Wave function evolution of an electron 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.
 A: It is possible for electrons to "just sit there" or "move in constant speed" in quantum mechanics, but this imposes some severe restrictions. In particular, both of those are statements about the electron's velocity (i.e. that the electron has a well-defined velocity, which is equal to zero in the first case) and therefore statements that it have a well-defined momentum, with a vanishing momentum uncertainty. And that means, because of the Heisenberg Uncertainty Principle in its form $\Delta x \geq \hbar/2\Delta p$, that the electron must occupy all of space. These are, mathematically speaking, reasonably-well-defined states, the plane waves
$$
\psi(\mathbf r,t) = \frac{1}{(2\pi\hbar)^{3/2}}e^{i\mathbf p\cdot \mathbf r/\hbar} e^{-i\frac{\mathbf p^2}{2m}\frac{t}{\hbar}},
$$
but they are not normalizable and they do not represent valid physical states. And, moreover, that 'wavepacket' cannot broaden because it is already as broad as it could possibly be.
If you have a physical wavepacket, on the other hand, with a finite extent, then the HUP demands that it have a nonzero momentum uncertainty, so it has components moving at different velocities which will naturally broaden the wavepacket.
