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

\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*}


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