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\begin{document}
\section{Conclusion}\label{sec:conclusion}
% We have simulated the two-dimensional time-dependent Schrödinger equation, to study 
% variations of the double-slit experiment. To derive a discretized equation 
% we applied the Crank-Nicolson scheme in 2+1 dimensions. 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 a deviation from conserved total probability on the 
% scale $10^{-14}$, for both the single and double slit setup. To illustrate the time 
evolution of the probability function, we created colormap plots for time steps 
$t = [0, 0.001, 0.002]$. We also included separate plots for each time step of 
Re$(u_{\ivec, \jvec})$ and Im$(u_{\ivec, \jvec})$. In addition, we determined the 
normalized particle detection probability $p(y \ | \ x=0.8, t=0.002)$, for single-, 
double- and triple-slit.
% Rewrite this section to differ from the abstract
We simulated the two-dimensional time-dependent Schrödinger equation, and studied 
variations of the double-slit experiment. To solve the partial differential equations 
we applied the Crank-Nicolson scheme in 2+1 dimensions, and derived a discretized 
equation. We used Dirichlet boundary conditions to simplify the equation, 
expressed the equation in matrix form and solved it using the sparse matrix solver 
\verb|superlu|. The total probability $\sum_{\ivec, \jvec} p_{\ivec, \jvec}^{n}$ 
deviated from $1.0$ by a factor of $10^{-14}$ for both the single and double slit 
setup. % Add something about computational accuracy?

We illustrated the time evolution of the probability function $p_{\ivec, \jvec}^{n} = u_{\ivec, \jvec}^{n*} u_{\ivec, \jvec}^{n}$, 
using colormap plots for time steps $t = [0, 0.001, 0.002]$. In addition, we included 
separate plots for each time step of Re$(u_{\ivec, \jvec})$ and Im$(u_{\ivec, \jvec})$, 
to show the components of the complex values. This resulted in a visible diffraction 
patterns for the double-slit experiment.

In addition, we studied the normalized particle detection probability $p(y \ | \ x=0.8, t=0.002)$, 
for single-, double- and triple-slit. We found that increasing the number of slits 
in the barrier, increased the number of areas of both high and low probability of 
particle detection. 
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