Sheyr,
Electrodynamics deals with charges, even in motion, through Maxwell's Equations, and on an individual basis.
Quantum Mechanics and Quantum Field Mechanics, however, deal with the situation from another point of view. It speakes about the probability of finding a charge (or a particle in general) per certain volume, within this system, at a certain position "at certain time (for time dependent fields)". So as to say it deals with the volumetric probability-density. But, it describes only an ensemble of events, rather being capable of speaking about only one event.
Of course, QM and ,more effectively, QFTs (especially QED) give significantly much more exact results rather than Electrodynamics, in certain domains of energy and dimensions.
Schrodinger, the founder of the wave-function concept and the mathematical structure of QM, had the idea that QM are covering some unknown property by the wave function. And, the right imagination, according to QM, come late in his age by a known scientist (I don't remeber his name) that its squared modulus introduces the probability density. This can be found in Shrodinger Original Papers, and he implicitly emphasized that this coincided with his thought about the nature of QM. Strange Papers! I will do my best to get them and share them together.
Schrodinger Equation was then made relativitistic by Dirac's for Fermions and By Klein and Gordon for Bosons.
However, there was something missing, the quantization of the field due to the quantized particles (or something like this). And, by the use of nature of Relativistic Quantum Mechanics and the equations of electromagnetic fields, QED was reformulated by Feynman and Tomonagi (And his partner), independently, at the same time! And, they got the noble prize after that for their work, and they deserve it. QED, in particular, is called by not-few scientists as the jewel of physics for its high accuracy.
Returning to the question, QED doesn't deal with forces. We can speak here about effects, in order to be able to compare it to Coulomb Force. For a very large domain of parameters' values, effects are the same (check atomic radii, ... by coulomb and by QM), but the problem is in the characteristic of the distribution of "Many Particles {not interacting mutually}" through space. Actually, these distributions are being graphed in QM, by Surface Graphs, not Curves.
By the way, whne you are talking about coulomb only, i.e. static electric field, then you are not in need for QED, you only need Dirac's QM.
Thank you very much for your patience {:-)] It was a good step in the way.
Yours,
Amr Morsi.
