Why are the quantum mechanical effects of sound observed in most solids but not most liquids?

Well, if liquids remained liquids down to low temperatures, then the quantum mechanical effects of sound would also occur in them as well. There is actually one example where these effects are important and this is in liquid helium.

Therefore the appropriate questions to ask then are: (i) when are quantum mechanical effects significant in the description of sound? and (ii) when does quantum mechanics have any observable consequences in matter at all?

The answer to this question is probably obvious to most people that read this blog. However, I would still think it needs to be reiterated every once in a while. When does the wave nature of “particles” become relevant? Usually, when the wavelength, \lambda, is on the order of some characteristic length, d:

\lambda \gtrsim d

What is this characteristic length in a liquid or solid? One can approximate this by the interparticle spacing, which one can take to be the inverse of the cube root of the density, n^{-1/3}. Therefore, quantum mechanical effects can be said to become important when:

d \sim n^{-1/3}

Now, lastly, we need an expression for the wavelength of the particles. One can use the deBroglie expression that relates the wavelength to the momentum:

\lambda \sim \frac{h}{p},

where h is Planck’s constant and p is the momentum. And one can approximate the momentum of a particle at temperature, T, by:

p \sim \sqrt{mk_BT}    (massive)    OR      p \sim k_BT/v_s     (massless),

where k_B is Boltzmann’s constan, m is the mass of the particle in question, and v_s is the speed of sound. Therefore we get that quantum mechanics becomes significant when:

n^{2/3}h^{2}/m \gtrsim k_BT   (massive)    OR     n^{1/3}h v_s \gtrsim k_BT     (massless).

Of course this expression is just a rough estimate, but it does tell us that most liquids end up freezing before quantum mechanical effects become relevant. Therefore sound, or phonons, express their quantum mechanical properties at low temperatures — usually below the freezing point of most materials. By the way, the most celebrated example of the quantum mechanical effects of sound in a solid is in the C_v \sim T^3 Debye model. Notice that the left hand side in formula above for massless particles is, within factors of order unity, the Boltzmann constant times the Debye temperature. Sound can exhibit quantum mechanical properties in liquids and gases, but these cases are rare: helium at low temperature is an example of a liquid, and Bose condensed sodium is an example of a gas.


One response to “Why are the quantum mechanical effects of sound observed in most solids but not most liquids?

  1. Also, um, little in the way of shear waves.


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