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In the end, I think this question is related to this one (which was never satisfactorily answered), but I extend slightly and therefore am asking a new question.

In discussing the semiclassical model of electron dynamics (appropriate the slowly varying EM fields and subject to some more restrictions), Ashcroft and Mermin state that

The wave vector ofan electron is only defined to within an additive reciprocal lattice vector $\mathbf{K}$. One cannot have two distinct electrons with the same band index $n$ and position $\mathbf{r}$, whose wave vectors $\mathbf{k}$ and $\mathbf{k}'$ differ by a reciprocal lattice vector $\mathbf{K}$; the labels $n$, $\mathbf{r}$, $\mathbf{k}$ and $n$, $\mathbf{r}$, $\mathbf{k} + \mathbf{K}$ are completely equivalent ways of describing the same electron.

OK, I can accept that. But what about $n$, $\mathbf{r}$, $\mathbf{k}$ and $n$, $\mathbf{r}'$, $\mathbf{k}$, for $\mathbf{r} \neq \mathbf{r}'$: can two (different) electrons have these states? If so, how does this make sense? I know that the semiclassical model is a very odd "massaging" of the theory of $N$ electrons in equilibrium in a periodic potential towards describing transport for such a system, but in that equilibrium theory there is explicitly no mention of localization precisely because the Bloch eigenstates are delocalized. I am therefore very confused as to how we can have the quantum numbers $n$ and $\mathbf{k}$ applying to two different electrons (at different points in space) when these quantum numbers only make sense in describing electrons which are delocalizaed throughout a given macroscopic crystal in the first place!

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I don't have A&M handy. I don't know why they are including $\textbf{r}$. It usually goes without saying.

You are right about delocalization. We say that the wave function of a single electron is an infinite plane wave, but we work with finite crystals. They must be thinking that $\textbf{r}$ and $\textbf{r'}$ are in different samples. It would be analogous to having two atoms. You can have two electrons both with the same quantum numbers if they are in separate hydrogen atoms.

Edit: Now I have A&M handy. Near the beginning of chapter 12, they define $\textbf{r}$ and $\textbf{k}$ as the mean values of a wavepacket. The wavepacket is a sum of free electron states. So the two "electrons" (really wavepackets) are not so delocalized. Because it is not an infinite sum, it will repeat in space, but I think they ignore that and assume it has one location.

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  • $\begingroup$ I think you may be referring to the rigorous discussion of band structure given in Chapters 8-11, whereas I am talking about smeiclassical transport as given in Chapter 12 (so that the two positions are in the same sample)? $\endgroup$
    – EE18
    Commented Jul 16, 2023 at 15:14
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A Bloch electron (fully delocalized, as expected) can be described by a wavefunction $\psi_{n,\mathbf{k}}(\mathbf{r}).$ If we have a two-electron wavefunction $\psi_{n,\mathbf{k},n',\mathbf{k}'}(\mathbf{r},\mathbf{r}'),$ and set $n=n'$ and $\mathbf{k}=\mathbf{k}',$ then for $\mathbf{r}=\mathbf{r}'$ this function must vanish: this follows from the exchange antisymmetry condition of fermions:

$$\psi_{n,\mathbf{k},n',\mathbf{k}'}(\mathbf{r},\mathbf{r}')=-\psi_{n',\mathbf{k}',n,\mathbf{k}}(\mathbf{r}',\mathbf{r}).$$

I think this is what A&M are trying to say (admittedly, with a rather sloppy wording).

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