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Finish first draft of introduction and method section.

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\begin{document}
\section{Theoretical background}\label{sec:theory}
Scientists have made use of light in devices leading to...
The use of light have facilitated many discoveries through history. The invention
of the compound microscope, which used lenses to focus light, led to the first
observation of animal cells. And the telescope, which led to a greater understanding
of the universe.
The nature of light has long been a subject of interest and discussion. Around year
1600 the first microscope was invented, which made use of light through lenses.
The observation made using the microscope led to a greater understanding of the
world at a microscopic scale. However, the study of light itself have led to (something).
Through the 1600s, the view shifted from particle to wave.
% Introduction?
The nature of light has long been a subject of discussion, from the 1500s
The nature of light has long been a subject of interest and discussion, from the 1500s
and the invention of microscopes, through the 1700s where both a particle theory
and a wave theory. In the late 1600s, Christiaan Huygens proposed the wave theory of light,
which was challenged by Isaac Newton's particle theory. The particle theory made
was the leading theory in the beginning of 1800s, when Thomas Young demonstrated
the interference of light, through his double-slit experiment, the wave theory found
which was challenged by Isaac Newton's particle theory. The particle theory was
the leading theory in the beginning of 1800s, when Thomas Young demonstrated
the interference of light, through his double-slit experiment. The wave theory found
new hold.
In the 1800s, the study of ideal black bodies done by Gustav R. Kirchhoff, lead to a
better understanding of heat radiation. Wilhelm Wien started working on determining the
spectral energy distribution, and Wien's law. The law did make sense for high frequencies,
however, there were inconsistencies when frequency were lower than a certain value.
however, as there were inconsistencies when the frequency were lower than a threshold value.
Wiens law led to an exponential curve, which disagree with the law of conservation.
Max Planck guessed a result which led to Plancks radiation law, which he later derived
using Boltzmanns statistical interpretation of the second law of thermodunamics.
Plancks findings gave rise to Einsteins quantum hypothesis, and later the wave-particle
Max Planck guessed a solution to the problem, which led to Planck's radiation law.
He later derived the radiation law using Boltzmanns statistical interpretation of
the second law of thermodynamics.
Plancks findings gave rise to the quantum hypothesis, and later Einsteins wave-particle
duality \cite{britannica:1998:planck}.
For small atoms classical mechanics are not able to explain the position of a particle,
For small atoms classical mechanics are not sufficient in describing the position of a particle,
and Heisenberg uncertainty priciple suggest that the particles have a wavelike behavior.
The wave-particle duality was later proposed to apply to particles by Louis de Broglie,
which inspired Erwin Schrödinger who proposed a wave function to describe the quantum
state of a particle, resulting in the wave equation.
% Methods?
% Schrödinger
The Schrödinger equation has a general form
\begin{align}
i \hbar \frac{\partial}{\partial t} | \Psi \rangle &= \hat{H} | \Psi \rangle \ ,
\label{eq:schrodinger_general}
\end{align}
where $i$ is the imaginary number, and $\hbar$ is Plancks constant. Here $| \Psi \rangle$
is the quantum state and $\hat{H}$ is a Hamiltonian operator.
For two-dimensional position space, the quantum state can be expressed using the
time-dependent complex-valued wave function $\Psi (x, y, t)$. The square modulus of the wave function $|\psi|^{2}$, predicts probability of finding the particle at position $(x, y)$ at time t.
% Segue to Born
The modulus of the wave function, is related to the probability density function
\begin{align}
p(x, y \ | \ t) &= |\Psi(x, y, t)|^{2} = \Psi^{*}(x, y, t) \Psi(x, y, t)
\label{eq:born_rule}
\end{align}
using the Born rule, where $^{*}$ denotes the complex conjugated wave function.
When the potential is time-independent, the Schrödinger equation can be expressed as
\begin{align*}
i \hbar \frac{\partial}{\partial t} \Psi (x, y, t) &= - \frac{\hbar^{2}}{2m} \bigg( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \bigg) \Psi (x, y, t) \\
& \quad + V(x, y, t) \Psi (x, y, t) \numberthis \ .
