(kx,ky,kz)=(0,0,0) solution for a free particle with PBC?

  • Context: Graduate 
  • Thread starter Thread starter Coffee_
  • Start date Start date
  • Tags Tags
    Free particle Particle
Click For Summary

Discussion Overview

The discussion revolves around the interpretation of the wavefunction for a free particle under periodic boundary conditions (PBC), specifically addressing the case where the wavevector components are all zero, (kx, ky, kz) = (0, 0, 0). Participants explore the implications of this scenario in contrast to Dirichlet boundary conditions, focusing on the nature of the wavefunction and its normalization.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes that under Dirichlet boundary conditions, a wavefunction that is zero at the boundaries leads to non-normalizable solutions, making (0,0,0) a non-physical situation.
  • Another participant explains that under periodic boundary conditions, the wavefunction can be expressed in terms of a basis spanned by ##e^{i\vec{k}.\vec{r}}##, where ##\vec{k}## must correspond to cubic periodicity, allowing for a non-trivial solution at ##\vec{k}=0## with zero energy.
  • A different participant challenges the notion of a momentum operator existing under Dirichlet boundary conditions, asserting that there is no self-adjoint operator for generating translations in this context.
  • One participant expresses initial confusion about the presence of a constant term in the wavefunction, but later acknowledges that it is not unusual, influenced by the reactions of others in the discussion.

Areas of Agreement / Disagreement

Participants exhibit differing views on the implications of boundary conditions for the wavefunction, with some asserting the non-physical nature of certain solutions under Dirichlet conditions, while others defend the validity of the constant term under periodic conditions. The discussion remains unresolved regarding the broader implications of these interpretations.

Contextual Notes

The discussion highlights the limitations of applying different boundary conditions and the resulting interpretations of wavefunctions, particularly concerning normalization and the existence of momentum operators. Specific assumptions about the nature of the wavefunction under different conditions are not fully explored.

Coffee_
Messages
259
Reaction score
2
When dealing with Dirichlet boundary conditions, that is asking for the wavefunction to be exactly zero at the boundaries, it can be clearly seen that (0,0,0) is not a physical situation as it is not normalizable. (Wavefunction becomes just 0 then)

However when dealing with periodic boundary conditions, the basis is spanned by ##e^{i\vec{k}.\vec{r}}## where the only condition on ##\vec{k}## is that it has to correspond with ##L^{3}## cubic periodicity.

The problem now is that ##\vec{k}=0## does seem to give a non trivial solution with zero energy ##\Psi=constant## which is periodic and noralizable.

How do I interpret this werid 'constant' term in the general wavefunction part?

Sources:

Kittel eigth edition, p137

http://people.umass.edu/bvs/pbc.pdf
 
Physics news on Phys.org
Why do you think it is weird?
 
To the contrary, it's not weird, but it's weird to think that with the Dirichlet boundary conditions you'd have a momentum operator. There is none, because there is no self-adjoint operator generating translations in this case. We have discussed this many times in this forum. Just search for it!
 
Oh I was thinking it was weird because I have never encountered such an extra constant term in the wavefunction so I was doubting my reasoning to arrive at the extra constant term. Based on your reactions I see that there is indeed nothing special about it, thanks.
 

Similar threads

Graduate Counting the states of a free particle (Periodic boundary conditions)
  • · Replies 4 ·
Replies
4
Views
4K
Understanding a math step in solving Schrödinger's equation for single particle in a box
  • · Replies 8 ·
Replies
8
Views
1K
Graduate Basic QM question: Why do we describe free particles using Aexp(ikx) ?
  • · Replies 15 ·
Replies
15
Views
10K
Graduate Dirichlet and Nuemann condition on the same boundary
  • · Replies 2 ·
Replies
2
Views
2K
Solve 3D TISE in Potential Well: Eigenvals/Fns & Normalize
  • · Replies 1 ·
Replies
1
Views
3K
Why Does the Spherical Box Partition Function Differ from the Cartesian Box?
  • · Replies 8 ·
Replies
8
Views
14K