Say we have N beads on a string along the x direction. The beads can only deviate from their equilibrium position in 1 dimension (say the y direction, transverse to x). Then there exist N normal modes in which the system can be. These normal modes can be seen as waves, each of which has their own k vector, so we can use k to label the normal modes of the system.
How can such a normal mode be seen as a harmonic oscillator?
According to Debye, the energy of such a wave, is given by the energy of the QHO where \omega=v_s|\mathbf{k}|.
Let’s say our system is in a state, such that the total energy is described by
This is a valid vibrational state of the string, all normal modes are in the ground state except for the one that corresponds to the lowest k vector, which is excited to the 6th level, that is, there are 6 phonons with energy \hbar\omega(\mathbf{k}_\mathrm{lowest}). (Of course there is nothing special about the lowest k vector or 6 phonons, but just to make it a bit more concrete).
How should I interpret this state of the system? Are there 6 waves with the same amplitude and phase that are superimposed on each other? Or should I see it as 1 wave with 6 times the amplitude of the wave with one quantum of energy. Does the phase need to be the same for all 6? If not then you could make them destructively interfere…
What I’m tending towards is understanding the phonons as a quantization of the amplitude of the waves that are the normal modes. Because I don’t see any other way in which the energy of a wave can be increased while the frequency remains the same.
If we are supposed to know already that waves can be described by a HO then maybe someone could point me to a resource that explains it, because I fail to see how.