Finish first draft of introduction and method section.
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latex/draft.tex
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latex/draft.tex
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\begin{document}
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\section{Theoretical background}\label{sec:theory}
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Scientists have made use of light in devices leading to...
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The use of light have facilitated many discoveries through history. The invention
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of the compound microscope, which used lenses to focus light, led to the first
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observation of animal cells. And the telescope, which led to a greater understanding
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of the universe.
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The nature of light has long been a subject of interest and discussion. Around year
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1600 the first microscope was invented, which made use of light through lenses.
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The observation made using the microscope led to a greater understanding of the
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world at a microscopic scale. However, the study of light itself have led to (something).
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Through the 1600s, the view shifted from particle to wave.
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% Introduction?
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The nature of light has long been a subject of discussion, from the 1500s
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The nature of light has long been a subject of interest and discussion, from the 1500s
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and the invention of microscopes, through the 1700s where both a particle theory
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and a wave theory. In the late 1600s, Christiaan Huygens proposed the wave theory of light,
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which was challenged by Isaac Newton's particle theory. The particle theory made
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was the leading theory in the beginning of 1800s, when Thomas Young demonstrated
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the interference of light, through his double-slit experiment, the wave theory found
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which was challenged by Isaac Newton's particle theory. The particle theory was
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the leading theory in the beginning of 1800s, when Thomas Young demonstrated
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the interference of light, through his double-slit experiment. The wave theory found
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new hold.
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In the 1800s, the study of ideal black bodies done by Gustav R. Kirchhoff, lead to a
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better understanding of heat radiation. Wilhelm Wien started working on determining the
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spectral energy distribution, and Wien's law. The law did make sense for high frequencies,
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however, there were inconsistencies when frequency were lower than a certain value.
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however, as there were inconsistencies when the frequency were lower than a threshold value.
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Wiens law led to an exponential curve, which disagree with the law of conservation.
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Max Planck guessed a result which led to Plancks radiation law, which he later derived
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using Boltzmanns statistical interpretation of the second law of thermodunamics.
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Plancks findings gave rise to Einsteins quantum hypothesis, and later the wave-particle
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Max Planck guessed a solution to the problem, which led to Planck's radiation law.
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He later derived the radiation law using Boltzmanns statistical interpretation of
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the second law of thermodynamics.
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Plancks findings gave rise to the quantum hypothesis, and later Einsteins wave-particle
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duality \cite{britannica:1998:planck}.
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For small atoms classical mechanics are not able to explain the position of a particle,
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For small atoms classical mechanics are not sufficient in describing the position of a particle,
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and Heisenberg uncertainty priciple suggest that the particles have a wavelike behavior.
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The wave-particle duality was later proposed to apply to particles by Louis de Broglie,
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which inspired Erwin Schrödinger who proposed a wave function to describe the quantum
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state of a particle, resulting in the wave equation.
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% Methods?
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% Schrödinger
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The Schrödinger equation has a general form
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\begin{align}
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i \hbar \frac{\partial}{\partial t} | \Psi \rangle &= \hat{H} | \Psi \rangle \ ,
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\label{eq:schrodinger_general}
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\end{align}
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where $i$ is the imaginary number, and $\hbar$ is Plancks constant. Here $| \Psi \rangle$
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is the quantum state and $\hat{H}$ is a Hamiltonian operator.
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For two-dimensional position space, the quantum state can be expressed using the
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time-dependent complex-valued wave function $\Psi (x, y, t)$. The square modulus of the wave function $|\psi|^{2}$, predicts probability of finding the particle at position $(x, y)$ at time t.
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% Segue to Born
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The modulus of the wave function, is related to the probability density function
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\begin{align}
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p(x, y \ | \ t) &= |\Psi(x, y, t)|^{2} = \Psi^{*}(x, y, t) \Psi(x, y, t)
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\label{eq:born_rule}
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\end{align}
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using the Born rule, where $^{*}$ denotes the complex conjugated wave function.
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When the potential is time-independent, the Schrödinger equation can be expressed as
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\begin{align*}
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i \hbar \frac{\partial}{\partial t} \Psi (x, y, t) &= - \frac{\hbar^{2}}{2m} \bigg( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \bigg) \Psi (x, y, t) \\
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& \quad + V(x, y, t) \Psi (x, y, t) \numberthis \ .
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\label{eq:schrodinger_special}
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\end{align*}
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When we scale Schrödinger equation by the dimensionful variables, we are left with
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a wave function $u$, potential $v$ and the dimensionless equation
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\begin{align}
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i \frac{\partial u}{\partial t} &= - \frac{\partial^{2} u}{\partial x^{2}} - \frac{\partial^{2} u}{\partial y^{2}} + v(x, y) u \ .
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\label{eq:schrodinger_dimensionless}
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\end{align}
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% Crank-Nicolson
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To evaluate the position of a single particle, we have to consider partial differential
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equations (PDE). To solve these numerically, we have to discretize Equation \eqref{eq:schrodinger_dimensionless}.
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We use the Crank-Nicolson method (CN), which combines the forward and backward finite
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difference method. %
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% Include the general $\theta$-method
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Using the $\theta$-rule \footnote{For $\theta \in [0, 1]$ we can derive Forward Euler using $\theta = 1$, Backward Euler using $\theta = 0$, and $\theta = 1/2$ gives Crank-Nicolson}
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with $\theta = 1/2$, CN can be written as
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\begin{align}
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\frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} &= \frac{1}{2} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \ .
