@@ -214,7 +214,7 @@ Due to the alternating hopping energies, we must treat two consecutive atoms as
??? hint
To this end, find expressions for $E \left< n,1 \vert \Psi \right> = \left< n,1 \right| H \left|\Psi \right>$ and $E \left< n,2 \vert \Psi \right> = \left< n,2 \right| H \left|\Psi \right>$.
3. Using the trial solutions $\phi_n = \phi_0 e^{ikna}$ and $\psi_n = \psi_0 e^{ikna}$, show that the Schödinger equation can be written in matrix form: $$\begin{pmatrix} \epsilon & t_1 + t_2 e^{-i k a} \\ t_1 + t_2 e^{i k a} & \epsilon \end{pmatrix} \begin{pmatrix} \phi_n\\\psi_n\end{pmatrix} = E \begin{pmatrix} \phi_n\\\psi_n\end{pmatrix}.$$
3. Using the trial solutions $\phi_n = \phi_0 e^{ikna}$ and $\psi_n = \psi_0 e^{ikna}$, show that the Schödinger equation can be written in matrix form: $$\begin{pmatrix} \epsilon & t_1 + t_2 e^{-i k a} \\ t_1 + t_2 e^{i k a} & \epsilon \end{pmatrix} \begin{pmatrix} \phi_0\\\psi_0\end{pmatrix} = E \begin{pmatrix} \phi_0\\\psi_0\end{pmatrix}.$$
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 and make a plot of it. Does your result makes sense when considering the band structure?
