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First, to imagine "individual electrons" as having well-defined values of $\vec p$ is in no way necessary. The electrons collectively occupy the whole band – which may be described as a subspace of the one-electron Hilbert space. But this subspace of the Hilbert space may be described by many different bases. For any basis, one may say that 1 electron is ...

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For any crystal, the First Brillouin Zone is found using the Wigner-Seitz construction for the reciprocal lattice. The high-symmetry points are labeled by certain letters mainly as a convention--like you said Gamma for (0,0,0) etc. The important thing to realize as far as the group theory, is that the group of the wavevector at the Gamma point has the full ...

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How photo electric effect happens? According to quantum theory (based upon which the photoelectric effect) is explained, the electrons in an atom are in quantized states. For a given photosensitive material the electrons in the valence shell are having unique quantized energy levels. When a light of suitable energy equal to the energy level of the electron ...

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If I'm understanding your question correctly, then the answer is that you actually can't justify the connection of your equations along the quantized direction. In fact, the "bands" in the direction perpendicular to your slab are going to be completely flat, which corresponds to an infinite effective mass in that direction. (The infinity comes from the ...

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I studied solid state physics last semester and we learned from "Solid State Physics" by Ashcroft and Mermin, but I assume that you can find these topics in every book about solid state physics.

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