Homework Help: Bose-Einstein Condensation & de Broglie wavelength

1. Apr 11, 2012

jncarter

1. The problem statement, all variables and given/known data
Bose-Einstein condensation of a fluid occurs when the de Broglie wavelength of a "typical" particle becomes greater than the average nearest-neighbor distance. One can interpret the momentum in the de Broglie equation as
$p=\sqrt{<p^{2}>}$
where $<p^{2}>$ means the thermal average for a single particle

(a) Write down the de Broglie equation and use the above to obtain an approximate expression for the Bose-Einstein condensation temperature $T_{B}$ in terms of the particle density (N/V), the particle mass m, Boltzmann's constant k and Planck's constant h

(b)For temperatures T less than the transition temperature $T_{B}$, the fraction, f, of particles in the ground state (f=N(0)/N) is a number between zero and one. Find the chemical potential, $\mu$ to order 1/N as a function of T when $T <<T_{B}$. Assume the ground state energy is zero.

2. Relevant equations
de Broglie momentum: $p= \frac{h}{\lambda}$
Boltzmann distribution function: $<n_{s}> = \frac{1}{e^{(\mu - \epsilon)\beta} - 1}$
where $\beta = \frac{1}{kT}$
Energy: $E = \frac{p^{2}}{2m}$
Expectation value: $<A> = \Sigma_{s} A <n_{s}>$

3. The attempt at a solution

I interpreted the average nearest-neighbor distance to be the ratio of the number of particles, N, and the volume, V, to the 1/3 power, $\lambda > \frac{N}{V^{1/3}}$.
Put this into our equation for momentum:
$p < \frac{h V^{1/3}}{N}$
But we have been told $p=\sqrt{<p^{2}>}$, thus $<p^{2}> <(\frac{h V^{1/3}}{N})^{2}$.
Now we can use the energy-momentum relationship to find $<E> < \frac{1}{2m}(\frac{h V^{1/3}}{N})^{2}$.
So the question comes down to how to approximate the expected energy for a single particle in state with possible states s. Normally, I would use the expected value function above and go through the approximation using density of states and find a function for the chemical potential and then finally solve for temperature. I just can't kick the feeling that there is an easier way. This was taken from a previous comprehensive exam, so I should be able to do it without notes in about an hour (or at least most of the problem). Is there a trick I'm missing?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution