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.

a) Calculate the density of states per unit energy range per unit length (hint: use periodic boundary conditions and positive and negative values of k).

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.

Solution Preview

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 ...

Solution Summary

We derive the solution to this problem from first principles.

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