\label{eq:schrodinger_special}
\end{align*}
When we scale Schrödinger equation by the dimensionful variables, we are left with
a wave function $u$, potential $v$ and the dimensionless equation
\begin{align}
i \frac{\partial u}{\partial t} &= - \frac{\partial^{2} u}{\partial x^{2}} - \frac{\partial^{2} u}{\partial y^{2}} + v(x, y) u \ .
\label{eq:schrodinger_dimensionless}
\end{align}
% Crank-Nicolson
To evaluate the position of a single particle, we have to consider partial differential
equations (PDE). To solve these numerically, we have to discretize Equation \eqref{eq:schrodinger_dimensionless}.
We use the Crank-Nicolson method (CN), which combines the forward and backward finite
difference method. %
% Include the general $\theta$-method
Using the $\theta$-rule \footnote{For $\theta \in [0, 1]$ we can derive Forward Euler using $\theta = 1$, Backward Euler using $\theta = 0$, and $\theta = 1/2$ gives Crank-Nicolson}
with $\theta = 1/2$, CN can be written as
\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] \ .
\label{eq:crank_nicolson}
\end{align}
To simplify notation and avoid confusion of indices with the imaginary number $i$,
we have used the notation $\ivec, \jvec$ in subscript to indicate the commonly named indices $i, j$
in x- and y-direction. In addition, the superscript $n, n+1$ indicate position in time.
We use CN to derive the discretized Schrödinger equation
\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} \numberthis \ .
\label{eq:schrodinger_discretized}
\end{align*}
The full derivation of both Equation \eqref{eq:crank_nicolson} and Equation \eqref{eq:schrodinger_discretized}
can be found in Appendix \ref{ap:crank_nicolson}
First, we Taylor
expand the wave equation $u$ around (position and time).
\section{Notes}\label{sec:notes}
\subsection*{Introduction - draft 2}
In classical mechanics we study the kinematics and dynamics of physical objects,
ignoring its intrinsic properties for simplicity. It allows us to describe the
forces acting on an object as well as the motion of the object. We can describe
a planets orbital movement \cite{britannica:2023:kepler}, calculate the ... necessary
to launch satellites into orbit, or simply figure out where a ball is going to land
when you throw it... However, when want to study an object at a microscopic level,
e.g. a single atom, classical mechanics falls short.
% Double-slit experiment
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
@@ -107,6 +52,292 @@ contradicted the principle of conservation of energy \cite{britannica:1998:planc
Max Planck assumed that the radiated energy consist of discrete values, or quanta,
to describe the peak in radiated energy.
% Reference: https://tex.stackexchange.com/questions/469109/how-to-change-arrowheads-to-lie-on-a-plane
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% LABELLING
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width=5.575cm,
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xmax=0.5,
ticks=none
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]
