Understanding Bloch's Theorem: Does 'n' Label Energy Bands?

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Discussion Overview

The discussion revolves around Bloch's theorem and its implications for understanding energy bands in a crystal. Participants explore whether the index 'n' in the wave function representation labels distinct energy bands and the role of the empty lattice approximation in constructing band diagrams.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant states that Bloch's theorem indicates the wave functions for electrons in a periodic potential take a specific form, questioning if 'n' in un corresponds to different energy bands.
  • Another participant agrees that 'n' does label different bands.
  • A different participant raises a concern about the empty lattice approximation, suggesting that while a band diagram can be created, the bands may not necessarily correspond to different 'n' values, expressing uncertainty about their understanding.
  • Another participant prompts for a written expression of un for the empty lattice, indicating a desire for further exploration of the topic.

Areas of Agreement / Disagreement

There is partial agreement on the role of 'n' in labeling energy bands, but uncertainty remains regarding the implications of the empty lattice approximation and its relationship to the bands.

Contextual Notes

Participants express uncertainty about the relationship between the empty lattice approximation and the labeling of bands, indicating potential limitations in their understanding of the concepts involved.

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Bloch's theorem states that the wave functions for electrons in a periodic potential have the form:

ψn,k(r) = un(r)exp(ik⋅r)

, where un has the same periodicity as the potential.
Bloch's theorem is used to calculate energy bands, and my question is:
Does the n in un label the different bands of the crystal?
 
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Yes, exactly
 
Okay but if remember correctly then a band diagram can be constructed using an empty lattice approximation, where all wave vectors outside the Brillouin zone are mapped back into the Brillouin zone. In this case you get a band diagram, but the different bands need not necessarily belong to different n? or am I wrong? maybe I am mixing up different things.
 
Try to wrire down the u-n for the empty lattice.
 

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