7.1: The Problem

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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 \(z_{i}=\sqrt{m_{i}} x_{i}\).)