Answer to Question #126381 in Electricity and Magnetism for christopher seebaran

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. Find an expression for the density of states for the infinite square well.

Answer 1

As per the given question,

Total number of particle =N

Total energy of the system=?

We know that, $E_n​=\frac{N\pi^2\hbar^2}{8mL}$

As per the question, n=2,

Hence,

$E_2​=\frac{N\pi^2\hbar^2}{4mL}$

Hence, Fermi energy, here, $n=\frac{N}{2}$

$E_F=\frac{\hbar^2}{8\pi^2m}(\frac{n\pi}{L})^2$

$E_F=\frac{\hbar^2}{8\pi^2m}(\frac{N}{2L})^2$