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 "" "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 +19,31 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmmi12.tfm" 1136765053 1524 4414a8315f39513458b80dfc63bff03a "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmmi6.tfm" 1136765053 1512 f21f83efb36853c0b70002322c1ab3ad "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmmi8.tfm" 1136765053 1520 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PDF statistics: - 50 PDF objects out of 1000 (max. 8388607) + 93 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..93ace2d 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..8676a1a 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..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 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