Abstract
A description is given of the dynamics of crystals by means of collective variables, namely the Fourier transforms of the divergence and curl of the current. It is shown that, in the harmonic approximation, the classical equations of motion are reduced to a set of three equations involving only the collective variables themselves. From these equations the usual result for the frequency of phonons is obtained. Furthermore, the quantum-mechanical equations of motion can again be reduced, in a suitable approximation, to a set of three coupled equations in the collective variables. A relationship between frequency and wave number is derived for the excitations of the system from them.
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