latex/sections/introduction.tex

@@ -3,45 +3,69 @@
 \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. %
+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, 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. 
 % 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 
+% 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. %
+% 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}. 
+% 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.
+% 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)} 
@@ -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.
 % 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

latex/sections/methods.tex

@@ -6,7 +6,13 @@
 % Add something that takes Planck to Schrödinger
 % In classical mechanics, we have Newton laws and conservation of energy. In quantum 
 % 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}
     i \hbar \frac{\partial}{\partial t} | \Psi \rangle &= \hat{H} | \Psi \rangle \ ,
     \label{eq:schrodinger_general}