The problem here is that the concept of 'motion' at the quantum scale is very different from the classical idea of motion. A quantum-mechanical object doesn't have a precise location or momentum. You can only know the probability of finding it in various locations. It can act as if it's 'in two places at once', having large probabilities at being found at two points in space separated by a large distance. That can even be the case in combination with an exactly zero probability of finding the particle in some intermediate location, meaning it can, in a sense, get from point A to point B without passing intermediate points.
Electrons moving around atoms don't follow the much-perpetuated picture where you have electrons wizzing around in elliptical 'orbits'. They can't move that way, because an orbit implies both a well-defined location and momentum. You can calculate the average position of an electron in an atom. You can calculate it's average momentum (which is zero of course, since an electron with a net average momentum would simply fly away from the atom!). You can calculate its average kinetic energy, or the average magnitude of the momentum though, and those values are non-zero.
Does an electron in an atom move? I would say yes, and I think most physicists would say yes. (in fact 'electron motion' is a quite established term among those who study it). But at the same time, it's understood that it's not 'moving' the classical sense of the word. You cannot predict where an electron may be located other than as a probability. Knowledge of where an electron is at one point in time does not enable you to predict where it's going and where it'll be measured next.
Now if you as "Why do electrons in an atom move?", that question only makes sense in terms of classical physics. Classically, an electron could simply lose all its kinetic energy; classical electrodynamics say that it actually must do so by itself, because an accelerating charge gives off radiation. But the mechanism by which it might lose its kinetic energy doesn't really matter - in principle it should be able to lose that kinetic energy and just sit still at the nucleus, where its potential energy is lowest - since the positive charge attracts it.
Quantum-mechanically, this is not permitted. You can rationalize that in terms of what I already said: A quantum mechanical object doesn't - and can't - have a definite location in space. The reason for this is that its probability distribution for position and momentum aren't independent of each other, unlike in classical mechanics where those are separate quantities. That's the basis of the famous uncertainty principle. If the electron was exactly located at one point in space (100% probability at x, 0% probability everywhere else), then it would have infinite momentum, and so, infinite kinetic energy. This is basically the 'wave-like' nature of the electrons in effect.
So in short, the more confined the electron is to a particular region of space, the higher its kinetic energy. On the other hand, the more spread out the electron is in space, the farther it is from the positively-charged nucleus. So you have potential energy working to 'pull' the electron in, so to speak, and kinetic energy working to 'push' it out. As a result, the only stable states of the electrons are where these two 'forces' balance each other out. (This is basically a verbal statement of the Schrödinger equation) And the lowest possible stable state is what we call the ground state. And unlike in classical mechanics, that state doesn't have zero kinetic energy, but merely the minimal amount of kinetic energy.
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