@@ -239,6 +239,7 @@ Suppose we have a vibrating 1D atomic chain with 3 different spring constants al
2. Derive the equations of motion for this chain.
3. By filling in the trial solutions for the equations of motion (which should be similar to the ones in eqs. (10.3) and (10.4) of the book), show that the eigenvalue problem is given by $$ \omega^2 \mathbf{x} = \frac{1}{m} \begin{pmatrix} \kappa_1 + \kappa_ 3 & -\kappa_ 1 & -\kappa_ 3 e^{i k a} \\ -\kappa_ 1 & \kappa_1+\kappa_2 & -\kappa_ 2 \\ -\kappa_ 3 e^{-i k a} & -\kappa_2 & \kappa_2 + \kappa_ 3 \end{pmatrix} \mathbf{x}$$
4. In general, the eigenvalue problem above cannot be solved analytically, and can only be solved in specific cases. Find the eigenvalues $\omega^2$ when $k a = \pi$ and $\kappa_ 1 = \kappa_ 2 = q$.
??? hint
To solve the eigenvalue problem quickly, make use of the fact that the Hamiltonian in that case commutes with the matrix $$ X = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}. $$ What can be said about eigenvectors of two matrices that commute?
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@@ -257,6 +258,7 @@ Due to the alternating hopping energies, we must treat two consecutive atoms as
1. Indicate the length of the unit cell 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 $\left<\Phi\right|H\left|\Phi\right> = N a_n \left<\phi_n\right|H\left|\Phi\right>$ for any $n$. Similar can be done for $\left<\Phi\right|H\left|\Psi\right>$ and $\left<\Psi\right|H\left|\Psi\right>$.
