I don't know how much experience you have with quantum mechanics, so apologies if I'm going over stuff you already know. For any quantum system, the Hamiltonian
H describes the energy of the system.
H, like just about everything else in quantum mechanics, is an operator, so for a given state of the system $$|\phi\rangle$$, applying
H will return another state that may or may not be a scalar multiple of the first. The states that do map to themselves under
H (ie. $$H|\phi\rangle=k|\phi\rangle$$ are called the eigenstates of
H. According to the Schrodinger equation, repeatedly applying
H tells us how the system evolves with time. Because of this, the eigenstates of
H can be thought of as the "steady states" of the system that do not change with time. For a more general state, time evolution can be determined by writing the state as a sum of eigenstates, then calculating how the different eigenstates interfere with each other as time progresses.
To see why light reflects in the ways that it does, we can look at the eigenstates of the electromagnetic Hamiltonian that governs light. In a vacuum, the eigenstates of
H are "plane waves", which oscillate as sine waves in the direction in which the light is propagating and don't vary at all in the other directions. The presence of an atom changes this Hamiltonian because an electromagnetic field interacts with the atom's electrons. (The field also interacts with the nucleus, but the nucleus is much heavier and the electrons shield it, so it's a good approximation to treat the nucleus as fixed and just think about how incoming light affects the electrons.) This new Hamiltonian has a set of spherically symmetric eigenstates that radiate outward from the atom.
We want to know what happens to a photon that comes in from far away, interacts with the atom, and leaves. In this scenario, both sets of eigenstates come into play, so we have to combine them into what are called "scattering" eigenstates. Scattering is a difficult topic (I'm actually taking a course this term that will hopefully refresh my memory on how to deal with it), but if you're really interested in the math it's covered in any graduate-level QM textbook. Qualitatively, the main thing you have to know is that one can determine how strongly the two sets of eigenstates mix together depending on two things: the strength of the light-matter interaction and the spatial overlap of the two eigenstates with each other. If these factors are small, the scattering eigenstates will look mostly like the free space eigenstates, with a small perturbation due to the atom. If the factors are large, the scattering eigenstates will be heavily perturbed and will have a lot of spherical character.
This all describes how light will scatter from a single atom. When we put many atoms together in a solid, all of their individual scattering patterns will interfere with each other, giving rise to a collective pattern. This means that the scattering properties of each atom and the spatial arrangement of all the atoms, taken together, tell us everything about how an object reflects light. In some cases, when the scattering effect is strong, it makes sense to talk about the atoms absorbing and re-emitting light. Any opaque or mirrored object is a good example of this. In such cases, the atoms on the surface of the object absorb incoming light and re-emit it in a spherical pattern. Light that is re-emitted away from the surface of the object goes on its merry way, while light emitted in the other direction hits the next layer of atoms and is scattered again. Each consecutive scattering exponentially decreases the probability that a given photon will still be travelling into the object, so a think enough object will scatter essentially all incoming light back the way it came. If the object's atoms are regularly spaced in such a way that their scattering patterns interfere coherently on a large scale, the patterns in the incoming light can be preserved during re-emission, and the surface will be reflective. If the atoms are not regularly spaced, re-emission will effectively randomize the incoming light, giving rise to a more "matte" appearance.
In other cases, when the scattering effect is weak, it makes more sense to think of the atoms as perturbing the trajectory of the light without ever actually absorbing it. (Just about everything does absorb light at some rate, but with weak scattering that isn't the dominant effect on incoming light.) This is the case in transparent materials. If the atoms of such a material are not regularly spaced, the weak scattering can lead to visible effects, even if very little light is absorbed; think of the way an object underwater appears to distort and ripple as the water moves around. If the atoms are regularly spaced in such a way that their collective scattering patterns interfere destructively, it's possible for light to pass through with almost no changes at all. This is what's happening in glass: the crystal structure of the atoms counteracts what little scattering is happening, so light passes straight through. Any disruption to the crystal pattern will break the interference and make the scattering visible again, which is why the surface of a glass is somewhat reflective.
As a final cool example, in some crystals the overlaps between the two kinds of eigenstates depends on the direction of the incoming electromagnetic field, so the scattering effect is different for different light polarizations. This gives rise to
birefringence. Hope that helps!
