Add draft for report, with a general setup of theoretical background and methods. Not ready for review!

latex/sections/appendices.tex

@@ -2,6 +2,47 @@
 
 \begin{document}
 \appendix
+\section{The Crank-Nicholson method}\label{ap:crank_nicolson}
+The Crank-Nicolson \(CN\) approach considers both the forward difference, an explicit scheme,
+\begin{equation*}
+    \frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} = F_{\ivec, \jvec}^{n} \ ,
+\end{equation*}
+and the backward difference, an implicit scheme, 
+\begin{equation*}
+    \frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} = F_{\ivec, \jvec}^{n+1} \ .
+\end{equation*}
+The result is a linear combination of the explicit and implicit scheme, given by 
+\begin{align*}
+    \frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} &= \theta F_{\ivec, \jvec}^{n+1} + (1 - \theta) F_{\ivec, \jvec}^{n} \ .
+\end{align*}
+The parameter $\theta$ is introduced for a general approach, however, for CN $\theta = 1/2$. 
+\begin{align*}
+    \frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} &= \frac{1}{2} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \\
+\end{align*}
+
+We need the first derivative in respect to both time and position, as well as the second derivative in respect to position. Taylor expanding will result in a discretized version, assume this is known...
+
+Schrödinger contain $i$ at the lhs, factor it as
+\begin{align*}
+    \frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} &= \frac{1}{2i} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \\
+    &= -\frac{i}{2} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] & \text{, where $\frac{1}{i} = -i$}
+\end{align*}
+
+Using Equation \eqref{eq:schrodinger_dimensionless}, we get
+\begin{align*}
+    u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n} & -\frac{i \Delta t}{2} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \\
+    &= -\frac{i \Delta t}{2} \bigg[ - \frac{u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1}}{2 \Delta x^{2}} \\
+    & \quad - \frac{u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1}}{2 \Delta y^{2}} + \frac{1}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\
+    & \quad - \frac{u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n}}{2 \Delta x^{2}} \\
+    & \quad - \frac{u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n}}{2 \Delta y^{2}} + \frac{1}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \bigg] \\
+\end{align*}
+We rewrite the expression, 
+\begin{align*}
+    & u_{\ivec, \jvec}^{n+1} - \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1} \big] \\
+    & - \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1} \big] + \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\
+    &= u_{\ivec, \jvec}^{n} + \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n} \big] \\
+    & \quad + \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n} \big] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \\
+\end{align*}
 
 
 \end{document}

latex/sections/introduction.tex

@@ -2,6 +2,13 @@
 
 \begin{document}
 \section{Introduction}\label{sec:introduction}
+% Light - particle or wave 
+% - double-slit, blackbodies radiation
 
+% Position space
+% - classical vs quantum mechanics
 
 \end{document}
+
+% crank-nicolson method!
+% wave equation

latex/sections/methods.tex

@@ -1,7 +1,29 @@
 \documentclass[../schrodinger_simulation.tex]{subfiles}
 
 \begin{document}
-\section{Methods}\label{sec:methods}
+\section{Methods}\label{sec:methods} % 
 
+\subsection{Double-slit experiment}\label{ssec:double_slit} % 
+In the beginning of the 1800s, the general view was that light consisted of particles. 
+However, in 1801 Thomas Young demonstrated the principle of interference of light 
+\cite{britannica:2023:young}, while postulating light as waves rather than particles. 
+% Thomas Young first performed the double-slit experiment in 1801, to demonstrate 
+% the principle of interference of light \cite{britannica:2023:young}, postulating 
+% light as waves. In the study of blackbodies, scientists were not able to describe 
+% the radiated intensity of increased frequencies using classical mechanichs, as they 
+% contradicted the principle of conservation of energy \cite{britannica:1998:planck}. 
+% Max Planck assumed that the radiated energy consist of discrete values, or quanta, 
+% to describe the peak in radiated energy. 
+The double-slit experiment
+
+\subsection{Schrödinger equation}\label{ssec:schrodinger} %
+
+\subsection{Crank-Nicolson}\label{ssec:crank_nicolson} %
+
+% Might be better to move theory on double-slit here and have a subsection of light 
+% property etc. as a first section or in introduction?
+\subsection{Implementation}\label{ssec:implementation} % 
+
+\subsection{Tools}\label{ssec:tools} %
 
 \end{document}