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pkirwin 2021-03-01 19:54:12 -07:00
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\newlabel{eq:numerical_E}{{7}{4}} \newlabel{eq:numerical_E}{{7}{4}}
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@ -389,9 +389,83 @@ H(N, 1) = -t0;
\item %2b \item %2b
Homogenous boundary conditions imply that the corner entries of \(\hat{H}\) will be 0. Homogenous boundary conditions imply that the corner entries of \(\hat{H}\) will be 0.
\item %1c \item %1c
Code:
\begin{lstlisting}
clear all;
%physical constants in MKS units
hbar = 1.054e-34;
q = 1.602e-19;
m = 9.110e-31;
epsilon_0 = 8.854e-12;
%generate lattice
N = 100; %number of lattice points
n = [1:N]; %lattice points
a = 0.1e-10; %lattice constant
r = 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 = -q^2./(4*pi*epsilon_0.*r) * (1/q); %potential at r in [eV]
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_2 = eigenvectors(:,2);
% find the probability densities of position for 1st and 50th eigenvectors
P_1 = phi_1 .* conj(phi_1);
P_2 = phi_2 .* conj(phi_2);
% Plot the probability densities for 1st and 2nd eigenvectors
figure(1); clf; h = plot(r,P_1,'k-');
grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
xlabel('RADIAL POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]');
yticks([0.02 0.04 0.06 0.08 0.10 0.12]);
legend('n=1');
axis([0 1e-9 0 0.12]);
figure(2); clf; h = plot(r,P_2,'k-');
grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
xlabel('RADIAL POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]');
yticks([0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04]);
legend('n=2');
axis([0 1e-9 0 0.04]);
\end{lstlisting}
\begin{figure}[H]
\centering
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{q2c_fig1_1s.png}
\caption{1s probability density.}
\end{subfigure}%
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{q2c_fig2_2s.png}
\caption{2s probability density.}
\end{subfigure}
\end{figure}
\item %1d \item %1d
\lipsum[1] For the 1s level, \(E = -13.4978\) eV.
\item %1e \item %1e
\lipsum[1] \lipsum[1]
\item %1f \item %1f

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@ -41,9 +41,6 @@ phi_2 = eigenvectors(:,2);
P_1 = phi_1 .* conj(phi_1); P_1 = phi_1 .* conj(phi_1);
P_2 = phi_2 .* conj(phi_2); P_2 = phi_2 .* conj(phi_2);
% 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 2nd eigenvectors % Plot the probability densities for 1st and 2nd eigenvectors
figure(1); clf; h = plot(r,P_1,'k-'); figure(1); clf; h = plot(r,P_1,'k-');
@ -58,19 +55,4 @@ grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
xlabel('RADIAL POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]'); xlabel('RADIAL POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]');
yticks([0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04]); yticks([0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04]);
legend('n=2'); legend('n=2');
axis([0 1e-9 0 0.04]); axis([0 1e-9 0 0.04]);
%{
% 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');
%}

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