From c046bc1b64aeb19569b80a3346a6be5665b985ac Mon Sep 17 00:00:00 2001 From: pkirwin Date: Mon, 1 Mar 2021 15:32:50 -0700 Subject: [PATCH 1/3] save --- PS2/doc.aux | 10 +++- PS2/doc.fdb_latexmk | 32 +++++++--- PS2/doc.fls | 48 ++++++++++++++- PS2/doc.log | 41 +++++++++---- PS2/doc.pdf | Bin 85838 -> 234173 bytes PS2/doc.synctex.gz | Bin 52018 -> 96971 bytes PS2/doc.tex | 134 ++++++++++++++++++++++++++++++++++++++++-- PS2/q1a.asv | 57 ++++++++++++++++++ PS2/q1a.m | 62 +++++++++++++++++-- PS2/q1a_1and50.png | Bin 0 -> 34155 bytes PS2/q1a_eigenvals.png | Bin 0 -> 23078 bytes 11 files changed, 353 insertions(+), 31 deletions(-) create mode 100644 PS2/q1a.asv create mode 100644 PS2/q1a_1and50.png create mode 100644 PS2/q1a_eigenvals.png diff --git a/PS2/doc.aux b/PS2/doc.aux index 3cb5a00..ee6ac14 100644 --- a/PS2/doc.aux +++ b/PS2/doc.aux @@ -1,2 +1,10 @@ \relax -\gdef \@abspage@last{7} +\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}} +\newlabel{fig:q3_ne}{{1a}{3}} +\newlabel{sub@fig:q3_ne}{{a}{3}} +\newlabel{fig:q3_current}{{1b}{3}} +\newlabel{sub@fig:q3_current}{{b}{3}} +\newlabel{eq:analytical}{{1}{3}} +\newlabel{eq:normalise}{{2}{3}} +\newlabel{eq:integral_ident}{{3}{3}} +\gdef \@abspage@last{8} diff --git a/PS2/doc.fdb_latexmk b/PS2/doc.fdb_latexmk index d4c7eaf..c054bc6 100644 --- a/PS2/doc.fdb_latexmk +++ b/PS2/doc.fdb_latexmk @@ -1,9 +1,12 @@ # Fdb version 3 -["pdflatex"] 1614634922 "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" 1614634924 +["pdflatex"] 1614637883 "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" 1614637886 + "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" 1614626913 80909 eab91d9745dd2edfd62a31d53cd5fe15 "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/jknappen/ec/tcrm1000.tfm" 993058908 1436 c7f957a372ef2fbe93c0982f96625e12 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1233951848 1004 54797486969f23fa377b128694d548df "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1233951848 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/cmextra/cmex9.tfm" 1234365220 996 a18840b13b499c08ac2de96a99eda4bc "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1233951854 916 f87d7c45f9c908e672703b83b72241a3 "" 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RhAvQrDzj@&ZRj%Y_&?$DKGFaH diff --git a/PS2/doc.tex b/PS2/doc.tex index d74219b..53c08b8 100644 --- a/PS2/doc.tex +++ b/PS2/doc.tex @@ -46,7 +46,7 @@ numbersep=9pt, % this defines how far the numbers are from the text emph=[1]{for,end,break},emphstyle=[1]\color{red}, %some words to emphasise %emph=[2]{word1,word2}, emphstyle=[2]{style}, - xleftmargin=5em + xleftmargin=2em } %\lstset{basicstyle=\small, % keywordstyle=\color{mauve}, @@ -81,18 +81,144 @@ \begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)] \item %1a - \lipsum[1] + Code: + \begin{lstlisting}[language=Matlab] +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'); + + \end{lstlisting} + % + \begin{figure}[H] + \centering + \begin{subfigure}{0.4\linewidth} + \centering + \includegraphics[width=\textwidth]{q1a_1and50.png} + \caption{probability densities for \(n=1\)\\and \(n=50\).} + \label{fig:q3_ne} + \end{subfigure}% + \begin{subfigure}{0.4\linewidth} + \centering + \includegraphics[width=\textwidth]{q1a_eigenvals.png} + \caption{Comparison of first 101 numerical and analytic eigenvalues.} + \label{fig:q3_current} + \end{subfigure} + \end{figure} \item %1b \begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)] \item %1bi The analytical solution is \begin{equation} - \Phi(x) = A sin (\frac{n \pi}{L} x) + \Phi(x) = A sin \left( \frac{n \pi}{L} x \right) + \label{eq:analytical} \end{equation} + + In order to normalise this equation it must conform to the following: + + \begin{equation} + \int_{0}^{L} \left| \Phi(x) \right| ^2 dx = 1 + \label{eq:normalise} + \end{equation} + + Using the following identity: + + \begin{equation} + foo + \label{eq:integral_ident} + \end{equation} + + Given