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Finished abstract and conclusion.

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Janita Willumsen
2023-12-30 15:09:34 +01:00
parent cb06e1af96
commit 36b91e5206
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latex/sections/appendices.tex
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@@ -25,10 +25,10 @@ We need the first derivative in respect to both time and position, as well as th
Schrödinger contain $i$ at the lhs, factor it as
\begin{align*}
\frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} &= \frac{1}{2i} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \\
&= -\frac{i}{2} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] & \text{, where $\frac{1}{i} = -i$}
&= -\frac{i}{2} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \ ,
\end{align*}
Using Equation \eqref{eq:schrodinger_dimensionless}, we get
where $1/i = -i$. Using Equation \eqref{eq:schrodinger_dimensionless}, which is
found in Section \ref{ssec:schrodinger}, we get
\begin{align*}
u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n} & -\frac{i \Delta t}{2} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \\
&= -\frac{i \Delta t}{2} \bigg[ - \frac{u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1}}{2 \Delta x^{2}} \\
@@ -94,4 +94,55 @@ $(M-2)^{2} \times (M-2)^{2} = 9 \times 9$ given by
0 & 0 & 0 & 0 & 0 & r & 0 & r & b_{8} \\
\end{bmatrix}
\end{align*}
\section{Figures}\label{ap:figures}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{images/color_map_0_real.pdf}
\caption{Re$(u_{\ivec, \jvec})$ at time $t=0$.}
\label{fig:colormap_0_real}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{images/color_map_1_real.pdf}
\caption{Re$(u_{\ivec, \jvec})$ at time $t=0.001$.}
\label{fig:colormap_1_real}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{images/color_map_2_real.pdf}
\caption{Re$(u_{\ivec, \jvec})$ at time $t=0.002$.}
\label{fig:colormap_2_real}
\end{subfigure}
\newline
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{images/color_map_0_imag.pdf}
\caption{Im$(u_{\ivec, \jvec})$ at time $t=0$.}
\label{fig:colormap_0_imag}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{images/color_map_1_imag.pdf}
\caption{Im$(u_{\ivec, \jvec})$ at time $t=0.001$.}
\label{fig:colormap_1_imag}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{images/color_map_2_imag.pdf}
\caption{Im$(u_{\ivec, \jvec})$ at time $t=0.002$.}
\label{fig:colormap_2_imag}
\end{subfigure}
\caption{The time evolution of the probability function $p_{\ivec, \jvec}^{n}$.}
\label{fig:colormap}
\end{figure*}
\end{document}