Add draft abstract and conclusion.

latex/references.bib

@@ -8,6 +8,14 @@
   year            = {2020}
 }
 
+@misc{mit:2004:physics,
+  author          = {Sen-ben Liao and Peter Dourmashkin and John W. Belcher},
+  title           = {Course notes in Physics 8.02 at MIT},
+  year            = {2004},
+  url             = {https://web.mit.edu/8.02t/www/802TEAL3D/visualizations/coursenotes/modules/guide14.pdf},
+  urldate         = {2023-12-22}
+}
+
 @misc{britannica:1999:light,
   author          = {The Editors of Encyclopaedia Britannica},
   title           = {Light},

latex/schrodinger_simulation.tex

@@ -88,7 +88,7 @@
 \noaffiliation                            % ignore this, but keep it.
 
 % Abstract
-% \subfile{sections/abstract}
+\subfile{sections/abstract}
 
 \maketitle
 

latex/sections/abstract.tex

@@ -2,6 +2,17 @@
 
 \begin{document}
 \begin{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, to derive 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, 
+we created colormap plots at 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}

latex/sections/conclusion.tex

@@ -2,6 +2,24 @@
 
 \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}

latex/sections/methods.tex

@@ -85,9 +85,35 @@ can be found in Appendix \ref{ap:crank_nicolson}.
 \subsection{The double-slit experiment}\label{ssec:double_slit} % 
 Thomas Young first performed the double-slit experiment in 1801 to demonstrate the  
 principle of interference of light \cite{britannica:2023:young}, while postulating 
-light as waves rather than particles. The double-slit experiment gives a diffraction 
-pattern, where constructive interference of light result in bright spots, and destructive 
-interference result in dark spots. 
+light as waves rather than particles. The double-slit experiment result in a diffraction 
+pattern on a detector screen, where constructive interference of light result in 
+bright spots, and destructive interference result in dark spots as showed in Figure 
+\ref{fig:youngs_double_slit}.
+\begin{figure}
+    \centering 
+    \includegraphics[width=\linewidth]{images/youngs_double_slit.pdf}
+    \caption{The setup of Thomas Young's double slit experiment, where $S_{0}$ denotes 
+    the light source, $S_{1}$ and $S_{2}$ denotes the slits in the wall.}
+    \label{fig:youngs_double_slit}
+\end{figure}
+
+After the wave passes through the two slits, the pattern observed is determined by 
+the path difference determined by 
+\begin{align*}
+    \delta = d \sin (\theta) = m \lambda \ ,
+\end{align*}
+where $\lambda$ is the wavelength and $m$ is called the order number. $d$ is the 
+distance between the center of the two slits, assuming that the distance between 
+the wall and the detector screen $L >> \delta$ \cite[p. 6]{mit:2004:physics}. In 
+this case, we observe constructive interference when 
+\begin{align*}
+    \delta = m \lambda && m = 0, \pm 1, \pm 2 \dots \ ,
+\end{align*} 
+and destructive interference when
+\begin{align*}
+    \delta = (m + \frac{1}{2}) \lambda && m = 0, \pm 1, \pm 2 \dots \ ,
+\end{align*} 
+
 % Something about Heisenberg uncertainty principle 
 
 \subsection{Implementation}\label{ssec:implementation} %