that $sin$ of a real value is always real, we can disregard the norm operation, and directly relate to the above identity. + Evaluating the integral gives us the following relationship + + \begin{equation*} + \frac{1}{A^2} = \frac{1}{2} L - \frac{L}{4n \pi} sin \left( \frac{2n \pi}{L} L \right) - \frac{1}{2} \cdot 0 + \frac{L}{4n \pi} sin(0) + \end{equation*} + + From this, we find: + + \begin{equation*} + A = \sqrt{\frac{2}{L}} + \end{equation*} + \item %1bii - \lipsum[1] + + {\em I really don't like this explanation}. + + The output of the numerical sum will change, as the value of $\alpha$ changes, by a factor if $\alpha$. + To correct for this, we need to factor the result of the summation by $\alpha$, which corresponds to + + % Idk how to explain this at all, I need to chew on it internally and come back later. + \begin{equation*} + foo + \end{equation*} + + This means that B must be + \begin{equation*} + B = \sqrt{\frac{2 \alpha}{L}} + \end{equation*} + \end{enumerate} \item %1c diff --git a/PS2/q1a.asv b/PS2/q1a.asv new file mode 100644 index 0000000..094029d --- /dev/null +++ b/PS2/q1a.asv @@ -0,0 +1,57 @@ +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) / ( +% 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]); + diff --git a/PS2/q1a.m b/PS2/q1a.m index 8885bd2..87df436 100644 --- a/PS2/q1a.m +++ b/PS2/q1a.m @@ -1,4 +1,4 @@ - +clear all; %physical constants in MKS units hbar = 1.054e-34; @@ -7,7 +7,59 @@ m = 9.110e-31; %generate lattice -N = 100; %number of lattice points -n = [1:N]; %lattice points -a = 1e-10; %lattice constant -x = a * [1:1:N]; \ No newline at end of file +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'); + + diff --git a/PS2/q1a_1and50.png b/PS2/q1a_1and50.png new file mode 100644 index 0000000000000000000000000000000000000000..4d332928870dc0677c8069ee18e94e214b79ecc9 GIT binary patch literal 34155 zcmagG1ys~u^gTL&bVzr%G)PHGgGfokfP_dXAT8a}ozmTnFm$)1q;yF)NaO$V{k{MD 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a/PS2/doc.tex b/PS2/doc.tex index 53c08b8..db73568 100644 --- a/PS2/doc.tex +++ b/PS2/doc.tex @@ -14,8 +14,8 @@ \usepackage{fancyhdr} \usepackage{titlesec} \usepackage{lipsum} - \usepackage{enumitem} +\usepackage{cancel} \titleformat{\subsection}[runin]{\normalfont \large \bfseries} {\thesubsection}{1em}{} @@ -171,29 +171,30 @@ legend({'Numerical','Analytical'},'Location','northwest'); The analytical solution is \begin{equation} - \Phi(x) = A sin \left( \frac{n \pi}{L} x \right) + \phi(x) = A sin \left( \frac{n \pi}{L} x \right) \label{eq:analytical} \end{equation} In order to normalise this equation it must conform to the following: \begin{equation} - \int_{0}^{L} \left| \Phi(x) \right| ^2 dx = 1 + \int_{0}^{L} \left| \phi(x) \right| ^2 dx = 1. \label{eq:normalise} \end{equation} - Using the following identity: + We use the following identity: \begin{equation} - foo + \int\sin^2{(\alpha x)}\:dx = \frac{1}{2} x -\frac{1}{4\alpha}\sin{(2\alpha x)}. \label{eq:integral_ident} \end{equation} - Given that $sin$ of a real value is always real, we can disregard the norm operation, and directly relate to the above identity. + Given that $sin$ of a real value is always real, we can disregard the norm operation, and directly relate (\ref{eq:analytical}) + to the above identity. Evaluating the integral gives us the following relationship \begin{equation*} - \frac{1}{A^2} = \frac{1}{2} L - \frac{L}{4n \pi} sin \left( \frac{2n \pi}{L} L \right) - \frac{1}{2} \cdot 0 + \frac{L}{4n \pi} sin(0) + \frac{1}{A^2} = \frac{1}{2} L - \cancel{\frac{L}{4n \pi} sin \left( \frac{2n \pi}{L} L \right)} - \cancel{\frac{1}{2} \cdot 0} + \cancel{\frac{L}{4n \pi} sin(0)} \end{equation*} From this, we find: @@ -224,7 +225,15 @@ legend({'Numerical','Analytical'},'Location','northwest'); \item %1c \begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)] \item %1ci - \lipsum[1] + + From the base form of $\phi _\ell = Bsin \left( \frac {n \pi}{L} a \ell \right)$, we can see that + $\phi _ {\ell + 1}$ and $\phi _ {\ell - 1}$ correspond to the trigonometric identities + $sin(\alpha + B) = sin(\alpha)cos(B) + cos(\alpha)sin(B)$ and + $sin(\alpha + B) = sin(\alpha)cos(B) + cos(\alpha)sin(B)$, respectively, where + $\alpha = \frac{n \pi \alpha \ell}{L}$ and $B = \frac{n \pi \ell}{L}$. + + Fro + \item %1cii \lipsum[1] \item %1ciii From 84cce755528c6bbbc1dfefe30b9f1d9b065ed7c9 Mon Sep 17 00:00:00 2001 From: pkirwin Date: Mon, 1 Mar 2021 16:18:33 -0700 Subject: [PATCH 3/3] commit --- PS2/doc.aux | 5 +++- PS2/doc.fdb_latexmk | 17 ++++++------ PS2/doc.fls | 5 ++-- PS2/doc.log | 18 ++++++------ PS2/doc.pdf | Bin 248293 -> 244757 bytes PS2/doc.synctex.gz | Bin 97998 -> 101941 bytes PS2/doc.tex | 65 ++++++++++++++++++++++++++++++++++++++------ 7 files changed, 80 insertions(+), 30 deletions(-) diff --git a/PS2/doc.aux b/PS2/doc.aux index ee6ac14..731303a 100644 --- a/PS2/doc.aux +++ b/PS2/doc.aux @@ -7,4 +7,7 @@ \newlabel{eq:analytical}{{1}{3}} \newlabel{eq:normalise}{{2}{3}} \newlabel{eq:integral_ident}{{3}{3}} -\gdef \@abspage@last{8} +\newlabel{eq:numerical_E}{{4}{4}} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Analytic solution to numeric system, plotted.\relax }}{4}{}\protected@file@percent } +\newlabel{fig:q1b_electrons}{{2}{4}} +\gdef \@abspage@last{9} diff --git a/PS2/doc.fdb_latexmk b/PS2/doc.fdb_latexmk index 82849c5..1f0c9a1 100644 --- a/PS2/doc.fdb_latexmk +++ b/PS2/doc.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1614638543 "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" 1614638561 +["pdflatex"] 1614640686 "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" 1614640702 "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 "" @@ -29,7 +29,6 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmsy6.tfm" 1136765053 1116 933a60c408fc0a863a92debe84b2d294 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmsy8.tfm" 1136765053 1120 8b7d695260f3cff42e636090a8002094 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmsy9.tfm" 1136765053 1116 25a7bf822c58caf309a702ef79f4afbb "" - "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmti10.tfm" 1136765053 1480 aa8e34af0eb6a2941b776984cf1dfdc4 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmbx10.pfb" 1247593066 34811 78b52f49e893bcba91bd7581cdc144c0 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmbx12.pfb" 1247593066 32080 340ef9bf63678554ee606688e7b5339d "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1247593067 30251 6afa5cb1d0204815a708a080681d4674 "" @@ -44,8 +43,8 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr9.pfb" 1247593066 33993 9b89b85fd2d9df0482bd47194d1d3bf3 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1247593067 32569 5e5ddc8df908dea60932f3c484a54c0d "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy7.pfb" 1247593066 32716 08e384dc442464e7285e891af9f45947 "" - "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmti10.pfb" 1247593067 37944 359e864bd06cde3b1cf57bb20757fb06 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/latxfont/line10.pfb" 1247593053 11493 4f5ed183a47d3197cf8cd322325db6de "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/symbols/msbm10.pfb" 1247593073 34694 870c211f62cb72718a00e353f14f254d "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/context/base/mkii/supp-pdf.mkii" 1580390158 71627 94eb9990bed73c364d7f53f960cc8c5b "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/generic/iftex/iftex.sty" 1583523400 6501 4011d89d9621e0b0901138815ba5ff29 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/generic/iftex/ifvtex.sty" 1583523400 1057 525c2192b5febbd8c1f662c9468335bb "" @@ -96,15 +95,15 @@ "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" 1614638561 334 906353ae3fc5daad100ce8b921730ac7 "" - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" 1614638542 10889 00e896f1f98c88bd40320d9e83a777cb "" - "doc.aux" 1614638561 334 