Momentum space representation for finite lattices - continued

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SUMMARY

The discussion focuses on the representation of momentum space for finite lattices, particularly in the context of Hamiltonian diagonalization under boundary conditions. It emphasizes the necessity of choosing k values as if periodic boundary conditions are applied (k=2πn/L) to ensure orthogonality and completeness in k-representation. The conversation also highlights the challenge of modeling finite size chains, which typically exhibit zero boundary conditions, and suggests using periodic boundary conditions to reconstruct physical wavefunctions effectively. The complexity of accurately representing nanometer-scale objects with limited atoms is acknowledged, indicating the need for a nuanced approach in selecting representations.

PREREQUISITES
  • Understanding of Hamiltonian mechanics
  • Familiarity with Fourier transforms in quantum mechanics
  • Knowledge of boundary conditions in physical systems
  • Concept of wavefunction reconstruction in quantum systems
NEXT STEPS
  • Study the implications of periodic boundary conditions in quantum mechanics
  • Explore Hamiltonian diagonalization techniques for finite systems
  • Research methods for wavefunction reconstruction in finite lattices
  • Investigate the impact of finite size effects on quantum representations
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Quantum physicists, researchers in condensed matter physics, and students studying the implications of boundary conditions in quantum systems will benefit from this discussion.

QuantumLeak
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I have been banned, maybe my nickname was not so kind. I let the topic continue here. I report my last comment:

"Ok, I got the point. thanks for replying!
It's just a change of basis that under boundary condition diagonalize the Hamiltonian. But then a subtle point:

In order for k-representation to be a good basis change (i.e. orthogonality and completeness properties) I guess that one has to choose the k values AS if there were periodic boundary conditions (i.e. k=2pi*n/L). Right?

Then another last point:
how do you really model a finite size chain? A finite size chain has zero boundary condition. In this case I use the k representation with k chosen as if there were periodic boundary condition and just then reconstruct the physical wavefunction by making a wavepacket that is zero on the borders?
"
 
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A finite object in real space will need an infinite number of Fourier components in reciprocal space to fully describe it.

When you have nanometer scale objects with a few 100s or so atoms in each direction, then it is not so obvious which representation will give you a more accurate picture.
 

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