diff --git a/PS2/doc.aux b/PS2/doc.aux index 913f089..9aad6e5 100644 --- a/PS2/doc.aux +++ b/PS2/doc.aux @@ -12,4 +12,4 @@ \newlabel{eq:numerical_E}{{7}{4}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Analytic solution to numeric system, plotted.\relax }}{5}{}\protected@file@percent } \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Sketch of appropriate energy levels\relax }}{7}{}\protected@file@percent } -\gdef \@abspage@last{11} +\gdef \@abspage@last{12} diff --git a/PS2/doc.fdb_latexmk b/PS2/doc.fdb_latexmk index c161cfd..aa584d0 100644 --- a/PS2/doc.fdb_latexmk +++ b/PS2/doc.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["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 +["pdflatex"] 1614653239 "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" 1614653243 "C:/Users/speed/AppData/Local/MiKTeX/fonts/pk/ljfour/jknappen/ec/dpi600/tcrm1000.pk" 1614620319 11548 693fc704971bedc6e913b37b081cf40e "" "C:/Users/speed/AppData/Local/MiKTeX/miktex/data/le/pdftex/pdflatex.fmt" 1612675286 9293900 83195ab665706b6f706823352908aa95 "" "C:/Users/speed/AppData/Local/MiKTeX/pdftex/config/pdftex.map" 1614638559 80909 eab91d9745dd2edfd62a31d53cd5fe15 "" @@ -97,10 +97,10 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/parskip/parskip.sty" 1592833306 4414 25631f9679adf750bac898538ededd61 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/setspace/setspace.sty" 1324294077 22913 a27d7908fc6f0385466454a966a316eb "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/titlesec/titlesec.sty" 1571240058 48596 2b6a95da931c07a430b1a61904aaa42d "" - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.aux" 1614651413 740 96d321d28a358a20184b7fe32068111f "" - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" 1614651410 17776 60a8990fb3ce32251e01cb0001a9ffb6 "" - "doc.aux" 1614651413 740 96d321d28a358a20184b7fe32068111f "pdflatex" - "doc.tex" 1614651410 17776 60a8990fb3ce32251e01cb0001a9ffb6 "" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.aux" 1614653242 740 9ca5a038ed4ac5ccabd4bb4170e5688b "" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" 1614653238 20515 ff9dd2e9b5270ae7d092432224a524d8 "" + "doc.aux" 1614653242 740 9ca5a038ed4ac5ccabd4bb4170e5688b "pdflatex" + "doc.tex" 1614653238 20515 ff9dd2e9b5270ae7d092432224a524d8 "" "q1a_1and50.png" 1614636283 34155 72f34c6abf061325ae1f162db0ffd06b "" "q1a_eigenvals.png" 1614637266 23078 9c99a5505761d0b964e665e35fd9c02a "" "q1cii.png" 1614640691 20581 6a6b3aa285fd3f5be369532a6d2a033d "" @@ -109,9 +109,11 @@ "q1di_fig1.png" 1614642205 23575 17449aebb9c23a5139435b3506cbca3d "" "q1di_fig2.png" 1614642205 13867 650413e942c9c7a655204496b4f3c21b "" "q1div.jpg" 1614643890 3254255 657a32881fc66c752988aa2be34fe320 "" + "q2c_fig1_1s.png" 1614652840 23989 50364061f25cfea80c735ce05ba391dd "" + "q2c_fig2_2s.png" 1614652829 29139 9c3bdf9df7dc973686f5105c04140b36 "" (generated) - "doc.pdf" - "doc.log" - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.log" - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf" "doc.aux" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf" + "doc.pdf" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.log" + "doc.log" diff --git a/PS2/doc.fls b/PS2/doc.fls index e9cfc3f..eda6ba3 100644 --- a/PS2/doc.fls +++ b/PS2/doc.fls @@ -557,6 +557,20 @@ INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1div.jpg INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1div.jpg INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1div.jpg +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2c_fig1_1s.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2c_fig1_1s.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2c_fig1_1s.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2c_fig1_1s.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2c_fig1_1s.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2c_fig1_1s.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2c_fig1_1s.