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What is the difference between Phonons and Normal mode? My professor told me that they are the same and that one can get the other from some derivation (I'm a bit unsure if he really said the 'derivation part' because he said more details around that which I did not fully grasp). If it is the case that there is no difference between Phonons and Normal mode then how come there exist two very different words for the same thing?

Below I will make some reference to what I have read that to me makes it seem that there is no difference between Phonons and Normal mode:

"The quanta of lattice vibrations are known as phonons" - Introductory to solid state physics Second Edition by H.P. Myers $\tag{1}$

"A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation." - Wikipedia $\tag{2}$

Related posts:
phonons-and-modes
what-is-the-difference-between-normal-mode-and-just-mode
difference-between-mechanical-modes-and-phonons - this one almost answers my question but then I emphasize on "If it is the case that there is no difference between Phonons and Normal mode then how come there exist two very different words for the same thing?"

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Normal mode describes the shape of the vibration pattern — how the system oscillates, and with what frequency. Phonons describe another aspect of the vibration: amplitude. In particular, in a coherent state mean number of phonons in a given normal mode is proportional to squared amplitude of vibration of this normal mode.

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  • $\begingroup$ To me, amplitude does not characterize a phonon. In interactions with electrons (or other phonons) the phonons come and go one at a time. $\endgroup$
    – garyp
    Commented Sep 14, 2021 at 11:00
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    $\begingroup$ @garyp Since the Hamiltonian of an eigenmode is basically a harmonic oscillator, there are classical boundaries of motion where potential energy becomes larger than total energy, and it's natural to call distance of these boundaries from equilibrium amplitude. And as number of phonons grows (in number states), this amplitude increases. As mean number of phonons grows in coherent states, amplitude, once again, increases. $\endgroup$
    – Ruslan
    Commented Sep 14, 2021 at 11:11
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    $\begingroup$ I think see what you are saying. Coherent state being as close as you can get to a "wave", and phonons are waves. I think this gets down to semantics and preference. To me, a phonon is a normal mode. But I accept that there can be other points of view. $\endgroup$
    – garyp
    Commented Sep 14, 2021 at 17:31

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