906353ae3fc5daad100ce8b921730ac7 "pdflatex" - "doc.tex" 1614638542 10889 00e896f1f98c88bd40320d9e83a777cb "" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.aux" 1614640688 560 a7f45c1ba582090fea381768d02972d1 "" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" 1614640701 13478 c506f224fad814ac5407ed9564ca556b "" + "doc.aux" 1614640688 560 a7f45c1ba582090fea381768d02972d1 "pdflatex" + "doc.tex" 1614640701 13478 c506f224fad814ac5407ed9564ca556b "" "q1a_1and50.png" 1614636283 34155 72f34c6abf061325ae1f162db0ffd06b "" "q1a_eigenvals.png" 1614637266 23078 9c99a5505761d0b964e665e35fd9c02a "" (generated) + "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.pdf" "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.log" diff --git a/PS2/doc.fls b/PS2/doc.fls index b8cecb6..cbb5f36 100644 --- a/PS2/doc.fls +++ b/PS2/doc.fls @@ -515,7 +515,6 @@ 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\q1a_eigenvals.png INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_eigenvals.png INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_eigenvals.png -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmti10.tfm 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 @@ -547,7 +546,7 @@ INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\c INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy10.pfb INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy7.pfb INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy7.pfb -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmti10.pfb -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmti10.pfb INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\latxfont\line10.pfb INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\latxfont\line10.pfb +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\symbols\msbm10.pfb +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\symbols\msbm10.pfb diff --git a/PS2/doc.log b/PS2/doc.log index 2dd77cd..def3426 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 15:42 +This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 21.1) (preloaded format=pdflatex 2021.2.6) 1 MAR 2021 16:18 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 @@ -332,19 +332,19 @@ File: q1a_eigenvals.png Graphic file (type png) Package pdftex.def Info: q1a_eigenvals.png used on input line 163. 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+ constant implies that \(N \to \infty\), since \(a=\frac{L}{N}\). + An integral is defined as the limit of a Riemann sum as follows: + \begin{equation*} + \int_a^bf(x)\:dx = \lim_{n \to \infty}\sum_{i=1}^n\Delta x\cdot f(x_i). + \end{equation*} + + + The output of the numerical sum will change, as the value of $\alpha$ changes, by a factor of $\alpha$. To correct for this, we need to factor the result of the summation by $\alpha$, which corresponds to % Idk how to explain this at all, I need to chew on it internally and come back later. @@ -226,18 +237,56 @@ legend({'Numerical','Analytical'},'Location','northwest'); \begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)] \item %1ci - From the base form of $\phi _\ell = Bsin \left( \frac {n \pi}{L} a \ell \right)$, we can see that + % TODO check that B is expressed correctly in all these equations (alpha, not ell, wait for pdf gen) + From the base form of $\phi _\ell = Bsin \left( \frac {n \pi}{L} \alpha \ell \right)$, we can see that $\phi _ {\ell + 1}$ and $\phi _ {\ell - 1}$ correspond to the trigonometric identities $sin(\alpha + B) = sin(\alpha)cos(B) + cos(\alpha)sin(B)$ and $sin(\alpha + B) = sin(\alpha)cos(B) + cos(\alpha)sin(B)$, respectively, where - $\alpha = \frac{n \pi \alpha \ell}{L}$ and $B = \frac{n \pi \ell}{L}$. + $\alpha = \frac{n \pi \alpha \ell}{L}$ and $B = \frac{n \pi \alpha}{L}$. - Fro + Plugging these identities into equation (7) from the assignment and simplifying, we get to this equation: + + \begin{equation*} + -t_0 B sin \left( \frac{n \pi \alpha \ell}{L} \right) + 