I need information on how to estimate the time (if any) needed for electromagnetic wave to relect from the border of two environments with different refractive indices from the point of view of quantum theory?
I guess it happens "almost" instantaneously, though the intensity of light that gets reflected and transmitted depends on the reflectance and transmittance coefficients of the interfaces respectively.
However, in order to be more precise, the time taken to reflect will depend on the incident frequency of the wave and the penetration depth at the interface. For example, at optical frequencies in a metal, the penetration depth will be on the order of several nanometers, and so the group delay of the wave will be of the order of attoseconds (10-18 s).
https://en.wikipedia.org/wiki/Penetration_depth
https://en.wikipedia.org/wiki/Mathematical_descriptions_of_opacity
Emission by a scatterer is proportional to the acceleration, so there is no time delay. But the refractive indices take time to form. Polarizability \alpha (\omega) = \int_0^\nfty \alpha(\tau) e^{i\omega \tau}. A charged particle hit by a field continues to vibrate and emit. So, for a very short pulse, the scattered wave would last longer. Both in classical abd quantum mechanics.
As far as I know the Kanth theory, I can assume that there is no delay at all. As for refractive indices - I have no opinion.
It is true that reflection of light is (or more correct, can be explained as) the consequent of the constructive superposition of the medium electrons radiation, vibrating by the incident wave and, naturally, it needs some time to occur. But it would not be forgotten that this is right the mechanism responsible for propagation of the light in the mediums that in turn, takes the same time to occur in each distance interval along the propagation direction. Of the interest, the lower speed of the light in physical mediums in respect to that of the vacuum can be interpreted based on this delay time associated to many absorption-emission interactions between the propagating light and the medium electrons. On other hand, the reflected radiation from the mediums boundary does not arise only from the second medium, in fact, it is the consequent of a constructive superposition of the fields simultaneously created by the both mediums electrons within a (relatively) very thin layer of each mediums, closed to their boundary surface. As is seen, the concept as well as the value of the requested time is strongly depended on the definition of the “boundary” of the mediums that is difficult to be spatially determined. However, it may be still possible to roughly estimate this time by assuming a (mathematically) ideal surface just between the mediums and then, to calculate the mean time it takes for light to penetrate and turn back within a “reasonable?” depth of the second medium (this field is to be superimposed with the initial field within the first medium to create the reflected beam). From the point of view of the quantum theory, in particular, it may be found some expressions to estimate the photon-atom interaction time but, as I know, those are ideal expressions based on the assumption of single, isolated and inert atoms and, as I know again, it is not possible to simply generalize them to such complex N-body systems.
I think It can be estimated basing on the phase shift of the reflected way, as Delta_phase = w*Delta_time. In turn, Delta_phase is obtained by solving the scattering problem.
Good luck,
Ioseph Gurwich
Thank you for the answers.
Dear m-r @K. Silakhori , what you will tell about the time for light reflection from a mirror?
If you mean “metal” mirror, it should be noted that the light penetration depth in conductive metals (of course, in optical region) is much lower than that of the dielectrics, while, this very narrow layer is also strongly absorptive. So that, it can be reasonably assumed that the reflected beam is mainly due to only a few layers of the metal surface free electrons. It implies that the time for reflection (if it can be essentially defined) from conductive metal mirrors is much lower than that can be assumed for dielectric ones and can be considered almost instantaneous or at least, more instantaneous than the dielectric case! I do not have any quantitative information, but I think it may be in the order of free electrons oscillation period which in turn, equals the incident wave period. Anyway, I think this time interval can be acceptably neglected in the most applicable cases, unless, probably, in the case of ultra-short few-cycle pulses.
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