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\label{eq:crank_nicolson}
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\end{align}
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To simplify notation and avoid confusion of indices with the imaginary number $i$,
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we have used the notation $\ivec, \jvec$ in subscript to indicate the commonly named indices $i, j$
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in x- and y-direction. In addition, the superscript $n, n+1$ indicate position in time.
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We use CN to derive the discretized Schrödinger equation
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\begin{align*}
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& u_{\ivec, \jvec}^{n+1} - \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1} \big] \\
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& - \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1} \big] + \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\
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&= u_{\ivec, \jvec}^{n} + \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n} \big] \\
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& \quad + \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n} \big] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \numberthis \ .
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\label{eq:schrodinger_discretized}
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\end{align*}
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The full derivation of both Equation \eqref{eq:crank_nicolson} and Equation \eqref{eq:schrodinger_discretized}
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can be found in Appendix \ref{ap:crank_nicolson}
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First, we Taylor
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expand the wave equation $u$ around (position and time).
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\section{Notes}\label{sec:notes}
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\subsection*{Introduction - draft 2}
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In classical mechanics we study the kinematics and dynamics of physical objects,
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ignoring its intrinsic properties for simplicity. 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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% Double-slit experiment
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Thomas Young first performed the double-slit experiment in 1801, to demonstrate
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the principle of interference of light \cite{britannica:2023:young}, postulating
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light as waves. In the study of blackbodies, scientists were not able to describe
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@ -107,6 +52,292 @@ contradicted the principle of conservation of energy \cite{britannica:1998:planc
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Max Planck assumed that the radiated energy consist of discrete values, or quanta,
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to describe the peak in radiated energy.
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% Reference: https://tex.stackexchange.com/questions/469109/how-to-change-arrowheads-to-lie-on-a-plane
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\begin{tikzpicture}[scale=1.25,every node/.append style={transform shape}]
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\draw (\x,-1) -- (\x,1);
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%\fill (2,0.5) circle (0.05) (2,-0.5) circle (0.05);
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\foreach \r in {0.25,0.5,...,2} {
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\draw (2,0.5) ++(-60:\r) arc (-60:60:\r);
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\draw (2,-0.5) ++(-60:\r) arc (-60:60:\r);
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}
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\draw[fill=black!10] (4,-2,-1) -- (4,-2,1) -- (4,2,1) -- (4,2,-1) -- (4,-2,-1);
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% LABELLING
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\begin{scope}[canvas is yz plane at x=2,rotate=-90]
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\node[circle,inner sep=0.5mm,fill,label=above:{S${}_1$}] at (0,0.5){};
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\draw[dash pattern=on 1.5pt off 1pt,thin] (-0.5,0.5) -- (0,0.5)
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(-0.5,-0.5) -- (0,-0.5);
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\pgflowlevelsynccm
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\draw[|<->|] (-0.5,0.5) -- (-0.5,-0.5) node[midway,right=-0.1cm] {d};
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\begin{axis}[
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width=5.575cm,
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\addplot [samples=1000,blue
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]
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{(cos(deg(5*pi*sin(deg(x)))))^(2)*((sin(deg(4*pi*sin(deg(x)))))/(4*pi*sin(deg(x))))^(2)};
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\draw[thin,densely dashed,blue] (2,0) -- +(-15:2.5);
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\draw[thin,densely dashed,blue] (2,0) -- +(32:2.5);
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\draw[thin,densely dashed,blue] (2,0) -- +(-32:2.5);
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\end{tikzpicture}
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% Author: Izaak Neutelings (June 2020)
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% Inspiration:
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% https://courses.physics.ucsd.edu/2011/Summer/session1/physics2c/diffraction.pdf
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% https://tex.stackexchange.com/questions/201830/periodic-shading-in-tikz
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% \begin{tikzpicture}[
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% nodal/.style={mylightgreen,dashed,very thin},
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% declare function={
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% xnode(\n,\dn,\lam,\f) = \lam/\f*sqrt( \n^2*(\f^2-\dn^2)+\n*\dn*(\f^2-\dn^2)+\dn^2*\f^2/2-(\f^4+\dn^4)/4 );
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% ynode(\n,\dn,\lam,\a) = (2*\n*\dn+\dn^2)*\lam/(2*\f);
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% intensity(\y,\lam,\a,\L) = cos(180*\a*\y/(2*\lam*sqrt(\L*\L+\y*\y)))^2;
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% }
