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\section{Conclusion}\label{sec:conclusion}
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 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 setups. We found that increasing the number of slits 
in the barrier, resulted in an increased number of areas of both high and low probability 
for particle detection. It also increased the variance of particle detection probability.
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