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recent study have invoked a question which I wish you could help with.

So in quantum that I learned that the wave function of electrons in a free-potential behave as plane waves, even in solids with small perturbations.

This tells you that in a free-potential space the electron could be found anywhere. I understand that this is an ideal case, and perfect plane waves don't occur in nature.

Do we say electrons in a macroscopic scale act as particles because plane waves converges and the wave function is local in a finite region? Otherwise, I don't see how zooming out on the length scale will demonstrate particle like behavior of a plane wave electron, as if its wave function is truly plane wave-like, it will be likely to be found anywhere even at the macro scale.

Thank you for your insight,

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    $\begingroup$ A localized electron is in a superposition of plane waves, a so-called "wave-packet". Just because the plane waves are the energy eigen"states" that doesn't mean that every free particle is purely in one of those. Is that what confuses you? $\endgroup$
    – ACuriousMind ♦
    Commented Jan 5, 2016 at 0:55
  • $\begingroup$ I didn't consider the possibility that an electron wave function could be in a superposition of plane waves, which perfectly explains for its locality. Thank you for the insight! $\endgroup$
    – Fineman
    Commented Jan 5, 2016 at 0:57
  • $\begingroup$ Whether electrons act like particles or not doesn't depend on the scale but on the strength of the position measurement vs. the total absolute momentum. If you are doing a weak position measurement that won't change the momentum much, then you see particle behavior, if you do strong measurements, then you need to apply the Born rule. Everything in-between requires a density matrix description of the experiment. $\endgroup$
    – CuriousOne
    Commented Jan 5, 2016 at 2:48

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As you said, plane wave is an ideal case. But if we follow your way of thinking macroscopically, in my opinion, we should bear in mind that the box normalization should be considered. With that condition we can confine the electron in a space that is microscopically large but still countable in macroscopic scale.

ps: I can't add a comment in your post. So I wrote my opinion here.

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In quantum physics the results of experiments on a particle can depend on all of the states the particle can be in during the experiment: this is called quantum interference.

For an electron attached to an atom these effects confine the electron to have a high probability of being found close to the nucleus on a macroscopic scale.

For an electron that isn't bound to an atom it may be the case that is is spread out on a relatively large scale and could undergo interference on that large scale but there are reasons that might not happen.

When information is copied out of a quantum system interference is suppressed and you will see the system in one of the possible measured states. This effect is called decoherence:

https://arxiv.org/abs/quant-ph/0306072

An electron is a charged particle so it will interact pretty strongly with the nearby objects and which would make it decohere:

https://arxiv.org/abs/2001.06154

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