How to Solve the Ripplons Quantum Stat Mech Problem?

[FillBill]

Homework Statement

Homework Equations

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!

 

[mark726]

Hi there! Your plan for solving this problem looks good so far. To answer your questions:

1. To find the chemical potential, you can use the given information about α and the fact that the system is in thermal equilibrium. This means that the chemical potential must be such that the number of particles in the system (given by the integral you derived) is equal to the total number of particles in the system. This will allow you to solve for μ.

2. As for the integral, it is a bit tricky to solve analytically. However, you can use numerical methods to approximate the integral and solve for the average energy. Alternatively, you can use a computer program such as Mathematica to help you solve the integral.

Hope this helps! Let me know if you have any further questions. Good luck with your problem!