Tight Binding Model: mathematics

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The discussion centers on understanding a specific summation in the context of the Tight Binding Model from Omar's textbook. Participants seek clarification on the transition from a double summation to a new expression involving a different index, with one contributor noting that Xj' is set to zero. The conversation highlights confusion regarding the mathematical reasoning behind the summation, particularly the implications of treating the sum over j' as identical summands. Additionally, there is uncertainty about the starting and ending values of the summation, prompting a suggestion to illustrate the process with an example for N=5. Overall, the thread reveals a collective struggle to grasp the mathematical concepts involved in the summation.
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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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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".
 
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
 
Make an example. Write down all summands for N=5 and see how it works out.
 
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