Interaction of e.m radiation: from classical to quantum behavior

[Gruxg]

(I am not sure what is the best subforum for this doubt, please move the thread if necessary)

When we study the interaction of the electromagnetic waves with a charged particle (let's say an electron) we can find two totally different approaches in the literature. If we are talking about light or lower frequency radiation, a classical view is often employed and it is said that the electron oscillates due to the electric field of the wave. However, for higher frequencies (gamma and X-rays) the e.m. field is viewed as particles that hit the electron (the Compton effect). Since both radiations are of the same nature and they differ only in the frequency, and all the intermediate frequencies are possible, it is reasonable to think that there should exist a smooth transition between the "electric-like" behavior and the "particle-like" behavior, but I have trouble to figure it out. I cannot imagine how it is.

For example, if we irradiate a conducting medium or a plasma with UV light (between visible and X-rays), is the interaction with the charges best described by the classical approach or by Compton scattering?. Are both effects present as different, simultaneous phenomena or there is something in between?

If we could irradiate the electrons with a coherent X-ray beam (I think it can be achieved in a synchrotron, but not sure), could we get an ordered oscillation instead of Compton effects?

If we have a visible or radio-frequency radiation very dim so that the photon flux is very low, would the interactions be "compton-like"?

Any explanation, reference or link that could help to understand this transition would be most appreciated!

 

[Gruxg]

I guess my doubt must be too difficult or too general or too stupid...

 

[fzero]

(I am not sure what is the best subforum for this doubt, please move the thread if necessary)

Your question is certainly not stupid. It might have gotten a faster response in the Quantum Physics subforum.

When we study the interaction of the electromagnetic waves with a charged particle (let's say an electron) we can find two totally different approaches in the literature. If we are talking about light or lower frequency radiation, a classical view is often employed and it is said that the electron oscillates due to the electric field of the wave. However, for higher frequencies (gamma and X-rays) the e.m. field is viewed as particles that hit the electron (the Compton effect). Since both radiations are of the same nature and they differ only in the frequency, and all the intermediate frequencies are possible, it is reasonable to think that there should exist a smooth transition between the "electric-like" behavior and the "particle-like" behavior, but I have trouble to figure it out. I cannot imagine how it is.

The size of quantum effects can typically be estimated by comparing energy scales. In the case of an EM wave interacting with some system, we would compare the energy scale of the EM wave, $h\nu = h/\lambda$ with the typical energy scale of the system.

For a hydrogen atom, the appropriate energy scale is the Rydberg constant $\alpha^2 m_e c^2/2 \sim 13.6~\mathrm{eV}$, which is equivalent to around $91~\mathrm{nm}$ (extreme UV spectrum). For wavelengths which are very large compared to this scale (IR and longer), the hydrogen atom looks classical. This is because the energy of the corresponding photons are not large enough to excite the electron in the ground state to an excited quantum state. If the H-atom absorbs a low-energy photon, it will just give the entire atom some momentum: for a gas of atoms, this means that we're just heating the sample. Classical physics is a good approximation to describe a hydrogen atom with a small velocity. As we get into the visible spectrum, a single photon has enough energy to cause transitions between excited states of the H-atom and we need quantum mechanics to describe the physics. At even higher energies, we can excite the ground state and for our $13.6~\mathrm{eV}$ photons, we have enough energy to ionize the atom, completely stripping the electron away.

At extremely high energies (the X-ray and gamma frequencies) we have enough energy to excite nuclei of atoms, so again we need quantum mechanics to describe the processes.

For example, if we irradiate a conducting medium or a plasma with UV light (between visible and X-rays), is the interaction with the charges best described by the classical approach or by Compton scattering?. Are both effects present as different, simultaneous phenomena or there is something in between?

The relevant energy scale for free nonrelativistic electrons would be the de Broglie wavelength= $h/p$. In a plasma, we can approximate this as $\lambda_B = h/(m v_T)$, where $v_T$ is the thermal velocity

$$v_T = \sqrt{\frac{k_B T}{m}}.$$

So interactions of photons with wavelengths much longer than $\lambda_B$ are classical. At higher energies, quantum effects would become relevant.

If we could irradiate the electrons with a coherent X-ray beam (I think it can be achieved in a synchrotron, but not sure), could we get an ordered oscillation instead of Compton effects?

If we have a visible or radio-frequency radiation very dim so that the photon flux is very low, would the interactions be "compton-like"?

Any explanation, reference or link that could help to understand this transition would be most appreciated!

I haven't found a particularly good link that would further explain the use of energy scales to estimate quantum vs classical behavior, but it's certainly something that is discussed in every QM text. Obviously the right place to look is the same place where they discuss the de Broglie wavelength and the physical significance.

 

[Gruxg]

Thanks fzero. You have explained how to find out in which cases the classical approach is a good approximation and in with cases the "particle like" behavior is dominant, depending if the photon energy is much higher or much lower than a characteristic energy of the system. However, what I attemp to understand is what happens in the transition zone, because I assume there is not a sharp change at a given energy (well, if we have an atom with its quantized energy levels there may be, but I was thinking in free electrons). How one kind of behevior arises from the other.

Perhaps the answer to the last two questions could bring some light to the matter.