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

\begin{document}
\section{Introduction}\label{sec:introduction}
% Light: wave particle
The nature of light has long been a subject of interest and discussion. In classical 
mechanics, we study the kinematics and dynamics of physical objects, while ignoring 
their intrinsic properties for simplicity. Elementary particles, such as photons 
and electrons, do not abide by the laws of classical mechanics. A solution was 
proposed by Max Planck with the radiation law, which he later derived using Boltzmann's 
statistical interpretation of the second law of thermodynamics. Planck's findings 
gave rise to the quantum hypothesis, and later Einstein's wave-particle duality \cite{britannica:1998:planck}.

The particle theory was the leading theory in the beginning of the 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. Kirchhoff's study of ideal blackbodies, 
showed that light exhibits 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 particles, 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 barriers with single-slit and triple-slits. 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 methods 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. 
% Many big discoveries have been made through observations, verified by mathematical 
% explanations. Classical physics is based on calculation predicting something we 
% 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?
% 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 
% 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 
% 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. 

% 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 
% 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. 
% This is known as the wave-particle duality, and was showed by Albert Einstein in 
% 1905.
% 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  
% Newtons particle-theory

% 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}. 

% 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.

% 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)} 
% dx \cdot dp \geq \frac{\hbar}{2}
% Fourier transform \Psi(x) \doublearrow hat{\Psi}(p)
% \Psi(x) = \int_{infty}^{infty} (alt sum) hat{\Psi}(p) \cos(px) dp sum of different wave forms
% Light - particle or wave 
% - double-slit, blackbodies radiation
% The nature of light.
% Position space
% - classical vs quantum mechanics
% Intuitions of the behavior of physical objects, predictable. Not like the quantum 
% when we scale down to the atom, our intuition are not as reliable and prediction 
% are not as perfect. 
% 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



% crank-nicolson method!
% wave equation