The refractive index is obtained owing to classical electrodynamics. Is there any way to know how much momentum the electron gains when light incident on a dielectric material? How does the momentum of the electron depend on the refractive index ?
But of course, this is well developed field of theoretical physics..
Answering the second question: negligible. The wavelength of light in visible region starts at approx 400 nm. This is huge compared to the size of atoms or interatomic distances. The wavelength of an electron is of order of interatomic distance. Therefore change of the electron's momentum is tiny. You have to go to the X-Ray of gamma to have Compton scattering, but in that frequency range refractive index is not relevant.
Answering the third question: These quantities are not related at all.
Does there exist a quantum model for the refractive index? -- The answer depends on what exactly this is supposed to mean: The refractive index as such is an entirely classical quantity which characterizes the dispersion relation of a classical field, namely the electromagnetic field in materials. In this sense, the refractive index cannot be "quantized", i.e. there is no "refractive index operator", the expectation value of which would correspond to the "classical refractive index". In another sense, however, there is indeed a quantum model for the refractive index and this means that the refractive index can (at least in principle) be calculated systematically from "first principles", i.e. from the many-body Schrödinger equation describing the system under scrutiny on the microscopic level. The precise procedure of course implies a number of conceptual questions, on which we have elaborated in this article:
https://www.researchgate.net/publication/282843974_Microscopic_Theory_of_the_Refractive_Index
A short summary can be found here:
https://www.researchgate.net/publication/313241686_Microscopic_Theory_of_the_Refractive_Index
Article Microscopic Theory of the Refractive Index
Presentation Microscopic Theory of the Refractive Index
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