Today will mostly be you doing the work. The class will be divided into three teams: each team will take one term in the Hamiltonian for our coupled operators. By expressing it in terms of creation and annihilation operators, you will show that the excitations of this chain obey a Schrodinger equation. Thus the collective excitations of a collection of harmonic oscillators obey the same equations as a quantum particle. Thus we can think of "phonons" as particles. This paradigm is used in high energy physics: there one models nature with a field theory (that is a theory of coupled harmonic oscillators). The particles we know and love (like the electron) are excitations of these fields.