Question

Consider a one-dimensional infinite square well with N particles. Given that all particles occupy the ground state, calculate the total energy of the system. Now assume that each energy level can hold no more than two particles. Calculate the energy of the ground state of the system and the maximum particle energy called the Fermi energy.

Answer 1

The total energy of the system for $N$ particles at ground level ($n=1$) is

$$E_{tot}=N\cdot\frac{n^2h^2}{8mL^2}=\frac{Nh^2}{8mL^2}.$$

Now the ground level is occupied by two particles, i.e., according to the previous expression,

$$E_{g.s.}=\frac{h^2}{4mL^2}.$$

The Fermi energy is (for two electrons for each state, i.e. $n=N/2$)

$$E_F=\frac{h^2}{8\pi^2 m}\cdot\Big(\frac{n\pi}{L}\Big)^2=\frac{h^2}{8 m}\cdot\Big(\frac{N}{2L}\Big)^2.$$