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% ]
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% \def\L{3.8} % distance between walls
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% \def\H{5.4} % total wall height
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% \def\h{2.8} % plane wave height
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% \def\t{0.15} % wall thickness
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% \def\a{1.15} % slit distance
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% \def\d{0.20} % slit size
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% \def\N{21} % number of waves
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% \def\lambd{0.20} % wavelength
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% \def\R{\N*\lambd} % wave radius
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% \def\Nlines{3} % number of nodal lines
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% \def\A{1.6} % amplitude
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% %\def\r{0.06} % point source radius
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% %\clip (-\t/2,0.7*\a) -- (0.6*\L,\H/2) -- (\L,\H/2) --
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% % (\L,-\H/2) -- (0.6*\L,-\H/2) -- (-\t/2,-0.7*\a) -- cycle;
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% % NODAL LINES
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% \f=\a/\lambd;
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% \nmin=2.5+0.2*\dn; %0.501*(-\dn+\f)
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% \nmax=10; %(NP0)
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% \c=int(\dn<\f);
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% \y=\L/sqrt((\a/(\lambd*\dn))^2-1);
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% }] in {1,...,\Nlines}{
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% \coordinate (NP+\dn) at (\L,\y); % to avoid "Dimension too large error"
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% \ifnum\c=1
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% -- (NP+\dn);
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% \draw[nodal,variable=\n,samples=\nsamples,smooth]
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% plot[domain=\nmin:\nmax] ({xnode(\n,\dn,\lambd,\f)},{-ynode(\n,\dn,\lambd,\f)})
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% -- (NP-\dn);
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% \fi
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% }
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% % WAVES
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% \ifodd\i
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% % WALL
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% (\t/2,\a/2-\d/2) rectangle (-\t/2,-\a/2+\d/2)
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% (\t/2,\a/2+\d/2) rectangle (-\t/2,\H/2)
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% (\t/2,-\a/2-\d/2) rectangle (-\t/2,-\H/2)
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% (\L,-\H/2) rectangle (\L+\t,\H/2);
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% % SHADES
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% \begin{scope}[shift={(1.08*\L,0)}]
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% \yn=\L/sqrt((\a/(2*\lambd*\n))^2-1);
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% \ym=\L/sqrt((\a/(2*\lambd*\m))^2-1);
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% (0,-\yn) rectangle (\A,-\ym);
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% }
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% \end{scope}
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% }
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% \modd=\i; %int(\i);
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% \meven=int(\i-1);
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% \y=\L/sqrt((\a/(\lambd*\i))^2-1);
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% }] in {1,...,\Nlines}{
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% \ifodd\i
|
||||
% \tick{0,-\y}{180} node[right=0,scale=0.85] {$m=-\frac{\modd}{2}$};
|
||||
% \tick{0,\y}{180} node[right=0,scale=0.85] {$m=+\frac{\modd}{2}$};
|
||||
% \else
|
||||
% \tick{0,-\y}{180} node[right=0,scale=0.85] {$m=-\meven$};
|
||||
% \tick{0,\y}{180} node[right=0,scale=0.85] {$m=+\meven$};
|
||||
% \fi
|
||||
% }
|
||||
% \end{scope}
|
||||
|
||||
% \end{tikzpicture}
|
||||
|
||||
% Schrödinger
|
||||
% -----------
|
||||
% The wave equation, Schrödinger equation has a general form
|
||||
% \begin{align}
|
||||
% i \hbar \frac{\partial}{\partial t} | \Psi \rangle &= \hat{H} | \Psi \rangle \ ,
|
||||
% \label{eq:schrodinger_general}
|
||||
% \end{align}
|
||||
% where $i$ is the imaginary number, and $\hbar$ is Plancks constant. Here $| \Psi \rangle$
|
||||
% is the quantum state and $\hat{H}$ is a Hamiltonian operator. % which represent the energy of the system
|
||||
% For two-dimensional position space, the quantum state can be expressed using the
|
||||
% time-dependent complex-valued wave function $\Psi (x, y, t)$. The square modulus of the wave function $|\psi|^{2}$, predicts probability of finding the particle at position $(x, y)$ at time t.
|
||||
% % Segue to Born
|
||||
% The modulus of the wave function, is related to the probability density function
|
||||
% \begin{align}
|
||||
% p(x, y \ | \ t) &= |\Psi(x, y, t)|^{2} = \Psi^{*}(x, y, t) \Psi(x, y, t)
|
||||
% \label{eq:born_rule}
|
||||
% \end{align}
|
||||
% using the Born rule, where $^{*}$ denotes the complex conjugated wave function.
|
||||
% When the potential is time-independent, the Schrödinger equation can be expressed as
|
||||
% \begin{align*}
|
||||
% i \hbar \frac{\partial}{\partial t} \Psi (x, y, t) &= - \frac{\hbar^{2}}{2m} \bigg( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \bigg) \Psi (x, y, t) \\
|
||||
% & \quad + V(x, y, t) \Psi (x, y, t) \numberthis \ .
|
||||
% \label{eq:schrodinger_special}
|
||||
% \end{align*}
|
||||
% When we scale Schrödinger equation by the dimensionful variables, we are left with
|
||||
% a wave function $u$, potential $v$ and the dimensionless equation
|
||||
% \begin{align}
|
||||
% i \frac{\partial u}{\partial t} &= - \frac{\partial^{2} u}{\partial x^{2}} - \frac{\partial^{2} u}{\partial y^{2}} + v(x, y) u \ .
|
||||
% \label{eq:schrodinger_dimensionless}
|
||||
% \end{align}
|
||||
|
||||
% Crank-Nicolson
|
||||
% --------------
|
||||
% To evaluate the position of a single particle, we have to consider partial differential
|
||||
% equations (PDE). To solve these numerically, we have to discretize Equation \eqref{eq:schrodinger_dimensionless}.
|
||||
% We use the Crank-Nicolson method (CN), which combines the forward and backward finite
|
||||
% difference method.
