Find the classical normal-mode frequencies and normal coordinates for the one-dimensional chain with Hamiltonian H = + = ($j - $j-1) Pj k 2mj (2.150) where at even sites m2;= m and at odd sites m2j+1 = M. Please sketch the dispersion curves. (b) What is the gap in the excitation spectrum? (c) Write the diagonalized Hamiltonian in second-quantized form and discuss how you might arrive at your final answer. You will now need two types of creation operator.