In many basic quantum mechanics books the wave packet of an electron is described. It will say that the wave packet will broaden as time evolves because of dispersion. But suppose the electron just sits there or moves with constant speed: how can the wave packet broaden? Because nothing real changed.
I think the real problem is, all the "plane wave", "wave packet" and "sit/moving electron" things are all concepts in our thoughts, the real important thing is experiment. Do we ever see evidence of the broaden packet or a sit/moving single electron? Maybe we only see a phenomenon of a lot of electrons. This question/discussion may have little meaning if there is no experiment evidence about anything mentioned above.