|
||||
% Include the general $\theta$-method
|
||||
% Using the $\theta$-rule \footnote{For $\theta \in [0, 1]$ we can derive Forward Euler using $\theta = 1$, Backward Euler using $\theta = 0$, and $\theta = 1/2$ gives Crank-Nicolson}
|
||||
% with $\theta = 1/2$, CN can be written as
|
||||
% \begin{align}
|
||||
% \frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} &= \frac{1}{2} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \ .
|
||||
% \label{eq:crank_nicolson}
|
||||
% \end{align}
|
||||
% To simplify notation and avoid confusion of indices with the imaginary number $i$,
|
||||
% we have used the notation $\ivec, \jvec$ in subscript to indicate the commonly named indices $i, j$
|
||||
% in x- and y-direction. In addition, the superscript $n, n+1$ indicate position in time.
|
||||
% We use CN to derive the discretized Schrödinger equation
|
||||
% \begin{align*}
|
||||
% & u_{\ivec, \jvec}^{n+1} - \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1} \big] \\
|
||||
% & - \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1} \big] + \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\
|
||||
% &= u_{\ivec, \jvec}^{n} + \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n} \big] \\
|
||||
% & \quad + \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n} \big] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \numberthis \ .
|
||||
% \label{eq:schrodinger_discretized}
|
||||
% \end{align*}
|
||||
% The full derivation of both Equation \eqref{eq:crank_nicolson} and Equation \eqref{eq:schrodinger_discretized}
|
||||
% can be found in Appendix \ref{ap:crank_nicolson}.
|
||||
|
||||
% Implementation
|
||||
The implementation of CN is simplified usein Dirichlet boundary conditions, which
|
||||
can be found in Table \ref{tab:dirichlet}. In addition, we use Gaussian wave packets
|
||||
for the initial wave function
|
||||
\begin{align}
|
||||
u(x, y, t=0) = e^{- \frac{(x-x_{c})^{2}}{2 \sigma_{x}^{2}} - \frac{(y-y_{c})^{2}}{2 \sigma_{y}^{2}} + ip_{x}x + ip_{y}y}
|
||||
\end{align}
|
||||
\begin{algorithm}[H]
|
||||
\caption{Crank-Nicolson scheme}
|
||||
\label{algo:cn_scheme}
|
||||
\begin{algorithmic}
|
||||
\Procedure{Crank-Nicolson}{$args$}
|
||||
\State Insert pseudo code $var \leftarrow \text{ some text}$
|
||||
\EndProcedure
|
||||
\end{algorithmic}
|
||||
\end{algorithm}
|
||||
|
||||
|
||||
\section{Notes}\label{sec:notes}
|
||||
\subsection*{Introduction - draft 2}
|
||||
In classical mechanics we study the kinematics and dynamics of physical objects,
|
||||
ignoring their intrinsic properties for simplicity. 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.
|
||||
|
||||
% Thomas Young first performed the double-slit experiment in 1801, to demonstrate
|
||||
% the principle of interference of light \cite{britannica:2023:young}, postulating
|
||||
% light as waves. In the study of blackbodies, scientists were not able to describe
|
||||
% the radiated intensity of increased frequencies using classical mechanichs, as they
|
||||
% contradicted the principle of conservation of energy \cite{britannica:1998:planck}.
|
||||
% Max Planck assumed that the radiated energy consist of discrete values, or quanta,
|
||||
% to describe the peak in radiated energy.
|
||||
|
||||
Light as particles -> waves -> particles/packets
|
||||
Erwin Schrödinger wanted to find a mathematical description of the wave characteristics
|
||||
of matter, supporting the wave-particle idea. He postulated a function which varies
|
||||
|
||||
@ -9,11 +9,71 @@
|
||||
}
|
||||
|
||||
@misc{britannica:1999:light,
|
||||
author = {The Editors of Encyclopaedia Britannica},
|
||||
title = {Light},
|
||||
publisher = {Britannica},
|
||||
url = {https://www.britannica.com/science/light},
|
||||
urldate = {2023-12-15},
|
||||
author = {The Editors of Encyclopaedia Britannica},
|
||||
title = {Light},
|
||||
publisher = {Britannica},
|
||||
url = {https://www.britannica.com/science/light},
|
||||
urldate = {2023-12-15},
|
||||
}
|
||||
|
||||
@article{sinha:2010:multi_order_inference,
|
||||
author = {Urbasi Sinha and Christophe Couteau and Thomas Jennewein and Raymond Laflamme and Gregor Weihs},
|
||||
journal = {Science},
|
||||
title = {Ruling Out Multi-Order Interference in Quantum Mechanics},
|
||||
volume = {329},
|
||||
number = {5990},
|
||||
pages = {418--421},
|
||||
year = {2010},
|
||||
doi = {10.1126/science.1190545},
|
||||
url = {https://www.science.org/doi/abs/10.1126/science.1190545},
|
||||
urldate = {2023-12-16}
|
||||
}
|
||||
|
||||
@article{young:1804:double_slit,
|
||||
author = {Thomas Young},
|
||||
title = {I. The Bakerian Lecture. Experiments and calculations relative to physical optics},
