Question

Electrostatics basically means dealing with time independent electric fields (which was produced by stationary charges)

Now consider a neutral conductor. We know that putting a net negative charge on the conductor, the charge will very quickly spread over the surface till electrostatic equilibrium is reached.

What does this exactly mean? does it mean that electrons on the surface are not "moving" anymore and they become stationary? So in this situation we have 2 kinds of electrons: 1-electrons inside the meat of the conductor whose dynamics is described by quantum mechanics 2-electron on the surface which are not moving in the classical sense?

Answer 1

All the electrons in the metal obey quantum mechanics at the microscopic level, regardless of whether it is the bulk or the surface. However, depending on the system you are studying, and the conditions you have created, you can safely employ a semiclassical description of electron motion. For most typical metals of macroscopic scale, the single-particle semiclassical description works well (Drude model is the simplest). My explanation below relies on this assumption:

Metals have half filled bands; or in other words, the chemical potential lies in between a band. As a result, the electrons near the chemical potential can move around the metal, while the electrons that are much deeper, i.e. way below the chemical potential, stay bound to the positive ion cores. If you introduce an excess negative charge, the extra electrons will go into the half filled band, and consequently the chemical potential will rise slightly; the band will still be half filled. You can safely ignore filling up of this band by the excess electrons. I am saying this because even if you introduce one extra electron per atom, and you have an Avogadro number of atoms, you will get some ridiculous values of net excess charge 6