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pkirwin
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# Fdb version 3
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["pdflatex"] 1614650523 "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.pdf" "doc" 1614650525
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["pdflatex"] 1614651411 "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.pdf" "doc" 1614651414
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"C:/Users/speed/AppData/Local/MiKTeX/pdftex/config/pdftex.map" 1614638559 80909 eab91d9745dd2edfd62a31d53cd5fe15 ""
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@ -97,10 +97,10 @@
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"C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/parskip/parskip.sty" 1592833306 4414 25631f9679adf750bac898538ededd61 ""
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"doc.aux" 1614650525 740 96d321d28a358a20184b7fe32068111f "pdflatex"
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"doc.tex" 1614650522 17710 726d1c26d86b0ac704aeac46e41f3d05 ""
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"doc.aux" 1614651413 740 96d321d28a358a20184b7fe32068111f "pdflatex"
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"doc.tex" 1614651410 17776 60a8990fb3ce32251e01cb0001a9ffb6 ""
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"q1a_1and50.png" 1614636283 34155 72f34c6abf061325ae1f162db0ffd06b ""
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"q1a_eigenvals.png" 1614637266 23078 9c99a5505761d0b964e665e35fd9c02a ""
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"q1cii.png" 1614640691 20581 6a6b3aa285fd3f5be369532a6d2a033d ""
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@ -111,7 +111,7 @@
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"q1div.jpg" 1614643890 3254255 657a32881fc66c752988aa2be34fe320 ""
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(generated)
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"doc.pdf"
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"d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.log"
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"doc.log"
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"doc.aux"
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"d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.log"
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"d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf"
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"doc.aux"
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@ -1,4 +1,4 @@
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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
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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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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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(d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex
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@ -375,7 +375,7 @@ Here is how much of TeX's memory you used:
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1141 hyphenation exceptions out of 8191
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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).
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Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (11 pages, 3616528 bytes).
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PDF statistics:
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124 PDF objects out of 1000 (max. 8388607)
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0 named destinations out of 1000 (max. 500000)
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BIN
PS2/doc.pdf
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@ -387,9 +387,9 @@ H(N, 1) = -t0;
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\hat{H}_{l(l\pm1)} = -\frac{\hbar^2}{2ma^2}.
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\end{equation*}
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\item %2b
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\lipsum[1]
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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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\lipsum[1]
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\item %1d
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\lipsum[1]
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\item %1e
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65
PS2/q2c.m
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65
PS2/q2c.m
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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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%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 = 1e-10; %lattice constant
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x = 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 = 0*x; %0 potential at all x
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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_50 = eigenvectors(:,50);
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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_50 = phi_50 .* conj(phi_50);
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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 50th eigenvectors
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figure(1); clf; h = plot(x,P_1,'kx',x,P_50,'k-');
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grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
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xlabel('POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]');
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legend('n=1','n=50');
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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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