The Science of Cables

I would argue that on a slightly handwaving way, you can study "electricity" using classical physics. After all, much of it can be understood through Maxwell's equations. However, if you do this, you simply have to consider the charge densities, dielectric constant, magnetic permeabilities, et cetera as black boxes.

Quantum mechanics kicks in if you want to understand why a material is a conductor or an insulator. In solids, you can treat these questions through the study of electronic band structures, which relies heavily on Bloch's theorem and also on the Fermi-Dirac statistics. These are all elements from quantum mechanics. Therefore, if you want to understand how these charge densities behave on a microscopic level, you need QM. In typical systems, we can recover more empirical notions of of electricity such as Ohm's law from quantum treatments.

Now, if you really want to study how electromagnetic fields and electrons interact with each other (in detail), you will have to go further and you need to consider quantum electrodynamics (QED). This will give you the most detailed description of how photons and electrons interact. I would argue, however, that QED is often a bit of an overkill for condensed matter or atomic physics problems (not always, but often). Therefore you will find many "effective models", which can significantly simplify things.

Among these effective models, you have for example the Hubbard model, which includes interactions between electrons, without explicitly including the fact that these interactions are mediated via the electromagnetic field. You have a whole zoo of similar models, so I will not go into all of them. The main point is that they usually focus on the behaviour of the electrons and do not explicitly consider couplings to the electromagnetic field. There are, however, models which study the response of the material to the electromagnetic field. Usually this leads you to models which treat quasi-particles, such as plasmons, polarons, and polaritons. I am a bit out of my field of expertise here, but I believe that these can be used to derive the parameters that go into Maxwell's equations. Note, however, that these models are still not explicitly considering full-scale quantum electrodynamics.

As I get the feeling that you are also interested in the radiation side of the story, let me shift gears a little. Radiation actually is a very old problem in quantum physics. It lies at the basis of the probabilistic interpretation of the theory and motivated Heisenberg to develop his matrix mechanics. Light-matter interactions in that time were narrowly connected to atomic physics and spectroscopy, later molecular and nuclear physics joined in, covering a range from microwaves to gamma-rays in the electromagnetic spectrum. Now, if we really want to understand in depth how the electrons (or nucleons for nuclear physics) in these systems interact with electromagnetic fields, we must again divert to QED.

Nevertheless, also on the side of the electromagnetic field, there are effective models. These can be found, for example, in quantum optics. In these models, you typically make serious simplifications on the level of the "matter" and focus on the electromagnetic field. Typically, the interaction between light and matter generates some type on nonlinear effects in the electromagnetic field, so I would argue that the vast majority of models in nonlinear optics are models where you had some type of interaction with matter, which you coarse-grain out. Note, however, that these effective descriptions do not even require quantum mechanics to make sense. You can usually do nonlinear optics using Maxwell's equations. If you want to see effective models of the quantum side of the electromagnetic field in action, you have to turn to quantum optics, where you usually include matter (like a "two-level atom") in a more explicit way, see for example the Jaynes-Cummings model.

With this little excursion into the realm of optics, you may notice that there was not a lot of "electricity". The reason why we did not really get into that, is because it is horribly difficult. The treatment of models in condensed-matter theory, which only deal with the interacting electrons are complicated on their own and so is the theory of the quantum and nonlinear effects in the electromagnetic field. There is, however, one additional playground which we can explore. You may wonder what happens when we consider quantum properties of the electromagnetic field and combine them with macroscopic conductors and insulators. This is done in what is called Macroscopic quantum electrodynamics, which can be used to study for example the Casimir effect.

To conclude, let me stress that genuine quantum effects in the electromagnetic field itself (so everything related to light et cetera) are quite rare in day to day life. The electromagnetic radiation effects that are related to electronics and electricity is described really well by Maxwell's equations. However, if you really want to understand what happens in materials through which your electricity flows, on a microscopic level, you will have to consider the quantum models of condensed-matter physics.

Disclaimer: None of these fields is really my speciality, so I would be happy if a condensed-matter physicist or a quantum optician could provide more details or corrections if necessary.