Finish first draft of introduction and method section.

latex/sections/introduction.tex

@@ -2,12 +2,63 @@
 
 \begin{document}
 \section{Introduction}\label{sec:introduction}
+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!