Tight Binding Model: mathematics

[Waxterzz]

From the textbook of Omar:

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

Thanks in advance.

 

[mfb]

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".

 

[Waxterzz]

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 X_j' = 0.

It's a function of the difference, X_j-X_j'. 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

 

[mfb]

Make an example. Write down all summands for N=5 and see how it works out.