Hamiltonian of a metal lattice

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SUMMARY

The discussion centers on calculating the density of states (DOS) for a lattice of NxN metal atoms using its Hamiltonian. The participants confirm that finding the eigenenergies and eigenvectors from the Hamiltonian is essential for deriving the DOS. They suggest that the potential can be approximated as a sum of terms related to the positions of nearest neighbor atoms, and that the reciprocal space and Brillouin zones are relevant for determining the DOS in energy. The conversation highlights the importance of understanding solid state physics concepts in this context.

PREREQUISITES
  • Understanding of Hamiltonian mechanics in quantum systems
  • Familiarity with eigenvalues and eigenvectors
  • Knowledge of solid state physics, particularly density of states
  • Concept of reciprocal space and Brillouin zones
NEXT STEPS
  • Research methods for calculating eigenenergies from a Hamiltonian
  • Study the relationship between eigenenergies and density of states
  • Explore solid state physics textbooks for detailed explanations of potential approximations
  • Learn about the implications of reciprocal space in solid state systems
USEFUL FOR

Physicists, materials scientists, and students studying solid state physics who are interested in calculating the density of states for metal lattices and understanding the underlying quantum mechanics.

Niles
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Hi guys

I have the Hamiltonian, which describes my lattice of NxN metal atoms, and their mutual coupling. What I need is the density of states of this lattice, and I am quite sure that there is a way to find it from my Hamiltonian; I just need to find out how.

What I thought was that I can of course find the eigenenergies and corresponding eigenvectors from my Hamiltonian. In order to find the DOS I assume I need to find an expression relating the eigenenergies and the number of possible states. Am I way off here?
 
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Your idea sounds correct. I am not the best at solid state (that's what you're doing, right?), but since no one else is chiming in ...

Can you write your Hamiltonian for us? Specifically, what kind of interaction are you talking about? If I remember correctly from solid state class, you typically approximate the potential as a sum of terms proportional to (xa2-xb2), xa and xb being the positions of some (nearest neighbor?) atoms, and if you go to reciprocal space, the reciprocal volume of the (Brillouin?) zones give you the density of states (in energy). (Is that because it is approximately linear?)

I don't know what to do if you consider higher-order details of the interaction potential (I've only taken one solid state class, and solid state is not my field of study). I'll think about it some more.
 

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