Exam 3 Problems

1) Bose-Einstein Condensation (from David Tong's Course)

a) Consider an ideal gas of bosons whose density of states is given by $g(E)=C{E}^{\alpha -1}$.
for some constants $C$ and $\alpha >1$. Derive an expression for the critical temperature $T_c$,
below which the gas experiences Bose-Einstein condensation.

b) Consider free bosons in a finite region in dimension less than 3. Will there be Bose-Einstein Condensation?

c) In BEC experiments, atoms are confined in magnetic traps which can be modeled
by a quadratic potential, i.e. $U(\mathbf{x})=\frac{1}{2}K|\mathbf{x}{|}^2$. Determine $T_c$ for bosons in a three dimensional trap. Show that bosons in a two-dimensional trap will condense at suitably low temperatures. In each case, calculate the number of particles in the condensate as a function of $T<T_c$. (See also Schroeder Problem 7.73)

2) Low-dimensional Electron Gas

a) Consider a 2D gas of free fermions confined to a finite region of space. Calculate the Fermi energy, the average internal energy, and the pressure at zero temperature. (see also Schroeder Problem 7.28)

b) Repeat (a) for fermions in 1D.

c) Now consider a 1D fermion system in a quadratic potential $U(x)=\frac{K}{2}{x}^2$. Find the Fermi Energy for $N$ electrons, the average internal energy, and the pressure at zero temperature.