- #1

fluidistic

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From what I understood, if one applies the momentum operator on the wavefunction of a Bloch electron, one doesn't get a constant times that wavefunction, which means that the wavefunction of the Bloch electron is not a momentum eigenstate. Then it is stated that the crystal momentum is [itex]\hbar \vec k[/itex]. My first question is:

1) Does this mean that the electrons have no well definite momentum, and thus, no well definite velocity?

Then the book arrives at the expression [tex]<\vec v_n(\vec k)>=\frac{1}{\hbar}\frac{\partial \varepsilon_n(\vec k)}{\partial \vec k}[/tex] (or something similar, I do not remember exactly).

I do not understand why it is associated to the speed of a Bloch electron in the nth band. I would think that the right hand side is related to the mean speed of a "quasiparticle" which would be the electron PLUS the interaction with the potential, whose momentum would be [itex]\hbar \vec k[/itex]. Not that of the electron itself, only.

2) Can someone explain me why the right hand side corresponds to the mean velocity of the Bloch electron? I've read the Appendix E and I still have that question.

It is stated that because this mean speed is finite and non zero, it means that the electrons move forever without any effects on the mean velocity despite the interaction with the ions, which goes in contrast with the Drude model electrons which bump into them. I do not understand the logic (the implication). In Drude model, aren't electrons also having a constant through time mean velocity? After each collision, the electron is magically assigned a random velocity. So that in average I'd tend to think its mean velocity remains constant through time. Thus I do not see why the property of having a constant through time mean velocity implies a "striking difference" with the electrons in Drude model.

3) Can someone tell me what I'm missing here?