{(cos(deg(5*pi*sin(deg(x)))))^(2)*((sin(deg(4*pi*sin(deg(x)))))/(4*pi*sin(deg(x))))^(2)};
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\draw[thin,densely dashed,blue] (2,0) -- +(32:2.5);
\draw[thin,densely dashed,blue] (2,0) -- +(-32:2.5);
\end{tikzpicture}
% Author: Izaak Neutelings (June 2020)
% Inspiration:
% https://courses.physics.ucsd.edu/2011/Summer/session1/physics2c/diffraction.pdf
% https://tex.stackexchange.com/questions/201830/periodic-shading-in-tikz
% \begin{tikzpicture}[
% nodal/.style={mylightgreen,dashed,very thin},
% declare function={
% %xnode(\n,\dn,\lam,\f) = sqrt( (\n^2+(\n+\dn)^2)*\lambd^2/2 - (\n^2-(\n+\dn)^2)^2*\lambd^4/(4*\a^2) - \a^2/4 );
% xnode(\n,\dn,\lam,\f) = \lam/\f*sqrt( \n^2*(\f^2-\dn^2)+\n*\dn*(\f^2-\dn^2)+\dn^2*\f^2/2-(\f^4+\dn^4)/4 );
% ynode(\n,\dn,\lam,\a) = (2*\n*\dn+\dn^2)*\lam/(2*\f);
% intensity(\y,\lam,\a,\L) = cos(180*\a*\y/(2*\lam*sqrt(\L*\L+\y*\y)))^2;
% }
% ]
% \def\L{3.8} % distance between walls
% \def\H{5.4} % total wall height
% \def\h{2.8} % plane wave height
% \def\t{0.15} % wall thickness
% \def\a{1.15} % slit distance
% \def\d{0.20} % slit size
% \def\N{21} % number of waves
% \def\lambd{0.20} % wavelength
% \def\R{\N*\lambd} % wave radius
% \def\Nlines{3} % number of nodal lines
% \def\A{1.6} % amplitude
% %\def\r{0.06} % point source radius
% %\def\nmax{10}
% \def\nsamples{100}
% \def\ang{62}
% \begin{scope}
% \clip (-\t/2,-\H/2) rectangle (\L,\H/2);
% %\clip (-\t/2,0.7*\a) -- (0.6*\L,\H/2) -- (\L,\H/2) --
% % (\L,-\H/2) -- (0.6*\L,-\H/2) -- (-\t/2,-0.7*\a) -- cycle;
% % NODAL LINES
% \draw[nodal]
% (0.08*\N*\lambd,0) -- (1.06*\R,0);
% \coordinate (NP0) at (\L,0); % to avoid "Dimension too large error"
% \foreach \dn [evaluate={
% \f=\a/\lambd;
% \nmin=2.5+0.2*\dn; %0.501*(-\dn+\f)
% \nmax=10; %(NP0)
% \c=int(\dn<\f);
% \y=\L/sqrt((\a/(\lambd*\dn))^2-1);
% }] in {1,...,\Nlines}{
% \coordinate (NP+\dn) at (\L,\y); % to avoid "Dimension too large error"
% \coordinate (NP-\dn) at (\L,-\y); % to avoid "Dimension too large error"
% \ifnum\c=1
% \draw[nodal,variable=\n,samples=\nsamples,smooth]
% plot[domain=\nmin:\nmax] ({xnode(\n,\dn,\lambd,\f)},{ynode(\n,\dn,\lambd,\f)})
% -- (NP+\dn);
% \draw[nodal,variable=\n,samples=\nsamples,smooth]
% plot[domain=\nmin:\nmax] ({xnode(\n,\dn,\lambd,\f)},{-ynode(\n,\dn,\lambd,\f)})
% -- (NP-\dn);
% \fi
% }
% % WAVES
% \foreach \i [evaluate={\R=\i*\lambd;}] in {1,...,\N}{
% \ifodd\i
% \draw[myblue,line width=0.8] (0,\a/2)++(\ang:\R) arc (\ang:-\ang:\R);
% \draw[myred,line width=0.8] (0,-\a/2)++(\ang:\R) arc (\ang:-\ang:\R);
% \else
% \draw[myblue!80,line width=0.1] (0,\a/2)++(\ang:\R) arc (\ang:-\ang:\R);
% \draw[myred!80,line width=0.1] (0,-\a/2)++(\ang:\R) arc (\ang:-\ang:\R);
% \fi
% }
% \end{scope}
% % PLANE WAVES
% \foreach \i [evaluate={\x=-\i*\lambd;}] in {0,...,5}{
% \ifodd\i
% \draw[myblue,line width=0.8] (\x,-\h/2) -- (\x,\h/2);
% \else
% \draw[myblue,line width=0.1] (\x,-\h/2) -- (\x,\h/2);
% \fi
% }
% % WALL
% \fill[wall]
% (\t/2,\a/2-\d/2) rectangle (-\t/2,-\a/2+\d/2)
