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

\begin{document}
\section{Conclusion}\label{sec:conclusion}
% Rewrite this section to differ from the abstract
We have simulated the two-dimensional time-dependent Schrödinger equation, to study 
variations of the double-slit experiment. To solve the partial differential equations 
we have applied the Crank-Nicolson scheme in 2+1 dimensions, and derived a discretized 
equation. In addition, we have used Dirichlet boundary conditions to express the 
equation in matrix form and solve it using the sparse matrix solver \verb|superlu|. 
Our implementation, and choice of solver method, resulted in conserved total 
probability $\sum_{\ivec, \jvec} p_{\ivec, \jvec}^{n}=1$ for both the single and 
double slit setup. 

To illustrate the time evolution of the probability function $p_{\ivec, \jvec}^{n} = u_{\ivec, \jvec}^{n*} u_{\ivec, \jvec}^{n}$, 
we created colormap plots at time steps $t = [0, 0.001, 0.002]$. Since we are working 
with complex numbers, we included separate plots for each time step of Re$(u_{\ivec, \jvec})$ 
and Im$(u_{\ivec, \jvec})$.
% We observed something...

In addition, we determined the normalized particle detection probability $p(y \ | \ x=0.8, t=0.002)$, 
for single-, double- and triple-slit.
% We observed something here as well...

\end{document}