|
||||
journal = {Philosophical Transactions of the Royal Society of London},
|
||||
volume = {94},
|
||||
pages = {1--16},
|
||||
year = {1804},
|
||||
doi = {10.1098/rstl.1804.0001},
|
||||
url = {https://royalsocietypublishing.org/doi/abs/10.1098/rstl.1804.0001},
|
||||
urldate = {2023-12-16},
|
||||
}
|
||||
|
||||
@book{benacquista:2018:classical_mechanics,
|
||||
author = {Matthew J. Benacquista and Joseph D. Romano},
|
||||
title = {Classical Mechanics},
|
||||
publisher = {Springer International Publishing : Imprint: Springer},
|
||||
year = {2018},
|
||||
edition = {1}
|
||||
}
|
||||
|
||||
@book{griffiths:2018:quantum_mechanics,
|
||||
author = {David J. Griffiths and Darrell F. Schroeter},
|
||||
title = {Introduction to quantum mechanics},
|
||||
publisher = {Cambridge University Press},
|
||||
year = {2018},
|
||||
edition = {3}
|
||||
}
|
||||
|
||||
@book{springer:2018:compendium_quantum_physics,
|
||||
author = {Daniel Greenberger and Klaus Hentschel and Friedel Weinert},
|
||||
title = {Compendium of Quantum Physics},
|
||||
publisher = {Springer Berlin, Heidelberg},
|
||||
year = {2009},
|
||||
edition = {1}
|
||||
}
|
||||
|
||||
@article{key,
|
||||
author = {Asher Peres and Daniel R. Terno},
|
||||
title = {Quantum Information and Relativity Theory},
|
||||
journal = {Reviews of Modern Physics},
|
||||
number = {1},
|
||||
volume = {76},
|
||||
doi = {10.1103/revmodphys.76.93},
|
||||
year = {2004},
|
||||
url = {http://dx.doi.org/10.1103/RevModPhys.76.93}
|
||||
}
|
||||
|
||||
# Schrodinger and Crank-Nicolson
|
||||
|
||||
@ -20,6 +20,7 @@
|
||||
%% I recommend downloading TeXMaker, because it includes a large library of the most common packages.
|
||||
|
||||
\usepackage{physics,amssymb} % mathematical symbols (physics imports amsmath)
|
||||
\usepackage[no-test-for-array]{nicematrix}
|
||||
|
||||
\usepackage{graphicx} % include graphics such as plots
|
||||
\graphicspath{.images/}
|
||||
@ -33,6 +34,29 @@
|
||||
\usepackage{algorithm}
|
||||
\usepackage[noend]{algpseudocode}
|
||||
\usepackage{tikz}
|
||||
% \usepackage{pgfplots}
|
||||
% \pgfplotsset{compat=1.18}
|
||||
% \usetikzlibrary{3d}
|
||||
\usepackage{xcolor}
|
||||
% \usepackage{etoolbox} %ifthen
|
||||
% \usetikzlibrary{calc}
|
||||
% \usetikzlibrary{arrows,arrows.meta}
|
||||
% \usetikzlibrary{decorations.markings}
|
||||
% \usetikzlibrary{angles,quotes}
|
||||
% \usetikzlibrary{fadings}
|
||||
% \tikzset{>=latex}
|
||||
% \colorlet{wall}{blue!30!black}
|
||||
% \colorlet{myblue}{blue!70!black}
|
||||
% \colorlet{myred}{red!70!black}
|
||||
% \colorlet{mydarkred}{red!50!black}
|
||||
% \colorlet{mylightgreen}{green!60!black!70}
|
||||
% \colorlet{mygreen}{green!60!black}
|
||||
% \colorlet{myredgrey}{red!50!black!80}
|
||||
% \colorlet{myshadow}{blue!30!black!90}
|
||||
% \tikzstyle{wave}=[myblue,thick]
|
||||
% \tikzstyle{mydashed}=[black!70,dashed,thin]
|
||||
% \tikzstyle{mymeas}=[{Latex[length=3,width=2]}-{Latex[length=3,width=2]},thin]
|
||||
% \tikzstyle{mysmallarr}=[-{Latex[length=3,width=2]}]
|
||||
% \usetikzlibrary{quantikz}
|
||||
% defines the color of hyperref objects
|
||||
% Blending two colors: blue!80!black = 80% blue and 20% black
|
||||
@ -44,7 +68,9 @@
|
||||
|
||||
% Biblio stuff
|
||||
% \def\biblio{\bibliographystyle{plain}\bibliography{../references/references}}
|
||||
|
||||
\newcommand\numberthis{\addtocounter{equation}{1}\tag{\theequation}}
|
||||
\newcommand{\rvline}{\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}}
|
||||
|
||||
% Defines indices i and j to avoid confusion with imaginary i
|
||||
\newcommand{\ivec}{\hat{\imath}}
|
||||
@ -67,10 +93,10 @@
|
||||
\maketitle
|
||||
|
||||
% Introduction
|
||||
% \subfile{sections/introduction}
|
||||
\subfile{sections/introduction}
|
||||
|
||||
% Methods
|
||||
% \subfile{sections/methods}
|
||||
\subfile{sections/methods}
|
||||
|
||||
% Results
|
||||
% \subfile{sections/results}
|
||||
@ -79,7 +105,7 @@
|
||||
% \subfile{sections/conclusion}
|
||||
|
||||
% Notes
|
||||
\subfile{draft}
|
||||
% \subfile{draft}
|
||||
|
||||
\clearpage
|
||||
\newpage
|
||||
|
||||
@ -3,7 +3,7 @@
|
||||
\begin{document}
|
||||
\appendix
|
||||
\section{The Crank-Nicholson method}\label{ap:crank_nicolson}
|
||||
The Crank-Nicolson \(CN\) approach considers both the forward difference, an explicit scheme,
|
||||
The Crank-Nicolson (CN) approach consider both the forward difference, an explicit scheme,
|
||||
\begin{equation*}
|
||||
\frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} = F_{\ivec, \jvec}^{n} \ ,
|
||||
\end{equation*}
|
||||
|
||||
@ -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!
|
||||
|
||||
@ -2,28 +2,231 @@
|
||||
|
||||
\begin{document}
|
||||
\section{Methods}\label{sec:methods} %
|
||||
\subsection{The Schrödinger equation}\label{ssec:schrodinger} %
|
||||
% 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
|
||||
\begin{align}
|
||||
i \hbar \frac{\partial}{\partial t} | \Psi \rangle &= \hat{H} | \Psi \rangle \ ,
|
||||
\label{eq:schrodinger_general}
|
||||
\end{align}
|
||||
where $i$ is the imaginary number, and $\hbar$ is Plancks constant. $\hat{H}$ is
|
||||
a Hamiltonian operator, which represent the energy for the system, and $| \Psi \rangle$
|
||||
is the quantum state. In two-dimensional position space, the quantum state can
|
||||
be expressed using the time-dependent complex-valued wave function $\Psi (x, y, t)$.
|
||||
Using Born rule, the square modulus of the wave function is proportional to the
|
||||
probability density of finding a particle at position $(x, y)$ at time t. The relation
|
||||
is given by
|
||||
\begin{align}
|
||||
p(x, y \ | \ t) &= |\Psi(x, y, t)|^{2} = \Psi^{*}(x, y, t) \Psi(x, y, t) \ ,
|
||||
\label{eq:born_rule}
|
||||
\end{align}
|
||||
where $\Psi^{*}$ denotes the complex conjugated wave function.
|
||||
% Add something about kinetic and potential energy, to introduce the potential V
|
||||
When the potential is time-independent, the Schrödinger equation can be expressed as
|
||||
\begin{align*}
|
||||
i \hbar \frac{\partial}{\partial t} \Psi (x, y, t) &= - \frac{\hbar^{2}}{2m} \bigg( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \bigg) \Psi (x, y, t) \\
|
||||
& \quad + V(x, y, t) \Psi (x, y, t) \numberthis \ .
|
||||
\label{eq:schrodinger_special}
|
||||
\end{align*}
|
||||
The partial derivatives (...) gives the kinetic energy, and the potental $V$ is
|
||||
the external environment. In this experiment we will only consider the case where
|
||||
the potential is time-independent, resulting in $V = V(x, y)$
|
||||
When we scale Schrödinger equation by the dimensionful variables, we are left with
|
||||
a wave function $u$, potential $v$ and the dimensionless equation
|
||||
\begin{align}
|
||||
i \frac{\partial u}{\partial t} &= - \frac{\partial^{2} u}{\partial x^{2}} - \frac{\partial^{2} u}{\partial y^{2}} + v(x, y) u \ .
|
||||
\label{eq:schrodinger_dimensionless}
|
||||
\end{align} %
|
||||
This gives us the Born rule
|
||||
\begin{align}
|
||||
p(x, y \ | \ t) &= |u(x, y, t)|^{2} = u^{*}(x, y, t) u(x, y, t) \ .
|
||||
\label{eq:born_rule_scaled}
|
||||
\end{align}
|
||||
|
||||
\subsection{Double-slit experiment}\label{ssec:double_slit} %
|
||||
In the beginning of the 1800s, the general view was that light consisted of particles.
|
||||
However, in 1801 Thomas Young demonstrated the principle of interference of light
|
||||
\cite{britannica:2023:young}, while postulating light as waves rather than particles.
|
||||
% Thomas Young first performed the double-slit experiment in 1801, to demonstrate
|
||||
% the principle of interference of light \cite{britannica:2023:young}, postulating
|
||||
% light as waves. In the study of blackbodies, scientists were not able to describe
|
||||
% the radiated intensity of increased frequencies using classical mechanichs, as they
|
||||
% contradicted the principle of conservation of energy \cite{britannica:1998:planck}.
|
||||
% Max Planck assumed that the radiated energy consist of discrete values, or quanta,
|
||||
% to describe the peak in radiated energy.
|
||||
The double-slit experiment
|
||||
|
||||
\subsection{Schrödinger equation}\label{ssec:schrodinger} %
|
||||
|
||||
\subsection{Crank-Nicolson}\label{ssec:crank_nicolson} %
|
||||
\subsection{The Crank-Nicolson scheme}\label{ssec:crank_nicolson} %
|
||||
When we evaluate a particle in position space, we have to consider partial differential
|
||||
equations (PDE). To solve these numerically, we have to discretize Equation \eqref{eq:schrodinger_dimensionless}.
|
||||
We use the $\theta$-rule \footnote{Using the $\theta$-rule, we can derive Forward Euler using $\theta = 1$, and Backward Euler using $\theta = 0$},
|
||||
to combine the forward (explicit) and backward (implicit) finite difference method.
|
||||
The result is a linear combination of the explicit and implicit scheme, given by
|
||||
\begin{align}
|
||||
\frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} &= \theta F_{\ivec, \jvec}^{n+1} + (1 - \theta) F_{\ivec, \jvec}^{n} \ ,
|
||||
\label{eq:theta_rule}
|
||||
\end{align} %
|
||||
where $\theta \in [0, 1]$.
