From c51bcb4369387dde5726f7aef18f4eb4fa6f0ee8 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Mon, 1 Jan 2024 13:05:09 +0100 Subject: [PATCH] Finished the introduction. --- latex/sections/introduction.tex | 76 +++++++++++++++++++++------------ latex/sections/methods.tex | 8 +++- 2 files changed, 55 insertions(+), 29 deletions(-) diff --git a/latex/sections/introduction.tex b/latex/sections/introduction.tex index b6f40b5..33f2675 100644 --- a/latex/sections/introduction.tex +++ b/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 \ No newline at end of file diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 52859bf..8f0e512 100644 --- a/latex/sections/methods.tex +++ b/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}