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pkirwin 2021-03-01 19:17:35 -07:00
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PS2/doc.fdb_latexmk
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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 19:02 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
entering extended mode entering extended mode
**d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex **d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex
(d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex (d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex
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68i,12n,75p,1278b,2408s stack positions out of 5000i,500n,10000p,200000b,50000s 68i,12n,75p,1278b,2408s stack positions out of 5000i,500n,10000p,200000b,50000s
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Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (11 pages, 3617145 bytes). Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (11 pages, 3616528 bytes).
PDF statistics: PDF statistics:
124 PDF objects out of 1000 (max. 8388607) 124 PDF objects out of 1000 (max. 8388607)
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@ -387,9 +387,9 @@ H(N, 1) = -t0;
\hat{H}_{l(l\pm1)} = -\frac{\hbar^2}{2ma^2}. \hat{H}_{l(l\pm1)} = -\frac{\hbar^2}{2ma^2}.
\end{equation*} \end{equation*}
\item %2b \item %2b
\lipsum[1] Homogenous boundary conditions imply that the corner entries of \(\hat{H}\) will be 0.
\item %1c \item %1c
\lipsum[1]
\item %1d \item %1d
\lipsum[1] \lipsum[1]
\item %1e \item %1e

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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');