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In band gap theory, why can we use periodic boundary conditions when finding the wave function of free electrons in a conductor? Why do you think it is smoothly connected at both ends of the conductor? Is that really so?

In my textbook, u(x+L)=u(x) (L is crystal length) is used as boundary condition to solve Schrödinger's equation of free electron in
well type potential of conductor crystal.

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  • $\begingroup$ The question is unclear. Are you asking about why one can use a Brioullan zone in a crystal? $\endgroup$ – Jon Custer Jun 28 '19 at 13:41
  • $\begingroup$ Sorry, I don't understand Brioullan zone. $\endgroup$ – Sano Jun 28 '19 at 13:43
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    $\begingroup$ In solid state physics, when working with crystalline materials, one uses the translational symmetry of the crystal and derives Bloch function solutions for the electron states. By definition of the crystal, these functions are periodic, since the crystal is periodic. This periodicity in real space corresponds to a periodicity in momentum space, meaning that you can 'fold' all the $E$ vs $k$ states into the first Brillouin zone. $\endgroup$ – Jon Custer Jun 28 '19 at 13:50
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Periodic boundary conditions are an approximation that greatly simplify the analysis. The size of a crystal is enormous compared to the size of an atom, so on a small, local scale the system appears to be periodic. In practice this is a very good approximation. It does leave out the physics that occurs in the vicinity of the surface. That region is handled separately, and is the large field of surface physics.

As I understand it (not too well, I'm not a theoretician) there are technical problem with periodic boundary conditions related to the fact that the position operator is unbounded, and strictly speaking, inadmissible. Years ago I had a question similar to yours, and looked into it a little. I have a memory, perhaps unreliable, of stumbling across a paper where a quantum mechanical treatment of a finite crystal was presented. The author was Klaus Fuchs.

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  • $\begingroup$ Why would the position operator be unbounded for a finite crystal with periodic BC? Its spectrum lies entirely in $[0,L]^n$ where $L$ is the size of the "crystal". In 1D such a system could be viewed as a model of a large-radius toroidal waveguide with small cross-section. In this case the role of position would be the angle. $\endgroup$ – Ruslan Jun 28 '19 at 14:31
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On the one hand, this boundary condition simplifies calculations, on the other hand, the results do not change much if you choose a different boundary condition (assuming $L>>a$, where $a$ is the lattice constant).

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