69 lines
3.9 KiB
TeX
69 lines
3.9 KiB
TeX
\documentclass[../schrodinger_simulation.tex]{subfiles}
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\begin{document}
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\section{Introduction}\label{sec:introduction}
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% Light: wave particle
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% Wave equation
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% Hyugens theory
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% Thomas Young
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The nature of light has long been a subject of interest and discussion. %
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% Important part of human behavior is observing and understanding our surroundings.
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% Many big discoveries have been made through observations, verified by mathematical
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% explanations. Classical physics is based on calculation predicting something we
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% verify by observation etc. But what happens when we move down to the microscopic
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% scale, can we still predict the position of a microscopic ball, also called an atom?
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In classical mechanics, we study the kinematics and dynamics of physical objects,
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while ignoring their intrinsic properties for simplicity. Newton's second law can be
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applied to an object to describe its trajectory. % It allows us to describe the
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% forces acting on an object as well as the motion of the object. We can describe
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% a planets orbital movement \cite{britannica:2023:kepler}, calculate the ... necessary
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% to launch satellites into orbit, or simply figure out where a ball is going to land
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% when you throw it... However, when want to study an object at a microscopic level,
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% e.g. a single atom, classical mechanics falls short.
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Elementary particles such as electrons, does not abide by the laws of classical mechanics.
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For several years, scientists did not agree on whether light was a particle or a
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wave. Through the study of interference of light, and radiation of ideal blackbodies,
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it has been shown that light has both wavelike and particle-like characteristics.
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This is known as the wave-particle duality, and was showed by Albert Einstein in
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1905. %
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% Thomas Young studied the interference of light, and found that light to showed
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% wavelike characteristics \cite{young:1804:double_slit}. This did not agree with
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% Newtons particle-theory
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Erwin Schrödinger wanted to find a mathematical description of the wave characteristics
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of matter, supporting the wave-particle idea. He postulated a wave function which varies
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with position, where the function squared can be interpreted as the probability
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of finding an electron at a given position. This resulted in the Schrödinger equation,
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a wave eqution of the energy levels for a hydrogen atom. It also shows how a quantum
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state evolves with time \cite[p. 81]{wu:2023:quantum}.
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We will simulate the time-dependent Schrödinger equation in two dimensions, to
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study the light wave interference in the double-slit experiment. In addition, we
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will include variations of walls such as single- and triple-slit. To solve the equation,
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we will apply the Crank-Nicolson method in 2+1 dimensions.
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% However, according to the Heisenberg uncertainty principle, we can't find dx and/or
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% dp = 0. dx = sqrt{Var(x)} "spread in position", dp = hat{\Psi}(p) = sqrt{Var(p)}
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% dx \cdot dp \geq \frac{\hbar}{2}
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% Fourier transform \Psi(x) \doublearrow hat{\Psi}(p)
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% \Psi(x) = \int_{infty}^{infty} (alt sum) hat{\Psi}(p) \cos(px) dp sum of different wave forms
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% Light - particle or wave
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% - double-slit, blackbodies radiation
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% The nature of light.
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% Position space
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% - classical vs quantum mechanics
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% Intuitions of the behavior of physical objects, predictable. Not like the quantum
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% when we scale down to the atom, our intuition are not as reliable and prediction
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% are not as perfect.
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% Instead of finding the path of a ball, we find all the possible paths a ball can take.
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% The world is not one-dimensional, and modelling it require partial diff eqs
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In Section \ref{sec:methods}, we will present the theoretical background for
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this experiment, as well as the algorithms and tools used in the implementation.
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Continuing with Section \ref{sec:results}, we will present our results and
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discuss our findings. Lastly, we will conclude our findings in Section \ref{sec:conclusion}.
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\end{document}
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% crank-nicolson method!
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% wave equation |