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pkirwin 2021-03-02 10:28:25 -07:00
parent 0ad1f62bed
commit f6b711a214
8 changed files with 249 additions and 94 deletions
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7
PS2/doc.aux
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\relax \relax
\providecommand \oddpage@label [2]{}
\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}} \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
\newlabel{fig:q3_ne}{{1a}{3}} \newlabel{fig:q3_ne}{{1a}{3}}
\newlabel{sub@fig:q3_ne}{{a}{3}} \newlabel{sub@fig:q3_ne}{{a}{3}}
@ -12,6 +13,6 @@
\newlabel{eq:Riemann_sum}{{6}{4}} \newlabel{eq:Riemann_sum}{{6}{4}}
\newlabel{eq:numerical_E}{{7}{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 {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 {5}{\ignorespaces Sketch of degenerate 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 } \@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Numerical result (black line) and analytical solution scaled by \(a\) (orange circles).\relax }}{11}{}\protected@file@percent }
\gdef \@abspage@last{13} \gdef \@abspage@last{14}

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@ -17,6 +17,7 @@
\usepackage{enumitem} \usepackage{enumitem}
\usepackage{cancel} \usepackage{cancel}
\usepackage{siunitx} \usepackage{siunitx}
\usepackage{adjustbox}
\titleformat{\subsection}[runin]{\normalfont \large \bfseries} \titleformat{\subsection}[runin]{\normalfont \large \bfseries}
{\thesubsection}{1em}{} {\thesubsection}{1em}{}
@ -77,7 +78,7 @@
\thispagestyle{empty} \thispagestyle{empty}
\singlespacing \singlespacing
\newpage \newpage
hello
\section*{Problem 1} \section*{Problem 1}
\begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)] \begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)]
@ -203,7 +204,7 @@ legend({'Numerical','Analytical'},'Location','northwest');
From this, we find: From this, we find:
\begin{equation*} \begin{equation*}
A = \sqrt{\frac{2}{L}} \boxed{A = \sqrt{\frac{2}{L}}}
\end{equation*} \end{equation*}
\item %1bii \item %1bii
@ -240,7 +241,7 @@ legend({'Numerical','Analytical'},'Location','northwest');
This means that B must be This means that B must be
\begin{equation*} \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{equation*}
\end{enumerate} \end{enumerate}
@ -268,20 +269,20 @@ legend({'Numerical','Analytical'},'Location','northwest');
With some minor rearranging, this leaves us with the final expression for $E$: With some minor rearranging, this leaves us with the final expression for $E$:
\begin{equation} \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} \label{eq:numerical_E}
\end{equation} \end{equation}
\item %1cii \item %1cii
We can see here that the "predicted" numerical response matches nearly exactly the actual
calculated numerical solution.
\begin{figure}[H] \begin{figure}[H]
\centering \centering
\includegraphics[width=\textwidth]{q1cii.png} \includegraphics[width=\textwidth]{q1cii.png}
\caption{Analytic solution to numeric system, plotted.} \caption{Analytic solution to numeric system, plotted.}
\end{figure} \end{figure}
We can see here that the "predicted" numerical response matches nearly exactly the actual
calculated numerical solution.
\item %1ciii \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) E = 2 t_0 \left( \frac{n^2 \pi^2 a^2}{2 L^2} \right)
\end{equation*} \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*} \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*} \end{equation*}
\item %1civ \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] \begin{figure}[H]
\centering \centering
\begin{subfigure}{0.5\textwidth} \begin{subfigure}{0.5\textwidth}
@ -314,11 +319,7 @@ legend({'Numerical','Analytical'},'Location','northwest');
\end{subfigure} \end{subfigure}
\end{figure} \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} \end{enumerate}
\item %1d \item %1d
@ -350,17 +351,19 @@ H(N, 1) = -t0;
\item %1dii \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 These eigenstates are degenerate because they both have the same
eigenvalue/energy (0.06). eigenvalue/energy.
\item %1diii \item %1diii
\begin{figure}[H] \begin{minipage}[t]{\linewidth}
\centering \centering
\includegraphics[width=\textwidth, angle=-90]{q1div.jpg} \adjustbox{valign=t}{
\caption{Sketch of appropriate energy levels} \includegraphics[width=\textwidth]{q1div.jpg}
\end{figure} }
\captionof{figure}{Sketch of degenerate energy levels.}
\end{minipage}
\item %1div \item %1div
@ -374,7 +377,7 @@ H(N, 1) = -t0;
\end{enumerate} \end{enumerate}
\newpage
\section*{Problem 2} \section*{Problem 2}
\begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)] \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} 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 + \frac{l_o(l_o + 1)\hbar^2}{2mr_l^2}\), the middle diagonal elements will have values
\begin{equation*} \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*} \end{equation*}
and the upper and lower diagonals elements will have values and the upper and lower diagonals elements will have values
\begin{equation*} \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*} \end{equation*}
\item %2b \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 \item %2c
Code: Code:
\begin{lstlisting} \begin{lstlisting}
@ -468,7 +471,7 @@ axis([0 1e-9 0 0.04]);
\end{subfigure} \end{subfigure}
\end{figure} \end{figure}
\item %2d \item %2d
For the 1s level, \(E = -13.4978\) eV. For the 1s level, \boxed{E = -13.4978\:\mathrm{eV}}.
\item %2e \item %2e
Beginning with equation (11) from the assignment, with \(l_o=0\): Beginning with equation (11) from the assignment, with \(l_o=0\):
\begin{align*} \begin{align*}
@ -487,7 +490,7 @@ axis([0 1e-9 0 0.04]);
We can then solve for \(E\): We can then solve for \(E\):
\begin{equation*} \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} 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*} \end{equation*}
This is very similar to the result in (d). This is very similar to the result in (d).
@ -506,6 +509,7 @@ axis([0 1e-9 0 0.04]);
\end{enumerate} \end{enumerate}
\newpage
\section*{Problem 3} \section*{Problem 3}
\begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)] \begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)]
@ -540,7 +544,7 @@ axis([0 1e-9 0 0.04]);
\lipsum[1] \lipsum[1]
\end{enumerate} \end{enumerate}
\newpage
\section*{Problem 4} \section*{Problem 4}
\begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)] \begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)]

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