Tight Binding Model: mathematics

In summary, the conversation is about understanding the summation in a textbook from Omar. The first person is asking for an explanation of the summation and the second person asks for clarification on which specific summation is being referred to. The second person then explains that the second equation is the same as the first one, but with a new index and the sum over j' being equivalent to N times the same summand. They also mention that Xj' is equal to 0 in the new expression. The first person still doesn't understand and asks for an example. The second person suggests writing out all the summands for N=5 to see how it works out.
  • #1
Waxterzz
82
0
From the textbook of Omar:

nkIVr83.jpg


Can someone explain me the summation of this? I simply don't get it?

Thanks in advance.
 
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  • #2
Which summation do you mean?
The second equation is the first one with the argument that the sum over j' looks like "something"+"something"+...+"something" (N times the same summand) = N*"something".
 
  • #3
mfb said:
Which summation do you mean?
The second equation is the first one with the argument that the sum over j' looks like "something"+"something"+...+"something" (N times the same summand) = N*"something".

I don't get how you go from the double summation to the new one with the new index etc.

Oh yeah i forgot to say: in the new expression they put Xj' = 0.

It's a function of the difference, Xj-Xj'. For each choice of j',the sum over j yields the same result. => I don't get it.

Also I don't get why you start from -N/2 and end with (N-1)/2
 
  • #4
Make an example. Write down all summands for N=5 and see how it works out.
 

1. What is the Tight Binding Model in mathematics?

The Tight Binding Model is a mathematical model used in solid state physics to describe the electronic structure of a crystal. It takes into account the interactions between neighboring atoms and their electrons, and is based on the principles of quantum mechanics.

2. How does the Tight Binding Model work?

The Tight Binding Model uses a linear combination of atomic orbitals to describe the electronic wavefunction of the crystal. The model takes into account the overlapping of orbitals between neighboring atoms, and uses this information to determine the electronic energy levels of the crystal.

3. What are the assumptions made in the Tight Binding Model?

The Tight Binding Model assumes that the electronic wavefunction of a crystal can be described as a linear combination of atomic orbitals, and that the orbitals of neighboring atoms overlap. It also assumes that the electronic energy levels are affected only by the interactions between neighboring atoms.

4. What are the limitations of the Tight Binding Model?

The Tight Binding Model is a simplified model and does not take into account the complex interactions between all atoms in a crystal. It also does not consider the effects of temperature, pressure, and other external factors on the electronic structure of a crystal.

5. How is the Tight Binding Model used in practical applications?

The Tight Binding Model is used in various areas of solid state physics, such as in the study of semiconductors, superconductors, and other materials with unique electronic properties. It is also used in the design of electronic devices, such as transistors and diodes.

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