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pkirwin 2021-03-02 10:41:36 -07:00
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@ -152,23 +152,26 @@ legend({'Numerical','Analytical'},'Location','northwest');
\end{lstlisting} \end{lstlisting}
% %
\begin{minipage}[t]{\linewidth}
\begin{figure}[H] \begin{figure}[H]
\centering \centering
\begin{subfigure}{0.4\linewidth} \begin{subfigure}{0.5\linewidth}
\centering \centering
\includegraphics[width=\textwidth]{q1a_1and50.png} \includegraphics[width=\textwidth]{q1a_1and50.png}
\caption{} \caption{}
\label{fig:q3_ne} \label{fig:q3_ne}
\end{subfigure}% \end{subfigure}%
\begin{subfigure}{0.4\linewidth} \begin{subfigure}{0.5\linewidth}
\centering \centering
\includegraphics[width=\textwidth]{q1a_eigenvals.png} \includegraphics[width=\textwidth]{q1a_eigenvals.png}
\caption{Comparison of first 101 numerical and analytic eigenvalues.} \caption{}
\label{fig:q3_current} \label{fig:q3_current}
\end{subfigure} \end{subfigure}
\caption{(a) Probability densities for \(n=1\) and \(n=50\). \caption{(a) Probability densities for \(n=1\) and \(n=50\).
(b) Comparison of first 101 numerical and analytic eigenvalues.} (b) Comparison of first 101 numerical and analytic eigenvalues.}
\end{figure} \end{figure}
\end{minipage}
\item %1b \item %1b
\begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)] \begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)]
\item %1bi \item %1bi
@ -274,15 +277,18 @@ legend({'Numerical','Analytical'},'Location','northwest');
\end{equation} \end{equation}
\item %1cii \item %1cii
\begin{minipage}[t]{\linewidth}
\centering
\adjustbox{valign=t}{
\includegraphics[width=\textwidth]{q1cii.png}
}
\captionof{figure}{Analytic solution to numeric system, plotted.}
\end{minipage}
We can see here that the "predicted" numerical response matches nearly exactly the actual We can see here that the "predicted" numerical response matches nearly exactly the actual
calculated numerical solution. calculated numerical solution.
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{q1cii.png}
\caption{Analytic solution to numeric system, plotted.}
\end{figure}
\item %1ciii \item %1ciii
@ -305,6 +311,9 @@ legend({'Numerical','Analytical'},'Location','northwest');
density function appears to be "squeezed" in the centre, for the $n = 50$ case. The \(n=50\) case is 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 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. plot in section (a), which has a low-frequency envelope around it.
\begin{minipage}[t]{\linewidth}
\begin{figure}[H] \begin{figure}[H]
\centering \centering
\begin{subfigure}{0.5\textwidth} \begin{subfigure}{0.5\textwidth}
@ -318,7 +327,7 @@ legend({'Numerical','Analytical'},'Location','northwest');
\caption{Comparison of first 101 numerical and analytic eigenvalues.} \caption{Comparison of first 101 numerical and analytic eigenvalues.}
\end{subfigure} \end{subfigure}
\end{figure} \end{figure}
\end{minipage}
\end{enumerate} \end{enumerate}
@ -335,6 +344,8 @@ H(1, N) = -t0;
H(N, 1) = -t0; H(N, 1) = -t0;
\end{lstlisting} \end{lstlisting}
\begin{minipage}[t]{\linewidth}
\begin{figure}[H] \begin{figure}[H]
\centering \centering
\begin{subfigure}{0.5\textwidth} \begin{subfigure}{0.5\textwidth}
@ -348,6 +359,7 @@ H(N, 1) = -t0;
\caption{Comparison of first 101 numerical and analytic eigenvalues.} \caption{Comparison of first 101 numerical and analytic eigenvalues.}
\end{subfigure} \end{subfigure}
\end{figure} \end{figure}
\end{minipage}
\item %1dii \item %1dii
@ -456,6 +468,7 @@ yticks([0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04]);
legend('n=2'); legend('n=2');
axis([0 1e-9 0 0.04]); axis([0 1e-9 0 0.04]);
\end{lstlisting} \end{lstlisting}
\begin{minipage}[t]{\linewidth}
\begin{figure}[H] \begin{figure}[H]
\centering \centering
@ -470,6 +483,7 @@ axis([0 1e-9 0 0.04]);
\caption{2s probability density.} \caption{2s probability density.}
\end{subfigure} \end{subfigure}
\end{figure} \end{figure}
\end{minipage}
\item %2d \item %2d
For the 1s level, \boxed{E = -13.4978\:\mathrm{eV}}. For the 1s level, \boxed{E = -13.4978\:\mathrm{eV}}.
\item %2e \item %2e
@ -499,11 +513,11 @@ axis([0 1e-9 0 0.04]);
as discussed in Problem 1. From (d), we also expect agreement in the curve shapes because the numerical and analytical energies for the 1s as discussed in Problem 1. From (d), we also expect agreement in the curve shapes because the numerical and analytical energies for the 1s
level are very similiar. We can see that the peak value of the analytic result is very slightly higher than that of the numerical result, level are very similiar. We can see that the peak value of the analytic result is very slightly higher than that of the numerical result,
which corresponds to the analytical result for the energy being slightly greater in magnitude (\(-13.6\) eV versus \(-13.4978\) eV). which corresponds to the analytical result for the energy being slightly greater in magnitude (\(-13.6\) eV versus \(-13.4978\) eV).
\begin{figure}[H] \begin{minipage}[t]{\linewidth}
\centering \centering
\includegraphics[width=\textwidth]{q2f_fig.png} \includegraphics[width=\textwidth]{q2f_fig.png}
\caption{Numerical result (black line) and analytical solution scaled by \(a\) (orange circles).} \captionof{figure}{Numerical result (black line) and analytical solution scaled by \(a\) (orange circles).}
\end{figure} \end{minipage}