Uncertainty Principle Examples in Solid State Physics

[mtmiec]

Hi, I'm working on a homework for my solid state physics class and I'm having trouble with this question:

Homework Statement

"We have seen many cases in solid state physics in which you can apply the concept of uncertainty principle to estimate some quantity. Describe two examples of a phenomenon where you can apply this concept, and show the calculation associated with that example. Choose the cases which were not covered in the lecture. Come up with your own examples."

The Attempt at a Solution

I have one potential answer, an estimation of the Fermi wavevector in the free electron model of a Fermi gas. Approximating Δx (of an electron) as (1/n)^1/3, where n is the charge carrier concetration, and Δp as hbar×k_f gives a result for k_f with the correct n dependence:

k_f=(1/2)×n^1/3

The actual answer is:

k_f=(3π²n)^1/3

I figure the uncertainty principle would more properly be used to estimate the rms electron momentum, but I think it would be better to compare my answer to an equation we used in class, and that's easy with k_f. How does this answer look?

I'm having some trouble coming up with a second example to use, because I think the standard applications of the uncertainty principle might be too elementary for this class (things like the size of the hydrogen atom or the linewidth of emission lines). Also, some good examples were already covered in class.

I feel like there should be a good example involving phonons, because momentum is so explicitly involved, but even so I can't think of a good way to apply the principle here. I'm not looking for any math or explicit answers, just some ideas.

Your help is greatly appreciated! Thanks!

 

[Nate810]

One example involving phonons is to consider the uncertainty in the momentum of a phonon. The uncertainty principle suggests that the momentum uncertainty of a phonon will be related to its wavelength. Therefore, one can estimate the momentum of a phonon by using its known wavelength. This can be done using the formula p=h/λ, where h is the Planck constant and λ is the wavelength. Additionally, one can also use the uncertainty principle to estimate the temperature of a phonon. The uncertainty principle states that ΔpΔx≥hbar, and thus one can estimate the temperature of a phonon by finding the product of (Δp)2 and (Δx)2 and then dividing it by hbar. This can then be converted into a temperature value.