1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Ripplons quantum stat mech problem

  1. Aug 29, 2012 #1
    1. The problem statement, all variables and given/known data

    lTPvz.png

    2. Relevant equations



    3. The attempt at a solution

    I have a vague idea of how to do this problem, but I'm not sure. Here's the plan I have for solving it:

    1. Find the dispersion relationship [itex]\epsilon(k)[/itex]
    2. Find the 2D density of states for wave vectors k: g(k)dk
    3. Using the dispersion relation, find the 2D density of states [itex]g(\epsilon)d\epsilon[/itex]
    4. Integrate this and the Bose Einstein occupation number and the energy to find the average energy...?
    5. Use this to find the heat capacity

    The dispersion relationship should be:

    [tex]\epsilon = \hbar \omega = \hbar(\alpha_0 k^3/\rho)^{1/2} \rightarrow k = (\rho/\alpha_0 \hbar^2)^{1/3}\epsilon^{2/3} \rightarrow dk = (2/3)(\rho/\alpha_0 \hbar^2)^{1/3} \epsilon^{-1/3} d\epsilon[/tex]

    Because the number of states is a circle in 2D, the number of states between k and k + dk should be:

    [tex]g(k) dk = (A/(2\pi)^2)2\pi k dk[/tex]

    Plugging the above in:

    [tex]g(\epsilon) d\epsilon = (A/2\pi)(2/3)(\rho/\alpha_0 \hbar^2)^{2/3} \epsilon^{1/3} d\epsilon[/tex]

    Now, we plug this into the integral with the B.E. occupation number:

    [tex]E = \int_0 ^\infty \epsilon n(\epsilon) g(\epsilon) d\epsilon = (A/2\pi)(2/3)(\rho/\alpha_0 \hbar^2)^{2/3} \int_0 ^\infty \frac{\epsilon^{4/3} d\epsilon}{e^{\beta(\epsilon - \mu)} - 1}[/tex]

    But here's where I'm stuck. First of all, I don't know how to analytically do this integral. Second, we were never given the chemical potential...am I supposed to figure it out from the first line of the problem, using α? I don't see how, though...

    Can anyone help me out? Thanks!
     
    Last edited: Aug 29, 2012
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: Ripplons quantum stat mech problem
  1. Stat Mech Question (Replies: 0)

  2. Stat Mech Question (Replies: 0)

Loading...