Find the Hamiltonian for the interaction between phonons and optical phonons


 * 1) Identify the Hamiltonians for phonons and optical phonons: \(\hat{\mathbf{H}}_1 = \sum_k c\left|\mathbf{k}\right|\hat{a}^{\dagger}(\mathbf{k})\hat{a}(\mathbf{k})\) \(\hat{\mathbf{H}}_2 = \sum_k \epsilon\hat{b}^{\dagger}(\mathbf{k})\hat{b}(\mathbf{k})\)
 * 2) Identify the interaction Hamiltonian: \(\hat{\mathbf{H}}_3 = \Delta \sum_k \hat{a}^{\dagger}(\mathbf{k})\hat{b}(\mathbf{k}) + \hat{b}^{\dagger}(\mathbf{k})\hat{a}(\mathbf{k})\)
 * 3) Sum the Hamiltonians together: \(\hat{\mathbf{H}} = \hat{\mathbf{H}}_1 + \hat{\mathbf{H}}_2 + \hat{\mathbf{H}}_3 = \sum_k \left( c\left|\mathbf{k}\right|\hat{a}^{\dagger}(\mathbf{k})\hat{a}(\mathbf{k}) + \epsilon\hat{b}^{\dagger}(\mathbf{k})\hat{b}(\mathbf{k}) + \Delta \hat{a}^{\dagger}(\mathbf{k})\hat{b}(\mathbf{k}) + \Delta \hat{b}^{\dagger}(\mathbf{k})\hat{a}(\mathbf{k})\right)\)
 * 4) Write in matrix form: \(\hat{\mathbf{H}} = \sum_k \begin{pmatrix} \hat{a}^{\dagger}(\mathbf{k}) & \hat{b}^{\dagger}(\mathbf{k})\end{pmatrix} \begin{pmatrix} c\left|\mathbf{k}\right| & \Delta \\ \Delta & \epsilon \end{pmatrix} \begin{pmatrix}\hat{a}(\mathbf{k}) \\ \hat{b}(\mathbf{k})\end{pmatrix}\)
 * 5) Diagonalise.
 * 6) Act on the Hamiltonian with a real rotation: \(\hat{\mathbf{H}} = \sum_k \begin{pmatrix} \hat{a}^{\dagger}(\mathbf{k}) & \hat{b}^{\dagger}(\mathbf{k})\end{pmatrix} \begin{pmatrix} c\left|\mathbf{k}\right| - E & \Delta \\ \Delta & \epsilon - E \end{pmatrix} \begin{pmatrix}\hat{a}(\mathbf{k}) \\ \hat{b}(\mathbf{k})\end{pmatrix}\)
 * 7) Force the determinant of the Hamiltonian to 0: \(\begin{vmatrix} c\left|\mathbf{k}\right| - E & \Delta \\ \Delta & \epsilon - E \end{vmatrix} = 0\)
 * 8) Expand: \(E^2 - \left(\epsilon + c\left|\mathbf{k}\right|\right)E + \epsilon c\left|\mathbf{k}\right| - \Delta^2=0\)
 * 9) Solve the quadratic equation for E: \(E=\frac{\epsilon + c\left|\mathbf{k}\right| \pm \sqrt{\left(\epsilon + c\left|\mathbf{k}\right|\right)^2 - 4\left(\epsilon c\left|\mathbf{k}\right| - \Delta^2\right)}}{2}\)
 * 10) Simplify: \(E=\frac{\epsilon + c\left|\mathbf{k}\right| \pm \sqrt{\left(\epsilon - c\left|\mathbf{k}\right|\right)^2 - 4\Delta^2}}{2}\)