2 t_0 \phi _{\ell} - t_0 sin \left( \frac{n \pi \alpha \ell}{L} \right) + \end{equation*} + + At this point, we notice that $\phi _ \ell = B sin \left( \frac {n \pi}{L} \alpha \ell \right)$, + so we can factor it out. + + With some minor rearranging, this leaves us with the final expression for $E$: + + \begin{equation} + E = 2t_0 \left( 1 - cos \left( \frac{n \pi \alpha}{L} \right) \right) + \label{eq:numerical_E} + \end{equation} \item %1cii - \lipsum[1] + + \begin{figure}[H] + \centering + \includegraphics[width=\textwidth]{q1cii.png} + \caption{Analytic solution to numeric system, plotted.} + \label{fig:q1b_electrons} + \end{figure} + + We can see here that the "predicted" numerical response matches nearly exactly the actual + calculated numerical solution. + \item %1ciii - \lipsum[1] + + Applying the approximation $cos(\Theta) = 1 - \frac{\Theta^2}{2}$ for small $\Theta$, + on equation (\ref{eq:numerical_E}), we get the following expression + + \begin{equation*} + E = 2 t_0 \left( \frac{n^2 \pi^2 \alpha^2}{2 L^2} \right) + \end{equation*} + + Fully substituting the known value of $t_0$, and we can get our final expression for $E$: + + \begin{equation*} + E = \frac{ \hbar^2 n^2 \pi^2 }{ 2 m L^2 } + \end{equation*} + \item %1civ \lipsum[1] \end{enumerate}

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Mon Sep 17 00:00:00 2001 From: pkirwin Date: Mon, 1 Mar 2021 15:43:09 -0700 Subject: [PATCH 2/3] commit --- PS2/doc.fdb_latexmk | 19 +++++++++++-------- PS2/doc.fls | 15 +++++++++++++++ PS2/doc.log | 18 ++++++++++-------- PS2/doc.pdf | Bin 234173 -> 248293 bytes PS2/doc.synctex.gz | Bin 96971 -> 97998 bytes PS2/doc.tex | 25 +++++++++++++++++-------- 6 files changed, 53 insertions(+), 24 deletions(-) diff --git a/PS2/doc.fdb_latexmk b/PS2/doc.fdb_latexmk index c054bc6..82849c5 100644 --- a/PS2/doc.fdb_latexmk +++ b/PS2/doc.fdb_latexmk @@ -1,8 +1,8 @@ # Fdb version 3 -["pdflatex"] 1614637883 "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" 1614637886 +["pdflatex"] 1614638543 "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" 1614638561 "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" 1614626913 80909 eab91d9745dd2edfd62a31d53cd5fe15 "" + "C:/Users/speed/AppData/Local/MiKTeX/pdftex/config/pdftex.map" 1614638559 80909 eab91d9745dd2edfd62a31d53cd5fe15 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/jknappen/ec/tcrm1000.tfm" 993058908 1436 c7f957a372ef2fbe93c0982f96625e12 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1233951848 1004 54797486969f23fa377b128694d548df "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1233951848 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -43,7 +43,9 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr7.pfb" 1247593067 32762 7fee39e011c23b3589931effd97b9702 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr9.pfb" 1247593066 33993 9b89b85fd2d9df0482bd47194d1d3bf3 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1247593067 32569 5e5ddc8df908dea60932f3c484a54c0d "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy7.pfb" 1247593066 32716 08e384dc442464e7285e891af9f45947 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmti10.pfb" 1247593067 37944 359e864bd06cde3b1cf57bb20757fb06 "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/latxfont/line10.pfb" 1247593053 11493 4f5ed183a47d3197cf8cd322325db6de "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/context/base/mkii/supp-pdf.mkii" 1580390158 71627 94eb9990bed73c364d7f53f960cc8c5b "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/generic/iftex/iftex.sty" 1583523400 6501 4011d89d9621e0b0901138815ba5ff29 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/generic/iftex/ifvtex.sty" 1583523400 1057 525c2192b5febbd8c1f662c9468335bb "" @@ -61,6 +63,7 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/base/article.cls" 1610120746 20145 aad8c3dd3bc36e260347b84002182bc2 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/base/omscmr.fd" 