\documentclass[../schrodinger_simulation.tex]{subfiles}

\begin{document}
\section{Results}\label{sec:results}
\subsection{Deviation}\label{ssec:deviation}
% Problem 3: Discuss approaches to solve Au^{n+1} = b, dealing with sparse matrix...
We used the \verb|superlu| solver, which is a solver for sparse matrices. It is 
generally used to solve nonsymmetric, sparse matrices. However, as the alternative 
solver \verb|lapack| converts a sparse matrix to a dense matrix, it will increase 
memory usage compared to \verb|superlu|.
% Problem 7: Consequenses of solver choice, in regards to accuracy of probability conserved
%            Add plot of deviation for both single- and double-slit
Since we used a solver for sparse matrices, we decrease the number of computations performed
compared to number of computations using a solver for dense matrices. 
We checked if the total probability was conserved over time, by plotting the deviation 
from $1.0$.
\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{images/probability_deviation.pdf}
    \caption{Deviation of total probability, for time $t \in [0, T]$ where $T=0.008$.}
    \label{fig:deviation}
\end{figure}
In Figure \ref{fig:deviation}, we observe a larger deviation of total probability 
for a barrier with double slits compared to no barrier. Interaction with the 
barrier result in a change ... of values, as the light passes through, 
The result is more prone to computational errors. Using no barrier, the light is not 
affected by obstacles resulting in a more stable deviation from the total probability. 
In addition, we have to consider the limitation of a computer, some computational 
error is to be expected.

\subsection{Time evolution}\label{ssec:time_evolution}
% Problem 8: Colormap, include plot of both Re and Im for different time steps
%            Account for color scale
We ran the simulation using the values in column 2 of Table \ref{tab:sim_setup}, found 
in Section \ref{ssec:implementation}. To study the time evolution of the probability 
function, we created colormap plots for different time steps. Figure \ref{fig:colormap_0_prob}, 
Figure \ref{fig:colormap_1_prob}, and Figure \ref{fig:colormap_2_prob} show the 
result for time steps $t=[0, 0.001, 0.002]$, respectively. In addition, we created 
separate plots for the real and imaginary part of $u_{\ivec, \jvec}$, for the same 
time steps. The result can be found in Appendix \ref{ap:figures}, in Figure \ref{fig:colormap}.
\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{images/color_map_0_prob.pdf}
    \caption{The probability function $p_{\ivec, \jvec}^{n}$, at time $t=0$.}
    \label{fig:colormap_0_prob}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{images/color_map_1_prob.pdf}
    \caption{The probability function $p_{\ivec, \jvec}^{n}$, at time $t=0.001$.}
    \label{fig:colormap_1_prob}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{images/color_map_2_prob.pdf}
    \caption{The probability function $p_{\ivec, \jvec}^{n}$, at time $t=0.002$.}
    \label{fig:colormap_2_prob}
\end{figure}
In Figure \ref{fig:colormap_1_prob}, the ... interacts with the double 
slitted barrier, which result in a wavelike pattern. However, the waves are more 
visible when we observe the real and imaginary part separately in Figure \ref{fig:colormap}.


\subsection{Particle detection}\label{ssec:particle_detection}
% Problem 9: Plot detection probability for single-, double- and triple-slit
We used the simulation from the previous section, assumed a detector screen 
located at $x=0.8$, and plotted the detection probability along the screen at time 
$t=0.002$. We adjusted the parameters to include single-, double-, and triple-slitted 
barrier. The results is found in Figure \ref{}, Figure \ref{}, and Figure \ref{fig:}.
\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{images/single_slit_detector.pdf}
    \caption{Probability of particle detection along a detector screen at time $t=0.002$, 
    when using a single-slit barrier.}
    \label{fig:particle_detection_single}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{images/double_slit_detector.pdf}
    \caption{Probability of particle detection along a detector screen at time $t=0.002$, 
    when using a double-slit barrier.}
    \label{fig:particle_detection_double}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{images/triple_slit_detector.pdf}
    \caption{Probability of particle detection along a detector screen at time $t=0.002$, 
    when using a triple-slit barrier.}
    \label{fig:particle_detection_triple}
\end{figure}

\end{document}