diff --git a/PS2/doc.aux b/PS2/doc.aux index d7e9143..87d0493 100644 --- a/PS2/doc.aux +++ b/PS2/doc.aux @@ -1,4 +1,5 @@ \relax +\providecommand \oddpage@label [2]{} \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}} \newlabel{fig:q3_ne}{{1a}{3}} \newlabel{sub@fig:q3_ne}{{a}{3}} @@ -12,6 +13,6 @@ \newlabel{eq:Riemann_sum}{{6}{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 } -\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Numerical result (black line) and analytical solution scaled by \(a\) (orange circles).\relax }}{10}{}\protected@file@percent } -\gdef \@abspage@last{13} +\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Sketch of degenerate 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PDF statistics: - 134 PDF objects out of 1000 (max. 8388607) + 137 PDF objects out of 1000 (max. 8388607) 0 named destinations out of 1000 (max. 500000) 56 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/PS2/doc.pdf b/PS2/doc.pdf index 2c4d1f3..bf2ab80 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 41d2003..cb47921 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 2f51ed3..3beef77 100644 --- a/PS2/doc.tex +++ b/PS2/doc.tex @@ -17,6 +17,7 @@ \usepackage{enumitem} \usepackage{cancel} \usepackage{siunitx} +\usepackage{adjustbox} \titleformat{\subsection}[runin]{\normalfont \large \bfseries} {\thesubsection}{1em}{} @@ -77,7 +78,7 @@ \thispagestyle{empty} \singlespacing \newpage - hello + \section*{Problem 1} \begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)] @@ -203,7 +204,7 @@ legend({'Numerical','Analytical'},'Location','northwest'); From this, we find: \begin{equation*} - A = \sqrt{\frac{2}{L}} + \boxed{A = \sqrt{\frac{2}{L}}} \end{equation*} \item %1bii @@ -240,7 +241,7 @@ legend({'Numerical','Analytical'},'Location','northwest'); This means that B must be \begin{equation*} - B = \sqrt{\frac{2 a}{L}} = \sqrt{a} \times A. + \boxed{B = \sqrt{\frac{2 a}{L}} = \sqrt{a} \times A.} \end{equation*} \end{enumerate} @@ -268,20 +269,20 @@ legend({'Numerical','Analytical'},'Location','northwest'); 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 a}{L} \right) \right) + \boxed{E = 2t_0 \left( 1 - cos \left( \frac{n \pi a}{L} \right) \right)} \label{eq:numerical_E} \end{equation} \item %1cii - + We can see here that the "predicted" numerical response matches nearly exactly the actual + calculated numerical solution. \begin{figure}[H] \centering \includegraphics[width=\textwidth]{q1cii.png} \caption{Analytic solution to numeric system, plotted.} \end{figure} - We can see here that the "predicted" numerical response matches nearly exactly the actual - calculated numerical solution. + \item %1ciii @@ -292,14 +293,18 @@ legend({'Numerical','Analytical'},'Location','northwest'); E = 2 t_0 \left( \frac{n^2 \pi^2 a^2}{2 L^2} \right) \end{equation*} - Fully substituting the known value of $t_0$, and we can get our final expression for $E$: + Fully substituting the known value of $t_0$, and we can get our final analytical expression for $E$: \begin{equation*} - E = \frac{ \hbar^2 n^2 \pi^2 }{ 2 m L^2 } + \boxed{E = \frac{ \hbar^2 n^2 \pi^2 }{ 2 m L^2 }} \end{equation*} \item %1civ - + With the decreased lattice spacing, and increased number of points, we can see the numerical solution + of eigenvalues matches the analytical solution much closer. As well, we can see that the probability + density function appears to be "squeezed" in the centre, for the $n = 50$ case. The \(n=50\) case is + now a constant-amplitude wave, which corresponds to the expected analytic result - in contrast to