What I know is that the classical picture of atomic structure can't predict its stability (also the discrete spectral lines) which led to inventing quantum mechanics that predicts stationary states for electrons around the atom. I have two questions: 1) How can we incorporate electromagnetism in this case? I mean, when dealing with electromagnetism we always assume interactions between charged "particles". But in QM we only have "wavefunctions". So how can one describe the electrostatic interaction between two wavefunctions for instance. 2) What can we get by applying EM to the problem of electron orbiting a nucleus within the quantum mechanics framework?
Maxwell's equations only determine the electromagnetic fields in terms of their sources (including the possibility of vacuum solutions in the absence of charge and current densities). As such, they are also microscopically valid; in particular even "in materials" -- see our review:
Article Ab initio materials physics and microscopic electrodynamics of media
On a more fundamental level, however, the (transverse parts of) electromagnetic fields have to be quantized, and this leads to quantum electrodynamics. By contrast, the stability of atoms has nothing to do with (quantum or classical) electrodynamics. Instead, the energy levels in atoms, molecules etc. result from a quantum theory of the "matter" degrees of freedom (as opposed to electromagnetic degrees of freedom). Furthermore, there is no electrostatic interaction between (one-particle) wave-functions. Rather, the transition to an interacting theory requires many-body wave-functions. The Coulomb interaction is then part of the fundamental Hamiltonian of electrons and nuclei -- see our review:
Article Response Theory of the Electron-Phonon Coupling
Finally, quantizing both matter and electromagnetic degrees of freedom corresponds to "full" quantum electrodynamics (as opposed to free quantum electrodynamics, where only electromagnetic fields are quantized while matter is absent or remains classical). Its application to energy levels in "matter" yields so-called "radiative corrections".
I recommend you study Quantum Electrodynamics, "QED" for a starter on how to solve problems such as this. All you'll get here on RG are promotions for new ideas from armchair physicists, like me.
The subatomic level (electrons and quarks and smaller) would require some experiment supported model of what charge is beyond (or below) what an electron has. Same for the magnetic field below what a moving charge induces(causes an electron to move).
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