# Ripplons quantum stat mech problem

1. Aug 29, 2012

### FillBill

1. The problem statement, all variables and given/known data

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 $\epsilon(k)$
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 $g(\epsilon)d\epsilon$
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:

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

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

$$g(k) dk = (A/(2\pi)^2)2\pi k dk$$

Plugging the above in:

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

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

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

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