Question

How can I get density of state, Fermi energy, and total energy in 1,2 dimension when we have N electrons without interaction.

Answer 1

I'll demonstrate the procedure for electrons confined to move in 1D potential well, this is, the potential is zero inside and infinite inside the linear dimension L (Sommerfeld model). The electron states are given by the Schrodinger equation, and to solve we use the 1D Born Von Karman periodic boundary conditions:

The solution is given by:

The boundary conditions dictate the allowed values for kx:

This states can be visualized as 1D grid points, with L/2Pi points per unit length. Each point can be occupied by 2 electrons (spins up and down according to Pauli's exclusion principle). All filled states corresponds to points that are separated a distance 2KF (Fermi points). All quantum states between Fermi points are occupied by electrons and outside are empty. The number of grid points in the Fermi region is L/2Pi x 2KF and the number of quantum states is twice that much. This quantity must equal the total number of electrons:

The largest momentum of the electrons is hkF then the largest energy of the electrons (Fermi Energy) is:

The integration over all points in the linear space is given by:

Since

  

Then using Maxwell Boltzmann's distribution for occupation probability of a quantum state we can write

Where

Is the density of states function. The Fermi Level is a function of temperature and decreases from EF as temperature increases. The energy density is:

For T=0K:

But before we arrived to the expression

So we can write for the total energy: