\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 superlu solver, which is a dedicated solver for sparse matrices. It is 
generally used to solve nonsymmetric, sparse matrices. However, as the lapack solver 
converts the sparse matrix to a dense matrix, it will increase memory usage compared 
to 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 use a solver for sparse matrices, we decrease number of computations performed 
compared to solver using dense matrix. We check if the total probability is conserved 
over time, by plotting the deviation $s$ as
\begin{align*}
    s^{n} = 1 - \sum_{\ivec , \jvec} p_{\ivec , \jvec}^{n} = 1 - \sum_{\ivec , \jvec} u_{\ivec , \jvec}^{n*} u_{\ivec , \jvec}^{n} \ .
\end{align*}
The deviation as a function of time is plotted in Figure \ref{fig:deviation}.
\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{images/probability_deviation.pdf}
    \caption{Deviation for $t \in [0, T]$ where $T=0.008$.}
    \label{fig:deviation}
\end{figure}

\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

\subsection{Particle detection}\label{ssec:particle_detection}
% Problem 9: Plot detection probability for single-, double- and triple-slit


\end{document}