1610120746 2470 59e43b502a8bea3a12f4954945302e40 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/base/size10.clo" 1610120746 8449 a72d5d4e612221b46000c3d71724e0ef "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/cancel/cancel.sty" 1388370247 7592 dd751af313a16a0308545d5bfd7aaaa2 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/caption/caption.sty" 1603736996 51746 e89c4da670ba533e6ab38e045ce6d1d9 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/caption/caption3.sty" 1603736996 67929 d50ae850e1d9604065e8337a75a41029 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/caption/subcaption.sty" 1603736996 6133 46bea384e36e9b74bd53a238fc2d0654 "" @@ -93,15 +96,15 @@ "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" 1614637886 334 906353ae3fc5daad100ce8b921730ac7 "" - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" 1614637882 10279 f262bef4de2ea276f6d0acd4c1d77d63 "" - "doc.aux" 1614637886 334 906353ae3fc5daad100ce8b921730ac7 "pdflatex" - "doc.tex" 1614637882 10279 f262bef4de2ea276f6d0acd4c1d77d63 "" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.aux" 1614638561 334 906353ae3fc5daad100ce8b921730ac7 "" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" 1614638542 10889 00e896f1f98c88bd40320d9e83a777cb "" + "doc.aux" 1614638561 334 906353ae3fc5daad100ce8b921730ac7 "pdflatex" + "doc.tex" 1614638542 10889 00e896f1f98c88bd40320d9e83a777cb "" "q1a_1and50.png" 1614636283 34155 72f34c6abf061325ae1f162db0ffd06b "" "q1a_eigenvals.png" 1614637266 23078 9c99a5505761d0b964e665e35fd9c02a "" (generated) - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf" - "doc.log" "doc.pdf" + "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.log" diff --git a/PS2/doc.fls b/PS2/doc.fls index fe1cd4c..b8cecb6 100644 --- a/PS2/doc.fls +++ b/PS2/doc.fls @@ -416,6 +416,17 @@ INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\enumitem\enumitem.s INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\enumitem\enumitem.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\enumitem\enumitem.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\enumitem\enumitem.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\listings\lstlang1.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\listings\lstlang1.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\listings\lstlang1.sty @@ -534,5 +545,9 @@ INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\c INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmr9.pfb INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy10.pfb INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy10.pfb +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy7.pfb +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy7.pfb INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmti10.pfb INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmti10.pfb +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\latxfont\line10.pfb +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\latxfont\line10.pfb diff --git a/PS2/doc.log b/PS2/doc.log index d1e4495..2dd77cd 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 15:31 +This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 21.1) (preloaded format=pdflatex 2021.2.6) 1 MAR 2021 15:42 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 @@ -230,6 +230,8 @@ Package: enumitem 2019/06/20 v3.9 Customized lists \enit@inbox=\box55 \enit@count@id=\count281 \enitdp@description=\count282 +) (C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/latex/cancel\cancel.sty +Package: cancel 2013/04/12 v2.2 Cancel math terms ) (C:\Users\speed\AppData\Local\Programs\MiKTeX\tex/latex/listings\lstlang1.sty File: lstlang1.sty 2020/03/24 1.8d listings language file ) (d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.aux) @@ -332,17 +334,17 @@ Package pdftex.def Info: q1a_eigenvals.png used on input line 163. 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