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

\begin{document}
\begin{abstract}
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.
\end{abstract}
\end{document}

% $| \sum_{\ivec, \jvec} p_{\ivec, \jvec}^{n} - 1 | \approx $ 