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In solving the Classical model for lattice dynamics [Rossler pag 38] we find that the lattice admits $$d\cdot N\cdot r = \#modes$$ where

$d=$dimension of the problem

$N=$ number of atoms

$r=$ number of atoms in the primitive cell

So, classically we have that only a finite number of modes (for a finite lattice) is possible.

When we quantize the Hamiltonian we get the hamiltonian of an HO

$$\hat{H}=\Sigma_s\Sigma_\vec{q}\hbar\omega_s(\vec{q})\left(a^\dagger_{s\vec{q}} a_{s\vec{q}}+\frac{1}{2}\right)$$

where $s$ is a branch index.

So, in the Quantum case it seems that we can have an infinite number of phonons.

Am I missing something? Since the creation of a phonon corresponds physically to excite a vibration mode in the lattice, it seem me strange that we can create an infinite number of phonons in QM while in classical mechanics we can have only a finite number of excitations.

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    $\begingroup$ In both cases number of modes is the same finite number and there is infinity of different states possible. What is different in the classical model is that energy of a mode is not limited to multiples of $\hbar \omega$. $\endgroup$ – Ján Lalinský Jul 30 '15 at 20:39
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Ok, I got the point: while classically I can (ideally) vary the energy of an oscillation mode by varying its amplitude from 0 to $\infty$, in QM I can vary the energy from $\frac{\hbar \omega}{2}$ to $\infty$ per steps of $\hbar\omega$.

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