diff --git a/PS2/doc.aux b/PS2/doc.aux index 3cb5a00..731303a 100644 --- a/PS2/doc.aux +++ b/PS2/doc.aux @@ -1,2 +1,13 @@ \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}} +\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 d4c7eaf..1f0c9a1 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"] 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" 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 "" + "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 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/symbols/msam5.tfm" 1233951854 924 9904cf1d39e9767e7a3622f2a125a565 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/symbols/msam7.tfm" 1233951854 928 2dc8d444221b7a635bb58038579b861a "" @@ -16,19 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1136765053 1292 21c1c5bfeaebccffdb478fd231a0997d "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmr9.tfm" 1136765053 1292 6b21b9c2c7bebb38aa2273f7ca0fb3af "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmsy10.tfm" 1136765053 1124 6c73e740cf17375f03eec0ee63599741 "" "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/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 "" + 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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 "" @@ -44,7 +60,9 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/amsmath/amsopn.sty" 1601804953 4233 32e6abd27229755a83a8b7f18e583890 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/amsmath/amstext.sty" 1601804953 2508 8ff93b1137020e8f21930562a874ae66 "" "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 "" @@ -77,13 +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 "" 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+ "q1a_1and50.png" 1614636283 34155 72f34c6abf061325ae1f162db0ffd06b "" + "q1a_eigenvals.png" 1614637266 23078 9c99a5505761d0b964e665e35fd9c02a "" (generated) - "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.pdf" + "doc.log" "doc.aux" diff --git a/PS2/doc.fls b/PS2/doc.fls index 46c6b0d..cbb5f36 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 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C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\listings\lstlang1.sty @@ -471,24 +482,71 @@ INPUT C:\Users\speed\AppData\Local\MiKTeX\pdftex\config\pdftex.map INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmr12.tfm INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmbx12.tfm INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmbx10.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\jknappen\ec\tcrm1000.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\base\omscmr.fd +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\base\omscmr.fd +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\base\omscmr.fd +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\base\omscmr.fd +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_1and50.png +INPUT 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PDF statistics: - 50 PDF objects out of 1000 (max. 8388607) + 104 PDF objects out of 1000 (max. 8388607) 0 named destinations out of 1000 (max. 500000) - 1 words of extra memory for PDF output out of 10000 (max. 10000000) + 11 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/PS2/doc.pdf b/PS2/doc.pdf index e398c14..b4853a9 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 564dc7c..b316ccf 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 d74219b..1e3993b 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}{} @@ -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}, @@ -81,28 +81,212 @@ \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} + + We use the following identity: + + \begin{equation} + \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 (\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 - \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: + + \begin{equation*} + A = \sqrt{\frac{2}{L}} + \end{equation*} + \item %1bii - \lipsum[1] + + Starting with the normalization condition for the numerical case: + \begin{align*} + a\sum_{l=1}^N|\phi_l|^2 = a \\ + a\sum_{l=1}^N|B\sin{(\frac{n\pi}{L}x_l)}|^2 = a, + \end{align*} + recalling that \(x = al\), and allowing \(a \to 0\) while holding \(L\) + 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. + \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 \begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)] \item %1ci - \lipsum[1] + + % 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 $B = \frac{n \pi \alpha}{L}$. + + 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} 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 0000000..4d33292 Binary files /dev/null and b/PS2/q1a_1and50.png differ diff --git a/PS2/q1a_eigenvals.png b/PS2/q1a_eigenvals.png new file mode 100644 index 0000000..9834520 Binary files /dev/null and b/PS2/q1a_eigenvals.png differ