_scattering

"Effect is significant because it demonstrates that light cannot be explained purely as a wave phenomenon.[4] Thomson scattering, the classical theory of an electromagnetic wave scattered by charged particles, cannot explain shifts in wavelength at low intensity: classically, light of sufficient intensity for the electric field to accelerate a charged particle to a relativistic speed will cause radiation-pressure recoil and an associated Doppler shift of the scattered light,[5] but the effect would become arbitrarily small at sufficiently low light intensities regardless of wavelength."

A classical, low intensity electromagnetic wave can only make a charged particle oscillate at the same frequency as the radiation. There is nothing that would produce radiation of a different frequency.

mfb said:

A classical, low intensity electromagnetic wave can only make a charged particle oscillate at the same frequency as the radiation. There is nothing that would produce radiation of a different frequency.

Okay, but what does low intensity have to do with anything? Intensity is the amount of photons that pass from one point in a time interval, right? So, what is the relationship between this definition and Thomson scattering?

In the classical theory, there was the possibility of light carrying momentum, and therefore exerting force when absorbed or reflected. Reflection could accelerate the reflector and therefore cause Doppler shifting of reflected light.
However, in classical non-quantum theory, absorption and reflection are continuous processes. Therefore at low intensity (although high frequency), the scattered light should cause a small absorbed energy at any unit time, small recoil momentum at any unit time, small Doppler shift...

snorkack said:

In the classical theory, there was the possibility of light carrying momentum, and therefore exerting force when absorbed or reflected. Reflection could accelerate the reflector and therefore cause Doppler shifting of reflected light.
However, in classical non-quantum theory, absorption and reflection are continuous processes. Therefore at low intensity (although high frequency), the scattered light should cause a small absorbed energy at any unit time, small recoil momentum at any unit time, small Doppler shift...

Okay but what about high intensity? what does change?

Take the famous formula given in the Wikipedia article
$$\lambda'-\lambda=\lambda_{\text{C}} (1-\cos \theta).$$
Here ##\lambda_{\text{C}}=h/(m_{\text{e}} c)## is the Compton wavelength and ##\theta## the scattering angle of the photon. The largest change you thus get at ##\theta=\pi##. Then the change in wavelength twice of the Compton wave-length, which is about ##2.4 \cdot 10^{-12} \text{m}##, which is in the ##\gamma##-ray range.

Visible light is at wave-lengths of 400-800 nm or ##(4-8)\cdot10^{-7} \text{m}##. Here the change in wave length (i.e., the momentum transfer to the electron) at the order of the Compoton wave-length of the electron is thus completely negligible and can be neglected. This is described by Thomson scattering.

All this has little to do with the intensity of the light but rather with his wavelength/frequency.

Also note that the Compton effect is not a proof for the quantization of the em. field. In fact it has been theoretically explained with modern quantum theory first in the semiclassical approximation, i.e., by treating only the electron quantum mechanically and keeping the em. field classical (Klein and Nishina 1929).

https://en.wikipedia.org/wiki/Klein–Nishina_formula

Contrary to the Wikipedia article. It's not a calculation in full QED but the treatment of the motion of an electron according to the Dirac equation (in the 1st-quantization formalism) in a plane-wave classical em. radiation field.