|
||||
To simplify notation and avoid confusion of indices with the imaginary number $i$,
|
||||
we have used the notation $\ivec, \jvec$ in subscript to indicate the commonly named indices $i, j$
|
||||
in x- and y-direction. In addition, the superscript $n, n+1$ indicate position in time.
|
||||
We get the Crank-Nicolson method (CN) when $\theta = 1/2$ gives Crank-Nicolson
|
||||
\begin{align}
|
||||
\frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} &= \frac{1}{2} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \ .
|
||||
\label{eq:crank_nicolson_method}
|
||||
\end{align} %
|
||||
Using CN, we derive the discretized Schrödinger equation given by
|
||||
\begin{align*}
|
||||
& u_{\ivec, \jvec}^{n+1} - \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1} \big] \\
|
||||
& - \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1} \big] + \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\
|
||||
&= u_{\ivec, \jvec}^{n} + \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n} \big] \\
|
||||
& \quad + \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n} \big] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \numberthis \ .
|
||||
\label{eq:schrodinger_discretized}
|
||||
\end{align*} %
|
||||
The full derivation of both Equation \eqref{eq:crank_nicolson_method} and Equation \eqref{eq:schrodinger_discretized}
|
||||
can be found in Appendix \ref{ap:crank_nicolson}.
|
||||
|
||||
|
||||
\subsection{The double-slit experiment}\label{ssec:double_slit} %
|
||||
Thomas Young first performed the double-slit experiment in 1801 to demonstrate the
|
||||
principle of interference of light \cite{britannica:2023:young}, while postulating
|
||||
light as waves rather than particles. The double-slit experiment gives a diffraction
|
||||
pattern, where constructive interference of light result in bright spots, and destructive
|
||||
interference result in dark spots.
|
||||
% Something about Heisenberg uncertainty principle
|
||||
|
||||
% Might be better to move theory on double-slit here and have a subsection of light
|
||||
% property etc. as a first section or in introduction?
|
||||
\subsection{Implementation}\label{ssec:implementation} %
|
||||
% Add tables of parameters used, initial conditions, notation etc.
|
||||
A, B are sparse csc matrix
|
||||
- theory of csc matrix?
|
||||
% \begin{equation*}
|
||||
% \begin{pNiceArray}{ccc|ccc}
|
||||
% \bullet & \bullet & & \bullet & & & & & \\
|
||||
% \bullet & \bullet & \bullet & & \bullet & & & & \\
|
||||
% & \bullet & \bullet & & & \bullet & & &
|
||||
% \end{pNiceArray}
|
||||
% \end{equation*}
|
||||
|
||||
\begin{equation*}
|
||||
\begin{bmatrix}
|
||||
\begin{matrix}
|
||||
\bullet & \bullet & \phantom{\bullet} \\
|
||||
\bullet & \bullet & \bullet \\
|
||||
\phantom{\bullet} & \bullet & \bullet
|
||||
\end{matrix}
|
||||
& \rvline &
|
||||
\begin{matrix}
|
||||
\bullet & \phantom{\bullet} & \phantom{\bullet} \\
|
||||
\phantom{\bullet} & \bullet & \phantom{\bullet} \\
|
||||
\phantom{\bullet} & \phantom{\bullet} & \bullet
|
||||
\end{matrix}
|
||||
& \rvline &
|
||||
\begin{matrix}
|
||||
\phantom{\bullet} & \phantom{\bullet} & \phantom{\bullet} \\
|
||||
\phantom{\bullet} & \phantom{\bullet} & \phantom{\bullet} \\
|
||||
\phantom{\bullet} & \phantom{\bullet} & \phantom{\bullet}
|
||||
\end{matrix} \\
|
||||
\hline
|
||||
\begin{matrix}
|
||||
\bullet & \phantom{\bullet} & \phantom{\bullet} \\
|
||||
\phantom{\bullet} & \bullet & \phantom{\bullet} \\
|
||||
\phantom{\bullet} & \phantom{\bullet} & \bullet
|
||||
\end{matrix}
|
||||
& \rvline &
|
||||
\begin{matrix}
|
||||
\bullet & \bullet & \phantom{\bullet} \\
|
||||
\bullet & \bullet & \bullet \\
|
||||
\phantom{\bullet} & \bullet & \bullet
|
||||
\end{matrix}
|
||||
& \rvline &
|
||||
\begin{matrix}
|
||||
\bullet & \phantom{\bullet} & \phantom{\bullet} \\
|
||||
\phantom{\bullet} & \bullet & \phantom{\bullet} \\
|
||||
\phantom{\bullet} & \phantom{\bullet} & \bullet
|
||||
\end{matrix} \\
|
||||
\hline
|
||||
\begin{matrix}
|
||||
\phantom{\bullet} & \phantom{\bullet} & \phantom{\bullet} \\
|
||||
\phantom{\bullet} & \phantom{\bullet} & \phantom{\bullet} \\
