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Wikipedia defines a delocalized electron inside a metal as one that is free to move from one atom to another. This state of not being bound to any metal ion is what allows it to conduct electricity and so forth. But the delocalized electron which follows the Bloch wavefunction is evenly spread throughout the entire macroscopic crystal which means that a single electron can at one time be on end of the metal and in next instant be on the extreme other side.

How is it possible to define "movement" for such a situation where the electron being totally delocalized can pop up anywhere in the crystal at any time ?

The plane waves of a delocalized electron does not restrict its position to any localized region at all, then how is it correct to say that conduction happens because of delocalized electrons "moving" ? Dont you need something thats at least a tiny bit localized like a wavepacket inorder to define things like drift velocity etc ?

How can an electron with a Bloch wavefunction have a drift velocity or a mean path length thats only a few atoms long when the wavefunction is entirely spread throughout the metal ?

Shouldnt there be something that prevents an electron moving at some rate from randomly appearing millions of atoms away inorder to make quantities like mean path length sensible ?

Edit : as someone has rightly pointed out I seem to have posted too many questions and that too in a rather haphazard manner for which i apologize, but my main concern was arent scattering events which are important to defining say the relaxation time and other quantities themselves localized events ? So how can delocalized wavefunctions of electrons scatter at all while remaining in such a delocalized state ?

Ajaykrishnan R
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    Since this is a bit of an unfocussed/rambling question, I'll pile on and also ask: Why would we ever expect that it is reasonable to describe interacting electrons in a solid in terms of single-particle non-interacting Bloch waves? – hft Nov 11 '22 at 21:31
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    @hft unfocussed is an understatement; there are two perfectly fine answers below I just do not know what question they answer... – hyportnex Nov 11 '22 at 23:28
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    @hyportnex My interpretation was, as reflected in my answer, that the questioner wants to know how drift velocity/electrical current can be modelled by bloch waves. It would be nice though, if the questioner can confirm this, because there are really a lot of questions for one single post. – hydra4jh Nov 12 '22 at 02:59

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Electrons are indistinguishable, they don’t have serial numbers or birthmarks or tattoos. There is an electron here now, and there was an electron way over there a split second ago. But there is no sense in which the electron here is the same individual electron as the electron there. (Also there is no sense that they are not the same)

When we talk about drift velocity we are not tracking the position of a single electron over time. What we are actually doing is closer to measuring the momentum at a single point and getting the speed from that. So there is no need to restrict or modify the wave function in the way you describe. The momentum is finite and hence the velocity and speed are also finite, regardless of the position that you measure.

Of course, in addition to the above, measuring the position would collapse the wavefunction to a different state anyway.

Dale
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  • But if we think about say the relaxation time then that is on average of an individual electron which therefore needs to be tracked (atleast conceptually) – Ajaykrishnan R Nov 11 '22 at 20:52
  • Not really. The relaxation time has to do with the momentum also. Assuming the collisions with the lattice are plastic, the average momentum is the result of accelerating from zero to some speed under the influence of the E field during the relaxation time. Since we know the E field and the average momentum we can obtain the relaxation time without tracking individual electrons – Dale Nov 11 '22 at 21:33
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The short answer is that it is the wrong model to describe currents.

One critical assumption of the Bloch model is that we assume a periodic potential with the same periodicity as the lattice itself. If we would apply a voltage on both ends of our solid body, that assumption wouldn´t hold anymore.

The reason why it is still useful for conduction phenomena is that it provides us with a tool to calculate the charge carrier density for a given crystal. This is what the ("stationary") energystructure also called bandstructure is used for. The charge carrier density can then be used for calculating the conductivity of the crystal.

hydra4jh
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