% Here we are going to solve the boxed matrix equation on page one of
% the 1D Weak Period Potential.
% Let K be the smallest reciprocal lattice vector.
% K vectors are integer multiples of K.
All other
K = 1;
% Rescale all energies by epsilon_K = hbar^2 K^2/2m .
% This means that U is actually U/epsilon_K and all energies
% are in units of epsilon_K.
% In this simple calculation we take all the U’s to be the same
% except for U_0 = 0.
U = 0.5;
% Keep (2*Nmax + 1) reciprocal lattice vectors.
% This can be changed to check for convergence.
Nmax = 5;
Kvalues = (-Nmax:1:Nmax)*K;
% The first Brillouin zone is between -K/2 and K/2.
kvalues = linspace(-K/2,K/2,1000);
energies = zeros(length(Kvalues),length(kvalues));
for counter = 1:length(kvalues);
k = kvalues(counter);
H = diag(((k - Kvalues).^2 -U));
H = H + U*ones(length(Kvalues));
energy = eig(H);
energies(:,counter) = energy;
end
plot(kvalues,energies)
axis([min(kvalues) max(kvalues) 0 5])
xlabel(’k/K’)
ylabel(’energy/epsilon_K’)
title(’U = 0.5, not small perturbation’)
U = 0, no perturbation
5
energy/epsilon_K
4
3
2
1
0
-0.4
-0.2
0
k/K
0.2
0.4
U = 0.02, very small perturbation
5
energy/epsilon_K
4
3
2
1
0
-0.4
-0.2
0
k/K
0.2
0.4
U = 0.1, small perturbation
5
energy/epsilon_K
4
3
2
1
0
-0.4
-0.2
0
k/K
0.2
0.4
U = 0.5, not small perturbation
5
energy/epsilon_K
4
3
2
1
0
-0.4
-0.2
0
k/K
0.2
0.4