@@ -238,6 +238,6 @@ Suppose we have a vibrating 1D atomic chain with 3 different spring constants al
...
@@ -238,6 +238,6 @@ Suppose we have a vibrating 1D atomic chain with 3 different spring constants al
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 $\kappa_ 1 = \kappa_ 2 = q$.
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
??? 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?
To solve the eigenvalue problem quickly, make use of the fact that the mass-spring matrix 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?
5. What will happen to the periodicity of the band structure if $\kappa_ 1 = \kappa_ 2 = \kappa_3$?
5. What will happen to the periodicity of the band structure if $\kappa_ 1 = \kappa_ 2 = \kappa_3$?
