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pkirwin
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a83a090aee
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@ -12,4 +12,4 @@
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\newlabel{eq:numerical_E}{{7}{4}}
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@ -1,5 +1,5 @@
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@ -557,6 +557,20 @@ INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\
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This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 21.1) (preloaded format=pdflatex 2021.2.6) 1 MAR 2021 19:16
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This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 21.1) (preloaded format=pdflatex 2021.2.6) 1 MAR 2021 19:47
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entering extended mode
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(d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex
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@ -365,19 +365,30 @@ File: q1div.jpg Graphic file (type jpg)
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@ -389,9 +389,83 @@ H(N, 1) = -t0;
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\item %2b
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Homogenous boundary conditions imply that the corner entries of \(\hat{H}\) will be 0.
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\item %1c
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Code:
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\begin{lstlisting}
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clear all;
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%physical constants in MKS units
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hbar = 1.054e-34;
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q = 1.602e-19;
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m = 9.110e-31;
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epsilon_0 = 8.854e-12;
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%generate lattice
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N = 100; %number of lattice points
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n = [1:N]; %lattice points
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a = 0.1e-10; %lattice constant
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r = a * n; %x-coordinates
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t0 = (hbar^2)/(2*m*a^2)/q; %encapsulating factor
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L = a * (N+1); %total length of consideration
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%set up Hamiltonian matrix
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U = -q^2./(4*pi*epsilon_0.*r) * (1/q); %potential at r in [eV]
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main_diag = diag(2*t0*ones(1,N)+U,0); %create main diagonal matrix
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lower_diag = diag(-t0*ones(1,N-1),-1); %create lower diagonal matrix
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upper_diag = diag(-t0*ones(1,N-1),+1); %create upper diagonal matrix
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H = main_diag + lower_diag + upper_diag; %sum to get Hamiltonian matrix
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[eigenvectors,E_diag] = eig(H); %"eigenvectors" is a matrix wherein each column is an eigenvector
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%"E_diag" is a diagonal matrix where the
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%corresponding eigenvalues are on the
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%diagonal.
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E_col = diag(E_diag); %folds E_diag into a column vector of eigenvalues
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% return eigenvectors for the 1st and 50th eigenvalues
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phi_1 = eigenvectors(:,1);
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phi_2 = eigenvectors(:,2);
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% find the probability densities of position for 1st and 50th eigenvectors
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P_1 = phi_1 .* conj(phi_1);
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P_2 = phi_2 .* conj(phi_2);
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% Plot the probability densities for 1st and 2nd eigenvectors
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figure(1); clf; h = plot(r,P_1,'k-');
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grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
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xlabel('RADIAL POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]');
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yticks([0.02 0.04 0.06 0.08 0.10 0.12]);
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legend('n=1');
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axis([0 1e-9 0 0.12]);
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figure(2); clf; h = plot(r,P_2,'k-');
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grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
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xlabel('RADIAL POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]');
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yticks([0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04]);
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legend('n=2');
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axis([0 1e-9 0 0.04]);
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\end{lstlisting}
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\begin{figure}[H]
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\centering
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\begin{subfigure}{0.5\textwidth}
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\centering
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\includegraphics[width=\textwidth]{q2c_fig1_1s.png}
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\caption{1s probability density.}
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\end{subfigure}%
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\begin{subfigure}{0.5\textwidth}
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\centering
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\includegraphics[width=\textwidth]{q2c_fig2_2s.png}
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\caption{2s probability density.}
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\end{subfigure}
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\end{figure}
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\item %1d
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\lipsum[1]
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For the 1s level, \(E = -13.4978\) eV.
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\item %1e
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\lipsum[1]
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\item %1f
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P_1 = phi_1 .* conj(phi_1);
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P_2 = phi_2 .* conj(phi_2);
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% Find first N analytic eigenvalues
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E_col_analytic = (1/q) * (hbar^2 * pi^2 * n.*n) / (2*m*L^2);
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% Plot the probability densities for 1st and 2nd eigenvectors
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figure(1); clf; h = plot(r,P_1,'k-');
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xlabel('RADIAL POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]');
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yticks([0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04]);
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legend('n=2');
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axis([0 1e-9 0 0.04]);
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%{
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% Plot numerical eigenvalues
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figure(2); clf; h = plot(n,E_col,'kx'); grid on;
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set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
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xlabel('EIGENVALUE NUMBER'); ylabel('ENERGY [eV]');
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axis([0 100 0 40]);
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% Add analytic eigenvalues to above plot
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hold on;
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plot(n,E_col_analytic,'k-');
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legend({'Numerical','Analytical'},'Location','northwest');
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%}
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axis([0 1e-9 0 0.04]);
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