Problem 27.1 (R.H. Swendsen): Bosons in two dimensions

Consider an ideal boson gas in two dimensions. The $N$ particles in the gas each have mass $m$ and are confined to a box of dimensions $L\times L$.

1. Calculate the density of states for the two-dimensional, ideal Bose gas.
2. Calculate the Einstein temperature in two dimensions in terms of the given parameters and a dimensionless integral. You do have to evaluate the dimensionless integral.

Problem 27.2 (R.H. Swendsen): Bosons in four dimensions

Consider an ideal boson gas in four dimensions. The $N$ particles in the gas each have mass $m$ and are confined to a box of dimensions $L\times L\times L\times L$.

1. Calculate the density of states for the four-dimensional, ideal Bose gas.
2. Calculate the Einstein temperature in four dimensions in terms of the given parameters and a dimensionless integral. You do not have to evaluate the dimensionless integral.
3. Below the Einstein temperature, calculate the occupation number of the lowest energy, single-particle state as a function of temperature, in terms of the Einstein temperature and the total number of particles.

Problem 27.3 (R. H. Swendsen): Energy of an ideal Bose gas in four dimensions

Again consider an ideal boson gas in four dimensions. The $N$ particles in the gas each have mass $m$ and are confined to a box of dimensions $L\times L\times L\times L$. Calculate the energy and the specific heat as functions of temperature below the Einstein temperature.

Problem 28.3 (R. H. Swendsen): Ideal Fermi gas in two dimensions

Consider an ideal Fermi gas in two dimensions. It is contained in an area of dimensions $L\times L$. The particle mass is $m$.

1. Calculate the density of states.
2. Using your result for the density of states, calculate the number of particles as a function of the chemical potential at zero temperature. ($\mu(T=0)=\epsilon_F$, the Fermi energy.)
3. Calculate the Fermi energy as a function of the number of particles.
4. Again using your result for the density of states, calculate the total energy of the system at zero temperature as a function of the Fermi energy, $\epsilon_F$.
5. Calculate the energy per particle as a function of the Fermi energy $\epsilon_F$.