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

\begin{document}
\section{Introduction}\label{sec:introduction}
% Light: wave particle
% Wave equation
% Hyugens theory
% Thomas Young
The nature of light has long been a subject of interest and discussion. %
% 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

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!
% wave equation