7.1: The Problem

7.1 The harmonic Hamiltonian

The Hamiltonian for lattice vibrations, in the harmonic approximation, is

$$\mathcal{H}=\frac{1}{2} \sum_{i=1}^{3 N} m_{i} \dot{x}_{i}^{2}+\frac{1}{2} \sum_{i=1}^{3 N} \sum_{j=1}^{3 N} x_{i} A_{i j} x_{j}.$$

Notice that this Hamiltonian allows the possibility that atoms at different lattice sites might have different masses. Accept the fact that any real symmetric matrix S can be diagonalized through an orthogonal transformation, i.e. that for any such S there exists a matrix B whose inverse is its transpose and such that

$$\mathrm{BSB}^{-1}$$

is diagonal. Show that the Hamiltonian can be cast into the form

$$\mathcal{H}=\frac{1}{2} \sum_{r=1}^{3 N}\left(\dot{q}_{r}^{2}+D_{r} q_{r}^{2}\right)$$

by a linear change of variables. (Clue: As a first step, introduce the change of variable $z_{i}=\sqrt{m_{i}} x_{i}$.)