There has recently been considerable interest in one-dimensional electrical conductors. In this problem, you are asked to calculate some free-electron properties for a system of length L containing N electrons. Thus, there are n =N/L electrons per unit length.
b) Calculate the Fermi energy e_f at T=0. Estimate e_f in electron volts for n = 10^8 /cm.
C) Calculate the energy of the gas at T =0. Give your answers in terms of n and e_f.
The wavefunction of an electron with wavevector k is:
psi(x) = 1/sqrt(L) exp(i k x)
The momentum of this state is
p_k = hbar k
The energy is
E_k = p_k^2/(2m) = hbar^2 k^2/(2m)
Imposing the periodic boundary condition gives:
psi(x+L) = psi(x) ----->
exp[i k (x+L)] = exp(i k x) ------------->
exp(i k L) = 1 ------------->
k L = 2 pi s
where s is some arbitrary integer (I use "s" instead of the more usual "n" because n is used to denote the density of electrons in this problem). So the allowed k values are:
k = 2 pi s/L (1)
To find the density of states, it is convenient to express s in terms of k:
s = k L/(2 pi) (2)
The idea is that all the allowed k values are labeled by some integer according to eq. (1). Then, since what we want to know is how many states there are between two k values, all we need to find out are the numbers corresponding to the two k values. The difference is then the number of allowed k values that ...
We derive the solution to this problem from first principles.