In subquestion 4.3
Calculating the eigenvalues of this using det(Matrix - lamda I) = 0 yields for the eigenvalues
lamda = E0 ± sqrt(E0^2 + gamma^2 - t^2)
I am not sure where I can use the approximation of gamma << t, I also tried guessing the eigenvectors by saying the electric field is so small that you can say | phi_1 | ^2 = |phi_2|^2 approximately but this also does not yield the result that is given in the solutions.
I look forward to your response
Hi! Your eigenvalues are unfortunately wrong… Look back in your derivation if you can spot the mistake.
You do not need to guess the eigenstates in this case, it is simpler to just compute them the old-fashioned way.
You can use the \gamma \ll t approximation to Taylor expand (or any other approximation) expressions that depend on t,\gamma, after you have simplified the expression as much as you can. As an example, take \sqrt{1+x}. For x \ll 1, we can expand this as 1+\frac12 x to first order. Do you see how you can apply this to our case?
Do also check out the video about diagonalizing matrices that I linked in here:
https://forum.solidstate.quantumtinkerer.tudelft.nl/t/lecture-5-lcao-model/3426
Computing eigenvalues is a basic operation that you can do without any effort if you know what to look at.
