Why don't stable atoms constantly emit light?

The electron cloud of a stable atom is the superposition of the electron at it's given energy level. A charge in motion produces light. If we look for the electron at one point, it's probability of being there will be the square of of the wave function. Let's say we find it here, then move over there, and find it there (lucky us!). It has moved, right? It was at one location, now it's at another, that's motion, right? So, when it moved, shouldn't it have emitted light? Did it emit light? If so, what happened to it's energy level? Did it change? What of some other atom. Surely, if we looked, we might find it's electron here or there, and since it's moving around, ought it to be emitting light, constantly?