Sample Test Problems

1. * Calculate the Fermi energy for $N$ particles of mass $m$ in a 3D cubic ``box'' of side $L$. Ignore spin for this problem.
   Answer
   The energy levels are given in terms of three quantum numbers.
   
   $$E={\pi^2\hbar^2\over 2mL^2}(n_x^2+n_y^2+n_z^2)$$
   
   
   The number of states with inside some $(n_x^2+n_y^2+n_z^2)_{max}$ (${1\over 8}$ of a a sphere in $n$ space) is
   
   $$N={1\over 8} {4\over 3}\pi (n_x^2+n_y^2+n_z^2)^{3\over 2}_{max}$$
   
   
   So for $N$ particles filling the levels,
   
   $$(n_x^2+n_y^2+n_z^2)^{3\over 2}_{max}={6N\over\pi} .$$
   
   
   
   
   $$(n_x^2+n_y^2+n_z^2)_{max}=\left({6N\over\pi}\right)^{2\over 3}$$
   
   
   The energy corresponding to this is the Fermi energy.
   
   $$E_F={\pi^2\hbar^2\over 2mL^2}\left({6N\over\pi}\right)^{2\over 3}$$
   
   

2. * We put $N$ fermions of mass $m$ into a (cold) one dimensional box of length $L$. The particles go into the lowest energy states possible. If the Fermi energy is defined as the energy of the highest energy particle, what is the Fermi energy of this system? You may assume that there are 2 spin states for these fermions.