# Bose Einstein condensation

1. Apr 17, 2015

### CAF123

1. The problem statement, all variables and given/known data
Explain the conditions under which Bose-Einstein condensation occurs and show this happens for density $\rho \equiv N/V > \rho_C(T)$, where $$\rho_C(T) = A(kT)^{3/2} \int_0^{\infty} \frac{x^{1/2}}{ e^x − 1} dx .$$

Suppose the energy of the particles on the lattice is now $\epsilon_j → \epsilon_{j\chi} = E\chi+ \frac{\hbar^2 k^2_j}{(2m)}$, where $\chi = 0, 1$ and $E > 0$. Obtain an expression for $\rho \equiv N/V$.

2. Relevant equations
$\rho = \rho_o + \rho_+$ where $\rho_o$ is density of ground state and $\rho_+$ density of all others.

3. The attempt at a solution

The first part is ok, in the integral, $x = \beta \epsilon$. In the second part, are we just shifting the ground state energy to $E\chi$? and then evalaute the integral $\rho_C(t)$ using that expression for $\epsilon_{j \chi}$? I'm not sure what the conditions $\chi = 0,1$ and $E>0$ imply yet either. Thanks!

2. Apr 20, 2015

### CAF123

Bump!

3. Apr 23, 2015

Bump