# Wave function evolution of an electron [closed]

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.

## closed as too broad by my2cts, GiorgioP, Jon Custer, Phonon, Cosmas ZachosMar 29 at 18:50

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• It is a free particle: en.wikipedia.org/wiki/Free_particle#Quantum_free_particle – K_inverse Mar 26 at 10:20
• You express doubt that certain quantum phenomena have ever been observed. Can you present arguments other than not knowing of any such observations? – my2cts Mar 26 at 22:13
• Behind the theory, there is a network of millions of experimental results. – peterh Mar 28 at 8:33

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.
• ...that the electron must occupy all of space. I feel like this isn't precise language. When $\Delta x$ is finite we don't say the electron exists within a length of $\Delta x$ of space, do we? – Aaron Stevens Mar 26 at 11:43