@@ -261,11 +261,6 @@ Due to the alternating hopping energies, we must treat two consecutive atoms as
1. Indicate the length of the unit cell $a$ in the figure.
2. What are the values of the matrix elements $\left<\phi_n\right|H\left|\phi_{n}\right>$, $\left<\psi_n\right|H\left|\psi_{n}\right>$, $\left<\psi_n\right|H\left|\phi_{n}\right>$, and $\left<\psi_n\right|H\left|\phi_{n+1}\right>$?
3. Using the $\{\left| \Phi\right>, \left| \Psi\right> \}$-basis, show that the Hamiltonian is given by $$H = \begin{pmatrix} \epsilon & t_1 + t_2 e^{i k a} \\ t_1 + t_2 e^{-i k a} & \epsilon \end{pmatrix}.$$
??? hint
Note that the coefficients of the trial LCAO's and the periodic boundary condition require that $\left< \Phi \right| H \left| \Phi \right> = N a_n \left< \Phi \right| H \left| \phi_n \right>$ for any $n$. Similar can be done for $\left< \Phi \right| H \left| \Psi \right>$ and $\left< \Psi \right| H \left| \Psi \right>$.
4. Derive the dispersion relation of this Hamiltonian. Does it look like the figure of the band structure shown on the [Wikipedia page](https://en.wikipedia.org/wiki/Peierls_transition#/media/File:Peierls_instability_after.jpg)? Does it reduce to the 1D, equally spaced atomic chain if $t_1 = t_2$?
5. Find an expression of the group velocity $v(k)$ and effective mass $m^*(k)$ of both bands.
6. Derive an expression for the density of states $g(E)$ of the entire band structure.
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6. Derive an expression for the density of states $g(E)$ of the entire band structure and make a plot of it. Does your result makes sense when considering the band structure?