% (\t/2,\a/2+\d/2) rectangle (-\t/2,\H/2)
% (\t/2,-\a/2-\d/2) rectangle (-\t/2,-\H/2)
% (\L,-\H/2) rectangle (\L+\t,\H/2);
% % SHADES
% \begin{scope}[shift={(1.08*\L,0)}]
% \def\yz{\L/sqrt((\a/\lambd)^2-1)} % m = +- 1/2
% \def\yZ{\L/sqrt((\a/\lambd/2)^2-1)} % m = +- 1
% \clip (0,-\H/2) rectangle (1.1*\A,\H/2);
% \fill[white] (0,-\H/2) rectangle++ (\A,\H); % to fill seams
% \foreach \i [evaluate={\n=0.5*\i;\yn=\L/sqrt((\a/(2*\lambd*\n))^2-1);
% }] in {1,...,\Nlines}{
% \ifodd\i % if even
% \fill[myshadow] (0,{-\yn-0.1}) rectangle++ (\A,0.2); % to fill seams
% \fill[myshadow] (0,{ \yn-0.1}) rectangle++ (\A,0.2); % to fill seams
% \fi
% }
% \path[left color=myshadow,right color=myshadow,middle color=white,shading angle={180}]
% (0,{-\yz}) rectangle (\A,{\yz});
% \foreach \i [evaluate={
% \n=0.5*\i;
% \m=0.5*(\i+1);
% \yn=\L/sqrt((\a/(2*\lambd*\n))^2-1);
% \ym=\L/sqrt((\a/(2*\lambd*\m))^2-1);
% \dang=mod(\i,2)*180;
% }] in {1,...,\Nlines}{
% \path[left color=myshadow,right color=white,shading angle={\dang}]
% (0,\yn) rectangle (\A,\ym);
% \path[left color=myshadow,right color=white,shading angle={180+\dang}]
% (0,-\yn) rectangle (\A,-\ym);
% }
% \end{scope}
% % INTENSITY
% \begin{scope}[shift={(1.1*\L+1.1*\A,0)}]
% \draw[->,thick] (-0.08*\A,0) -- (1.3*\A,0) node[right=-2] {$\expval{I}$}; % I axis
% \draw[->,thick] (0,-0.52*\H) -- (0,0.54*\H) node[right] {$y$}; % y axis
% \draw[nodal] (NP0) --++ (0.15*\L+2.1*\A,0); % green nodal lines
% \foreach \i [evaluate={\y=\L/sqrt((\a/(\lambd*\i))^2-1)}] in {1,...,\Nlines}{ % green nodal lines
% \draw[nodal] (NP+\i) --++ ({0.15*\L+1.1*\A+\A*intensity(\y,\lambd,\a,\L)},0);
% \draw[nodal] (NP-\i) --++ ({0.15*\L+1.1*\A+\A*intensity(\y,\lambd,\a,\L)},0);
% }
% \draw[myred,thick,variable=\y,samples=\nsamples,smooth,domain=-\H/2:\H/2]
% plot({\A*intensity(\y,\lambd,\a,\L)},\y);
% \foreach \i [evaluate={ % ticks
% \modd=\i; %int(\i);
% \meven=int(\i-1);
% \y=\L/sqrt((\a/(\lambd*\i))^2-1);
% }] in {1,...,\Nlines}{
% \ifodd\i
% \tick{0,-\y}{180} node[right=0,scale=0.85] {$m=-\frac{\modd}{2}$};
% \tick{0,\y}{180} node[right=0,scale=0.85] {$m=+\frac{\modd}{2}$};
% \else
% \tick{0,-\y}{180} node[right=0,scale=0.85] {$m=-\meven$};
% \tick{0,\y}{180} node[right=0,scale=0.85] {$m=+\meven$};
% \fi
% }
% \end{scope}
% \end{tikzpicture}
% Schrödinger
% -----------
% The wave equation, Schrödinger equation has a general form
% \begin{align}
% i \hbar \frac{\partial}{\partial t} | \Psi \rangle &= \hat{H} | \Psi \rangle \ ,
% \label{eq:schrodinger_general}
% \end{align}
% where $i$ is the imaginary number, and $\hbar$ is Plancks constant. Here $| \Psi \rangle$
% is the quantum state and $\hat{H}$ is a Hamiltonian operator. % which represent the energy of the system
% For two-dimensional position space, the quantum state can be expressed using the
% time-dependent complex-valued wave function $\Psi (x, y, t)$. The square modulus of the wave function $|\psi|^{2}$, predicts probability of finding the particle at position $(x, y)$ at time t.
