The wave function at the border of Wigner-Seitz cell

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Discussion Overview

The discussion centers on the behavior of the wave function at the border of the Wigner-Seitz cell in the context of band structure calculations. It explores the implications of symmetry and differentiability conditions imposed by the Schrödinger equation.

Discussion Character

  • Technical explanation, Conceptual clarification

Main Points Raised

  • One participant questions why the normal derivative of the wave function must be zero at the Wigner-Seitz cell's border.
  • Another participant suggests that this requirement is due to mirror symmetry at the faces of the Wigner-Seitz cells, which would be violated if the wave function had a non-zero derivative there.
  • A third participant notes that this condition specifically applies to states with k=0.
  • A later reply expresses gratitude to previous contributors for their insights.

Areas of Agreement / Disagreement

Participants present differing viewpoints regarding the conditions under which the normal derivative of the wave function is zero, but there is no explicit consensus on the necessity or implications of these conditions.

Contextual Notes

The discussion does not clarify the assumptions underlying the symmetry argument or the specific implications for wave functions with k≠0.

hokhani
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In the cellular method of calculating the band structure:
Why we have to take zero the normal derivative of the wave function at the Wigner-Seitz cell's border?
 
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Because of symmetry. At the faces of the Wigner-Seitz cells there is a mirror symmetry, and if the wavefunction has a non-zero derivative at that point (normal to the face) that would violate the requirement of the Schrödinger equation that the wavefunction be twice-differentiable in all space.
 
Also take in mind that this only holds for the states with k=0.
 
I thank both of you daveyrocket and DrDu.
 

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