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Finished the introduction.

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latex/sections/introduction.tex
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\begin{document} \begin{document}
\section{Introduction}\label{sec:introduction} \section{Introduction}\label{sec:introduction}
% Light: wave particle % Light: wave particle
% Wave equation The nature of light has long been a subject of interest and discussion. In classical
% Hyugens theory mechanics, we study the kinematics and dynamics of physical objects, while ignoring
% Thomas Young their intrinsic properties for simplicity. Elementary particles, such as photons
The nature of light has long been a subject of interest and discussion. % and electrons, does not abide by the laws of classical mechanics. A solution was
proposed by Max Planck in the radiation law, which he later derived 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}.
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
while postulating light as waves \cite{young:1804:double_slit}. The study of
interference of light, and Gustav R. Kirchhoffs study of ideal blackbodies,
showed that light exibits both wavelike and particle-like characteristics. 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.
We will simulate the time-dependent Schrödinger equation in two dimensions, to
study the light wave interference in the double-slit experiment. In addition, we
will include variations of walls such as single- and triple-slit. To solve the equation,
we will apply the Crank-Nicolson method in 2+1 dimensions.
In Section \ref{sec:methods}, we will present the theoretical background for
this experiment, as well as the algorithms and tools used in the implementation.
Continuing with Section \ref{sec:results}, we will present our results and
discuss our findings. Lastly, we will conclude our findings in Section \ref{sec:conclusion}.
\end{document}
% Important part of human behavior is observing and understanding our surroundings. % Important part of human behavior is observing and understanding our surroundings.
% Many big discoveries have been made through observations, verified by mathematical % Many big discoveries have been made through observations, verified by mathematical
% explanations. Classical physics is based on calculation predicting something we % explanations. Classical physics is based on calculation predicting something we
% verify by observation etc. But what happens when we move down to the microscopic % verify by observation etc. But what happens when we move down to the microscopic
% scale, can we still predict the position of a microscopic ball, also called an atom? % scale, can we still predict the position of a microscopic ball, also called an atom?
In classical mechanics, we study the kinematics and dynamics of physical objects, % In classical mechanics, we study the kinematics and dynamics of physical objects,
while ignoring their intrinsic properties for simplicity. Newton's second law can be % while ignoring their intrinsic properties for simplicity. Newton's second law can be
applied to an object to describe its trajectory. % It allows us to describe the % applied to an object to describe its trajectory. It allows us to describe the
% forces acting on an object as well as the motion of the object. We can describe % 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 % 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 % 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, % when you throw it... However, when want to study an object at a microscopic level,
% e.g. a single atom, classical mechanics falls short. % e.g. a single atom, classical mechanics falls short.
Elementary particles such as electrons, does not abide by the laws of classical mechanics. % Elementary particles such as electrons, does not abide by the laws of classical mechanics.
For several years, scientists did not agree on whether light was a particle or a % For several years, scientists did not agree on whether light was a particle or a
wave. Through the study of interference of light, and radiation of ideal blackbodies, % wave. Through the study of interference of light, and radiation of ideal blackbodies,
it has been shown that light has both wavelike and particle-like characteristics. % it has been shown that light has both wavelike and particle-like characteristics.
This is known as the wave-particle duality, and was showed by Albert Einstein in % This is known as the wave-particle duality, and was showed by Albert Einstein in
1905. % % 1905.
% Thomas Young studied the interference of light, and found that light to showed % Thomas Young studied the interference of light, and found that light to showed
% wavelike characteristics \cite{young:1804:double_slit}. This did not agree with % wavelike characteristics \cite{young:1804:double_slit}. This did not agree with
% Newtons particle-theory % Newtons particle-theory
Erwin Schrödinger wanted to find a mathematical description of the wave characteristics % Erwin Schrödinger wanted to find a mathematical description of the wave characteristics
of matter, supporting the wave-particle idea. He postulated a wave function which varies % of matter, supporting the wave-particle idea. He postulated a wave function which varies
with position, where the function squared can be interpreted as the probability % with position, where the function squared can be interpreted as the probability
of finding an electron at a given position. This resulted in the Schrödinger equation, % of finding an electron at a given position. This resulted in the Schrödinger equation,
a wave eqution of the energy levels for a hydrogen atom. It also shows how a quantum % a wave eqution of the energy levels for a hydrogen atom. It also shows how a quantum
state evolves with time \cite[p. 81]{wu:2023:quantum}. % state evolves with time \cite[p. 81]{wu:2023:quantum}.
We will simulate the time-dependent Schrödinger equation in two dimensions, to % We will simulate the time-dependent Schrödinger equation in two dimensions, to
study the light wave interference in the double-slit experiment. In addition, we % study the light wave interference in the double-slit experiment. In addition, we
will include variations of walls such as single- and triple-slit. To solve the equation, % will include variations of walls such as single- and triple-slit. To solve the equation,
we will apply the Crank-Nicolson method in 2+1 dimensions. % we will apply the Crank-Nicolson method in 2+1 dimensions.
% However, according to the Heisenberg uncertainty principle, we can't find dx and/or % However, according to the Heisenberg uncertainty principle, we can't find dx and/or
% dp = 0. dx = sqrt{Var(x)} "spread in position", dp = hat{\Psi}(p) = sqrt{Var(p)} % dp = 0. dx = sqrt{Var(x)} "spread in position", dp = hat{\Psi}(p) = sqrt{Var(p)}
@ -59,11 +83,7 @@ we will apply the Crank-Nicolson method in 2+1 dimensions.
% Instead of finding the path of a ball, we find all the possible paths a ball can take. % Instead of finding the path of a ball, we find all the possible paths a ball can take.
% The world is not one-dimensional, and modelling it require partial diff eqs % The world is not one-dimensional, and modelling it require partial diff eqs
In Section \ref{sec:methods}, we will present the theoretical background for
this experiment, as well as the algorithms and tools used in the implementation.
Continuing with Section \ref{sec:results}, we will present our results and
discuss our findings. Lastly, we will conclude our findings in Section \ref{sec:conclusion}.
\end{document}
% crank-nicolson method! % crank-nicolson method!
% wave equation % wave equation

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latex/sections/methods.tex
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% Add something that takes Planck to Schrödinger % Add something that takes Planck to Schrödinger
% In classical mechanics, we have Newton laws and conservation of energy. In quantum % In classical mechanics, we have Newton laws and conservation of energy. In quantum
% mechanics, we have Schrödinger equation. % mechanics, we have Schrödinger equation.
The Schrödinger equation has a general form Erwin Schrödinger wanted to find a mathematical description of the wave characteristics
of matter, supporting the wave-particle idea. He postulated a wave function which varies
with position, where the function squared can be interpreted as the probability
of finding an electron at a given position. This resulted in the Schrödinger equation,
a wave eqution of the energy levels for a hydrogen atom. It also shows how a quantum
state evolves with time \cite[p. 81]{wu:2023:quantum}. The Schrödinger equation
has a general form
\begin{align} \begin{align}
i \hbar \frac{\partial}{\partial t} | \Psi \rangle &= \hat{H} | \Psi \rangle \ , i \hbar \frac{\partial}{\partial t} | \Psi \rangle &= \hat{H} | \Psi \rangle \ ,
\label{eq:schrodinger_general} \label{eq:schrodinger_general}