clear all;
%physical constants in MKS units

hbar = 1.054e-34;
q = 1.602e-19;
m = 9.110e-31;

%generate lattice

N  = 100;                   %number of lattice points
n  = [1:N];                 %lattice points
a  = 1e-10;                 %lattice constant
x  = a * n;                 %x-coordinates
t0 = (hbar^2)/(2*m*a^2)/q;  %encapsulating factor
L  = a * (N+1);             %total length of consideration

%set up Hamiltonian matrix 

U = 0*x; %0 potential at all x
main_diag  = diag(2*t0*ones(1,N)+U,0); %create main diagonal matrix
lower_diag = diag(-t0*ones(1,N-1),-1); %create lower diagonal matrix
upper_diag = diag(-t0*ones(1,N-1),+1); %create upper diagonal matrix

H = main_diag + lower_diag + upper_diag; %sum to get Hamiltonian matrix

[eigenvectors,E_diag] = eig(H); %"eigenvectors" is a matrix wherein each column is an eigenvector
                                %"E_diag" is a diagonal matrix where the
                                %corresponding eigenvalues are on the
                                %diagonal.
                                
E_col = diag(E_diag); %folds E_diag into a column vector of eigenvalues

% return eigenvectors for the 1st and 50th eigenvalues

phi_1  = eigenvectors(:,1);
phi_50 = eigenvectors(:,50);

% find the probability densities of position for 1st and 50th eigenvectors

P_1 = phi_1 .* conj(phi_1);
P_50 = phi_50 .* conj(phi_50);

% Find first N analytic eigenvalues
E_col_analytic = (1/q) * (hbar^2 * pi^2 * n.*n) / (2*m*L^2);

% Plot the probability densities for 1st and 50th eigenvectors

figure(1); clf; h = plot(x,P_1,'kx',x,P_50,'k-');
grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
xlabel('POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]');
legend('n=1','n=50');

% Plot numerical eigenvalues
figure(2); clf; h = plot(n,E_col,'kx'); grid on;
set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
xlabel('EIGENVALUE NUMBER'); ylabel('ENERGY [eV]');
axis([0 100 0 40]);

% Add analytic eigenvalues to above plot

hold on;
plot(n,E_col_analytic,'k-');
legend({'Numerical','Analytical'},'Location','northwest');