the + plot in section (a), which has a low-frequency envelope around it. \begin{figure}[H] \centering \begin{subfigure}{0.5\textwidth} @@ -314,11 +319,7 @@ legend({'Numerical','Analytical'},'Location','northwest'); \end{subfigure} \end{figure} - With the decreased lattice spacing, and increased number of points, we can see the numerical solution - of eigenvalues matches the analytical solution much closer. As well, we can see that the probability - density function appears to be "squeezed" in the centre, for the $n = 50$ case. The \(n=50\) case is - now a constant-amplitude wave, which corresponds to the expected analytic result - in contrast to the - plot in section (a), which has a low-frequency envelope around it. + \end{enumerate} \item %1d @@ -350,17 +351,19 @@ H(N, 1) = -t0; \item %1dii - The energy levels for eigenvalues number 4 and 5 are both $0.06eV$. + The energy levels for eigenvalues number 4 and 5 are both \boxed{0.06\:\mathrm{eV}}. These eigenstates are degenerate because they both have the same - eigenvalue/energy (0.06). + eigenvalue/energy. \item %1diii - \begin{figure}[H] + \begin{minipage}[t]{\linewidth} \centering - \includegraphics[width=\textwidth, angle=-90]{q1div.jpg} - \caption{Sketch of appropriate energy levels} - \end{figure} + \adjustbox{valign=t}{ + \includegraphics[width=\textwidth]{q1div.jpg} + } + \captionof{figure}{Sketch of degenerate energy levels.} + \end{minipage} \item %1div @@ -374,7 +377,7 @@ H(N, 1) = -t0; \end{enumerate} - + \newpage \section*{Problem 2} \begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)] @@ -383,14 +386,14 @@ H(N, 1) = -t0; Since \(t_0 = \frac{\hbar^2}{2ma^2}\), and \(U_l = -\frac{q^2}{4\pi\varepsilon_0r_l} + \frac{l_o(l_o + 1)\hbar^2}{2mr_l^2}\), the middle diagonal elements will have values \begin{equation*} - \hat{H}_{ll} = \frac{\hbar^2}{ma^2}-\frac{q^2}{4\pi\epsilon_0r_l} + \frac{l_o(l_o + 1)\hbar^2}{2mr_l^2}, + \boxed{\hat{H}_{ll} = \frac{\hbar^2}{ma^2}-\frac{q^2}{4\pi\epsilon_0r_l} + \frac{l_o(l_o + 1)\hbar^2}{2mr_l^2},} \end{equation*} and the upper and lower diagonals elements will have values \begin{equation*} - \hat{H}_{l(l\pm1)} = -\frac{\hbar^2}{2ma^2}. + \boxed{\hat{H}_{l(l\pm1)} = -\frac{\hbar^2}{2ma^2}.} \end{equation*} \item %2b - Homogenous boundary conditions imply that the corner entries of \(\hat{H}\) will be 0. + Homogenous boundary conditions imply that the corner entries of \(\hat{H}\) will be \boxed{0}. \item %2c Code: \begin{lstlisting} @@ -468,7 +471,7 @@ axis([0 1e-9 0 0.04]); \end{subfigure} \end{figure} \item %2d - For the 1s level, \(E = -13.4978\) eV. + For the 1s level, \boxed{E = -13.4978\:\mathrm{eV}}. \item %2e Beginning with equation (11) from the assignment, with \(l_o=0\): \begin{align*} @@ -487,7 +490,7 @@ axis([0 1e-9 0 0.04]); We can then solve for \(E\): \begin{equation*} E\:\mathrm{[eV]} = -\frac{1}{q}\cdot\frac{\hbar^2}{2ma_0^2} = -\frac{1}{q}\cdot\frac{(\SI{1.054e-34}{\joule\cdot\second})^2}{2(\SI{9.110e-31}{\kilogram})(\SI{0.0529}{\nm})^2} - = - 13.6\:\mathrm{[eV]} + = \boxed{- 13.6\:\mathrm{eV}.} \end{equation*} This is very similar to the result in (d). @@ -505,7 +508,8 @@ axis([0 1e-9 0 0.04]); \end{enumerate} - + + \newpage \section*{Problem 3} \begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)] @@ -540,7 +544,7 @@ axis([0 1e-9 0 0.04]); \lipsum[1] \end{enumerate} - + \newpage \section*{Problem 4} \begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)] diff --git a/PS2/q1div.jpg b/PS2/q1div.jpg index d366334..c2a9ad5 100644 Binary files a/PS2/q1div.jpg and b/PS2/q1div.jpg differ