% % Segue to Born
% The modulus of the wave function, is related to the probability density function
% \begin{align}
% p(x, y \ | \ t) &= |\Psi(x, y, t)|^{2} = \Psi^{*}(x, y, t) \Psi(x, y, t)
% \label{eq:born_rule}
% \end{align}
% using the Born rule, where $^{*}$ denotes the complex conjugated wave function.
% When the potential is time-independent, the Schrödinger equation can be expressed as
% \begin{align*}
% i \hbar \frac{\partial}{\partial t} \Psi (x, y, t) &= - \frac{\hbar^{2}}{2m} \bigg( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \bigg) \Psi (x, y, t) \\
% & \quad + V(x, y, t) \Psi (x, y, t) \numberthis \ .
% \label{eq:schrodinger_special}
% \end{align*}
% When we scale Schrödinger equation by the dimensionful variables, we are left with
% a wave function $u$, potential $v$ and the dimensionless equation
% \begin{align}
% i \frac{\partial u}{\partial t} &= - \frac{\partial^{2} u}{\partial x^{2}} - \frac{\partial^{2} u}{\partial y^{2}} + v(x, y) u \ .
% \label{eq:schrodinger_dimensionless}
% \end{align}
% Crank-Nicolson
% --------------
% To evaluate the position of a single particle, we have to consider partial differential
% equations (PDE). To solve these numerically, we have to discretize Equation \eqref{eq:schrodinger_dimensionless}.
% We use the Crank-Nicolson method (CN), which combines the forward and backward finite
% difference method.
% Include the general $\theta$-method
% Using the $\theta$-rule \footnote{For $\theta \in [0, 1]$ we can derive Forward Euler using $\theta = 1$, Backward Euler using $\theta = 0$, and $\theta = 1/2$ gives Crank-Nicolson}
% with $\theta = 1/2$, CN can be written as
% \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] \ .
% \label{eq:crank_nicolson}
% \end{align}
% To simplify notation and avoid confusion of indices with the imaginary number $i$,
% we have used the notation $\ivec, \jvec$ in subscript to indicate the commonly named indices $i, j$
% in x- and y-direction. In addition, the superscript $n, n+1$ indicate position in time.
% We use CN to derive the discretized Schrödinger equation
% \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} \numberthis \ .
% \label{eq:schrodinger_discretized}
% \end{align*}
% The full derivation of both Equation \eqref{eq:crank_nicolson} and Equation \eqref{eq:schrodinger_discretized}
% can be found in Appendix \ref{ap:crank_nicolson}.
% Implementation
The implementation of CN is simplified usein Dirichlet boundary conditions, which
can be found in Table \ref{tab:dirichlet}. In addition, we use Gaussian wave packets
for the initial wave function
\begin{align}
u(x, y, t=0) = e^{- \frac{(x-x_{c})^{2}}{2 \sigma_{x}^{2}} - \frac{(y-y_{c})^{2}}{2 \sigma_{y}^{2}} + ip_{x}x + ip_{y}y}
\end{align}
\begin{algorithm}[H]
\caption{Crank-Nicolson scheme}
\label{algo:cn_scheme}
\begin{algorithmic}
\Procedure{Crank-Nicolson}{$args$}
\State Insert pseudo code $var \leftarrow \text{ some text}$
\EndProcedure
\end{algorithmic}
\end{algorithm}
\section{Notes}\label{sec:notes}
\subsection*{Introduction - draft 2}
In classical mechanics we study the kinematics and dynamics of physical objects,
ignoring their intrinsic properties for simplicity. It allows us to describe the
forces acting on an object as well as the motion of the object. We can describe
a planets orbital movement \cite{britannica:2023:kepler}, calculate the ... necessary
to launch satellites into orbit, or simply figure out where a ball is going to land
when you throw it... However, when want to study an object at a microscopic level,
e.g. a single atom, classical mechanics falls short.
% 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.
Light as particles -> waves -> particles/packets
Erwin Schrödinger wanted to find a mathematical description of the wave characteristics
of matter, supporting the wave-particle idea. He postulated a function which varies