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2c_fig2_2s.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2c_fig2_2s.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2c_fig2_2s.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2c_fig2_2s.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2c_fig2_2s.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2c_fig2_2s.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2c_fig2_2s.png INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.aux INPUT C:\Users\speed\AppData\Local\MiKTeX\fonts\pk\ljfour\jknappen\ec\dpi600\tcrm1000.pk INPUT C:\Users\speed\AppData\Local\MiKTeX\fonts\pk\ljfour\jknappen\ec\dpi600\tcrm1000.pk diff --git a/PS2/doc.log b/PS2/doc.log index d4b8435..d95fea4 100644 --- a/PS2/doc.log +++ b/PS2/doc.log @@ -1,4 +1,4 @@ -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 +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 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 @@ -365,19 +365,30 @@ File: q1div.jpg Graphic file (type jpg) Package pdftex.def Info: q1div.jpg used on input line 358. (pdftex.def) Requested size: 505.89pt x 379.41656pt. - [6 ] [7 ] [8] [9] [10] [11] (d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.aux) ) + [6 ] [7 ] [8] + +File: q2c_fig1_1s.png Graphic file (type png) + +Package pdftex.def Info: q2c_fig1_1s.png used on input line 458. +(pdftex.def) Requested size: 252.94499pt x 189.7102pt. + +File: q2c_fig2_2s.png Graphic file (type png) + +Package pdftex.def Info: q2c_fig2_2s.png used on input line 463. +(pdftex.def) Requested size: 252.94499pt x 189.7102pt. + [9 ] [10] [11] [12] (d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.aux) ) Here is how much of TeX's memory you used: - 8374 strings out of 479618 - 125825 string characters out of 2865027 + 8398 strings out of 479618 + 126163 string characters out of 2865027 981750 words of memory out of 3000000 - 25432 multiletter control sequences out of 15000+200000 + 25454 multiletter control sequences out of 15000+200000 412866 words of font info for 64 fonts, out of 3000000 for 9000 1141 hyphenation exceptions out of 8191 68i,12n,75p,1278b,2408s stack positions out of 5000i,500n,10000p,200000b,50000s -Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (11 pages, 3616528 bytes). +Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (12 pages, 3676324 bytes). PDF statistics: - 124 PDF objects out of 1000 (max. 8388607) + 129 PDF objects out of 1000 (max. 8388607) 0 named destinations out of 1000 (max. 500000) - 41 words of extra memory for PDF output out of 10000 (max. 10000000) + 51 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/PS2/doc.pdf b/PS2/doc.pdf index 6f28268..7c2e79e 100644 Binary files a/PS2/doc.pdf and b/PS2/doc.pdf differ diff --git a/PS2/doc.synctex.gz b/PS2/doc.synctex.gz index 7b66400..bc94896 100644 Binary files a/PS2/doc.synctex.gz and b/PS2/doc.synctex.gz differ diff --git a/PS2/doc.tex b/PS2/doc.tex index a512529..7dae119 100644 --- a/PS2/doc.tex +++ b/PS2/doc.tex @@ -389,9 +389,83 @@ H(N, 1) = -t0; \item %2b Homogenous boundary conditions imply that the corner entries of \(\hat{H}\) will be 0. \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 - \lipsum[1] + For the 1s level, \(E = -13.4978\) eV. \item %1e \lipsum[1] \item %1f diff --git a/PS2/q2c.m b/PS2/q2c.m index 9b69784..d1e40d1 100644 --- a/PS2/q2c.m +++ b/PS2/q2c.m @@ -41,9 +41,6 @@ phi_2 = eigenvectors(:,2); P_1 = phi_1 .* conj(phi_1); 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 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]'); 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]); - -%{ -% 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'); -%} - +axis([0 1e-9 0 0.04]); \ No newline at end of file diff --git a/PS2/q2f_fig.png b/PS2/q2f_fig.png new file mode 100644 index 0000000..1f36dce Binary files /dev/null and b/PS2/q2f_fig.png differ