Phonons in a lattice

1. Apr 3, 2007

clicheophobe

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.

2. Apr 3, 2007

Staff: Mentor

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.

3. Apr 3, 2007

clicheophobe

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?

4. Apr 4, 2007

5. Apr 4, 2007

clicheophobe

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.

6. Apr 4, 2007

christianjb

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?

7. Apr 5, 2007

clicheophobe

I mean the latter, energy levels.

8. Apr 5, 2007

Gokul43201

Staff Emeritus
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.

Last edited: Apr 5, 2007
9. Apr 5, 2007

ensabah6

Do you think elementary particles could arise in this way?

10. Apr 5, 2007

christianjb

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.

Last edited: Apr 5, 2007
11. Apr 8, 2007

clicheophobe

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?

12. Apr 8, 2007

christianjb

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.