|
||||
\phantom{\bullet} & \phantom{\bullet} & \phantom{\bullet}
|
||||
\end{matrix}
|
||||
& \rvline &
|
||||
\begin{matrix}
|
||||
\bullet & \phantom{\bullet} & \phantom{\bullet} \\
|
||||
\phantom{\bullet} & \bullet & \phantom{\bullet} \\
|
||||
\phantom{\bullet} & \phantom{\bullet} & \bullet
|
||||
\end{matrix}
|
||||
& \rvline &
|
||||
\begin{matrix}
|
||||
\bullet & \bullet & \phantom{\bullet} \\
|
||||
\bullet & \bullet & \bullet \\
|
||||
\phantom{\bullet} & \bullet & \bullet
|
||||
\end{matrix}
|
||||
\end{bmatrix}
|
||||
\end{equation*}
|
||||
|
||||
Notations:
|
||||
In addition, we use an equal step size in x- and y-direction, $h$ such that
|
||||
\begin{align*}
|
||||
x \in [0, 1] && x \rightarrow x_{\ivec} = \ivec h && \ivec = 0, 1, \dots, M-1 \\
|
||||
y \in [0, 1] && y \rightarrow y_{\jvec} = \jvec h && \jvec = 0, 1, \dots, M-1 \\
|
||||
t \in [0, T] && t \rightarrow t_{n} = n \Delta t && n = 0, 1, \dots, N_{t}-1
|
||||
\end{align*}
|
||||
And simplify indices such that
|
||||
\begin{align*}
|
||||
u(x, y, t) \rightarrow u(\ivec h, \jvec h, n \Delta t) \equiv u_{\ivec, \jvec}^{n} \\
|
||||
v(x, y) \rightarrow u(\ivec h, \jvec h) \equiv v_{\ivec, \jvec}
|
||||
\end{align*}
|
||||
which gives a matrix $U^{n}$ that contains elements $u_{\ivec, \jvec}^{n}$, and
|
||||
a matrix $V$ that contains elements $v_{\ivec, \jvec}$.
|
||||
\begin{table}[H]
|
||||
\centering
|
||||
\begin{tabular}{l l l} % @{\extracolsep{\fill}}
|
||||
\hline
|
||||
Position & Value \\
|
||||
\hline
|
||||
$u(x=0, y, t)$ & $0$ \\
|
||||
$u(x=1, y, t)$ & $0$ \\
|
||||
$u(x, y=0, t)$ & $0$ \\
|
||||
$u(x, y=1, t)$ & $0$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\caption{Boundary conditions in the xy-plane, also known as Dirichlet boundary conditions.}
|
||||
\label{tab:boundary_conditions}
|
||||
\end{table}
|
||||
|
||||
For the general setup of the wall, we used
|
||||
\begin{table}[H]
|
||||
\centering
|
||||
\begin{tabular}{l l} % @{\extracolsep{\fill}}
|
||||
\hline
|
||||
Parameter & Value \\
|
||||
\hline
|
||||
Wall thickness & $0.02$ \\
|
||||
Wall position & $0.5$ \\
|
||||
Separator length & $0.05$ \\
|
||||
Slit aperture & $0.05$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\caption{Wall setup.}
|
||||
\label{tab:wall_setup}
|
||||
\end{table}
|
||||
|
||||
\begin{table}[H]
|
||||
\centering
|
||||
\begin{tabular}{l l l} % @{\extracolsep{\fill}}
|
||||
\hline
|
||||
Simulation & $1$ & $2$ \\
|
||||
\hline
|
||||
$h$ & $0.005$ & $0.005$ \\
|
||||
$\Delta t$ & $2.5 \times 10^{-5}$ & $2.5 \times 10^{-5}$ \\
|
||||
$T$ & $0.008$ & $0.002$ \\
|
||||
$x_{c}$ & $0.25$ & $0.25$ \\
|
||||
$\sigma_{x}$ & $0.05$ & $0.05$ \\
|
||||
$p_{x}$ & $200$ & $200$ \\
|
||||
$y_{c}$ & $0.5$ & $0.5$ \\
|
||||
$\sigma_{y}$ & $0.05$ & $0.20$ \\
|
||||
$p_{y}$ & $0$ & $0$ \\
|
||||
$v_{0}$ & $0$ & $1 \times10^{10}$ \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\caption{Wall setup.}
|
||||
\label{tab:sim_setup}
|
||||
\end{table}
|
||||
|
||||
\subsection{Tools}\label{ssec:tools} %
|
||||
|
||||
The double-slit experiment is implemented in C++. We use the Python library
|
||||
\verb|matplotlib| \cite{hunter:2007:matplotlib} to produce all the plots, and
|
||||
\verb|seaborn| \cite{waskom:2021:seaborn} to set the theme in the figures.
|
||||
\end{document}
|
||||
|
||||
@ -2,6 +2,17 @@
|
||||
|
||||
\begin{document}
|
||||
\section{Results}\label{sec:results}
|
||||
\subsection{Deviation}\label{ssec:deviation}
|
||||
% Problem 3: Discuss approaches to solve Au^{n+1} = b, dealing with sparse matrix...
|
||||
% Problem 7: Consequenses of solver choice, in regards to accuracy of probability conserved
|
||||
% Add plot of deviation for both single- and double-slit
|
||||
|
||||
\subsection{Time evolution}\label{ssec:time_evolution}
|
||||
% Problem 8: Colormap, include plot of both Re and Im for different time steps
|
||||
% Account for color scale
|
||||
|
||||
\subsection{Particle detection}\label{ssec:particle_detection}
|
||||
% Problem 9: Plot detection probability for single-, double- and triple-slit
|
||||
|
||||
|
||||
\end{document}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user