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As per de Broglie's Law, every wave has a particle nature associated with it. I tried to find what sound particles are called, and I got that they are phonons.

Now, I am curious about what it's energy will be. We have for light:

$$E_{\text{photon}}=h\nu$$

I suspect a similar energy equation for phonons must exist.

$$E_{\text{phonon}}=k\nu$$

where $k$ is a proportionality constant.

What could this $k$ be?

Qmechanic
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Pritt Balagopal
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  • You quantize lattice vibrations with the standard canonical algebra, hence $k=h$. – Sunyam Oct 25 '17 at 17:03
  • @confused Unfortunately, I don't know what "canonical algebra" is supposed to mean, so I can't understand your comment. – Pritt Balagopal Oct 25 '17 at 17:06
  • $[x_i,x_j]=0$, $[p_i,p_j]=0$ and $[x_i,p_j]=\delta_{ij}$. – Sunyam Oct 25 '17 at 17:12
  • Thanks, but you only made me more @confused (just like your name!). Btw I am a first year college student, who knows little math other than what I've learnt in school. – Pritt Balagopal Oct 25 '17 at 17:15
  • @confused "You quantize lattice vibrations with the standard canonical algebra, hence k=hk=h" What do you mean, lattice vibrations? I'm talking about sound, just sound that travels through air or any other medium. – Pritt Balagopal Oct 25 '17 at 17:29
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    Typo : $[x_i,p_j]=i\hbar\delta_{ij}$. Thanks for complement ;). To simplify, phonons are nothing but quantized excitations of crystal lattice vibrations. Lattice vibrations are treated as collection of harmonic oscillators, when quantized you get a set of quantized harmonic oscillators, excitations of which are called phonons. This quantisation process is carried out using $\hbar$, which reflects in the fact that energies of these oscillatos are $\hbar v$. Sound needs medium to propagate (you are right), vibrations of constituents of the medium i reffered to as lattice vibrations. – Sunyam Oct 25 '17 at 17:31

2 Answers2

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Sound waves are typically made up of a large amount of phonons.

The energy of a phonon is E = hf where h is planks constant and f is the frequency of the wave whether transverse or longitudinal.

Let me know if you need any clarifications.

CuriousOne
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You just use the same Planck constant.

Photons are the quantized excitations of the electromagnetic field. Phonons (which generally only occur in solids) are excitations of the displacement field of the nuclei. As such, they're really not very different at all, and they're treated in exactly the same way by quantum mechanics, including of course the basic identification of frequency with energy - otherwise just known as the Schrödinger equation, $$ i\hbar \frac{\partial}{\partial t}\psi = H\psi, $$ where $i\frac{\partial }{\partial t}$ measures the temporal frequency of the wavefunction $\psi$, and $H$ is known as the hamiltonian and gives the energy of the system. This equation is universal to quantum mechanics, and the constant that links the two is always the same Planck constant $\hbar$.

Emilio Pisanty
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