# Phonons in a lattice

I'm looking for some physical intuition as to why sound waves are quantized.

I know two mathematical procedures for deriving phonons in a lattice: 1) impose the canonical commutation relations on the system ad-hoc, and 2) apply the Schrodinger equation to the lattice. But neither of these gives me any sense of what could be the mechanism by which sound is quantized in the lattice.

Any reference material would be greatly appreciated. I've looked around for a while on this.

Astronuc
Staff Emeritus
Matter, the medium in which sounds propagates, is quantized.

The lattice is composed of atoms, each of which has approximately the same mass, and which are approximately regularly spaced (lattice parameter).

Sound propagates by transfer of momentum and energy through successive atoms.

If the fact that the lattice is made of particles explains sound quantization, then does that imply that phonons could have been discovered prior to the advent of the Schrodinger equation?

Have you already checked this http://en.wikipedia.org/wiki/Phonon" [Broken]?

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I did look at the wiki article, thanks. I found it to be mathematically consistent with texts, but I didn't find the physical sense I'm trying to get. It gives a visual representation of the propagation of sound, but not quantization as far as I can tell.

Discretization could mean many things here.

For a finite lattice, phonons can only have discrete k values.

There are several w(k) branches for each value of k.

Phonons can have quantized energy levels like a particle in a harmonic (or anharmonic) well.

Which one are you talking about?

Discretization could mean many things here.

For a finite lattice, phonons can only have discrete k values.

There are several w(k) branches for each value of k.

Phonons can have quantized energy levels like a particle in a harmonic (or anharmonic) well.

Which one are you talking about?

I mean the latter, energy levels.

Gokul43201
Staff Emeritus
Gold Member
The quantization of phonon energies arises not only out of the quantization of matter but more importantly, out of the boundary conditions that are applied to the differential equation describing the displacements.

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Matter, the medium in which sounds propagates, is quantized.

The lattice is composed of atoms, each of which has approximately the same mass, and which are approximately regularly spaced (lattice parameter).

Sound propagates by transfer of momentum and energy through successive atoms.

Do you think elementary particles could arise in this way?

Quantization of energy levels for phonons occurs for just about the same reason that quantization of energy levels for a particle in a potential well --except that a phonon mode corresponds to atomic vibrations across the whole crystal.

For a harmonic system (and crystals are typically near-harmonic) Schrodinger's eqn. becomes separable into 3N non-interacting harmonic oscillator wells, where each well corresponds to a particular normal-mode of vibration. The energy in a each well is just (n+1/2) hbar omega, where omega is the frequency in the that well.

Oh- and the distinction between phonon modes and normal modes (say in a molecule) is that phonons have wave-vectors corresponding to the periodicity of the lattice.

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Quantization of energy levels for phonons occurs for just about the same reason that quantization of energy levels for a particle in a potential well --except that a phonon mode corresponds to atomic vibrations across the whole crystal.

For a harmonic system (and crystals are typically near-harmonic) Schrodinger's eqn. becomes separable into 3N non-interacting harmonic oscillator wells, where each well corresponds to a particular normal-mode of vibration. The energy in a each well is just (n+1/2) hbar omega, where omega is the frequency in the that well.

So when the phonon energy increases, the reason it does so incrementally (i.e., quantized) is because the vibrational modes of the crystal have disrcrete energies? Or is it because the number of oscillators (molecules) is increasing by one, additional individual oscillators thus adding their vibrational energies to the mix?

So when the phonon energy increases, the reason it does so incrementally (i.e., quantized) is because the vibrational modes of the crystal have disrcrete energies? Or is it because the number of oscillators (molecules) is increasing by one, additional individual oscillators thus adding their vibrational energies to the mix?

The first. The vibrational modes are quantized. In a perfectly harmonic crystal, the vibrational mode corresponding to a well of frequency w0 would have energy levels (n+1/2)hbar w0. The 0-1 transition corresponds to a phonon of energy hbar w0, the 0-2 transition corresponds to a phonon of frequency 2hbar w0, which classically corresponds to an overtone.