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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.

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

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

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  • $\begingroup$ I know what you said which are in the books,and by sit/move I meant sit/move as a wave packet,that's not the point here.But now I think the real problem is ,all the "plane wave/wave packet/and sit/moving electron" thing 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?The question/discussion have little meaning if there are no experiment evidence. $\endgroup$ – jw_ Mar 26 at 11:35
  • $\begingroup$ ...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? $\endgroup$ – Aaron Stevens Mar 26 at 11:43
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    $\begingroup$ @jw_ "do we ever see evidence of the broaden packet or a sit/moving single electron?" ─ yes, absolutely. Did you think that your textbook was just telling you fantasies without experimental support? The observation that wavepackets broaden is empirical, and there has yet to be a reproducible report of a particle that sits, unconfined, without its probability density spreading as per QM's predictions. $\endgroup$ – Emilio Pisanty Mar 26 at 12:07
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    $\begingroup$ Frankly, I'm not particularly interested in chasing moving goalposts ─ if you're prepared to disbelieve the fact that QM is grounded in experimental reality, and that there isn't a single experiment that is inconsistent with it, without any real understanding of what the experimental basis is, then you're already on the wrong track. This broadening is routinely observed (when commissioning and benchmarking BEC and ion-trap experiments, in particular) but it is no longer routinely reported, and I don't have time to dig into old publications (particularly given the moving goalposts). $\endgroup$ – Emilio Pisanty Mar 26 at 12:44
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    $\begingroup$ @jw_ Ask that separately. $\endgroup$ – Emilio Pisanty Mar 27 at 6:56

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