36 lines
1.6 KiB
TeX
36 lines
1.6 KiB
TeX
\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}
|