Add draft abstract and conclusion.

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} %