.gitattributes

@@ -1 +0,0 @@
-. !text !filter !merge !diff

.gitignore

@@ -50,5 +50,3 @@ test
 
 # Score-p
 scorep*
-
-data/animation.txt

latex/references.bib

@@ -187,29 +187,6 @@
   pages     = {3021}
 }
 
-@article{harris:2020:numpy,
-  title     = {Array programming with {NumPy}},
-  author    = {Charles R. Harris and K. Jarrod Millman and St{\'{e}}fan J.
-               van der Walt and Ralf Gommers and Pauli Virtanen and David
-               Cournapeau and Eric Wieser and Julian Taylor and Sebastian
-               Berg and Nathaniel J. Smith and Robert Kern and Matti Picus
-               and Stephan Hoyer and Marten H. van Kerkwijk and Matthew
-               Brett and Allan Haldane and Jaime Fern{\'{a}}ndez del
-               R{\'{i}}o and Mark Wiebe and Pearu Peterson and Pierre
-               G{\'{e}}rard-Marchant and Kevin Sheppard and Tyler Reddy and
-               Warren Weckesser and Hameer Abbasi and Christoph Gohlke and
-               Travis E. Oliphant},
-  year      = {2020},
-  month     = sep,
-  journal   = {Nature},
-  volume    = {585},
-  number    = {7825},
-  pages     = {357--362},
-  doi       = {10.1038/s41586-020-2649-2},
-  publisher = {Springer Science and Business Media {LLC}},
-  url       = {https://doi.org/10.1038/s41586-020-2649-2}
-}
-
 # C++ libraries
 @misc{openmp:2018,
   author  = {OpenMP},

latex/schrodinger_simulation.tex

@@ -82,8 +82,8 @@
 
 \begin{document}
 
-\title{Simulating the Schrödinger wave equation using the Crank-Nicolson scheme in 2+1 dimensions}  % self-explanatory
-\author{Cory Alexander Balaton \& Janita Ovidie Sandtrøen Willumsen \\ \faGithub \, \url{https://gitea.balaton.dev/FYS3150/Project-5}} % self-explanatory
+\title{Simulating the Schrödinger wave equation using the Crank-Nicolson method in 2+1 dimensions}  % self-explanatory
+\author{Cory Alexander Balaton \& Janita Ovidie Sandtrøen Willumsen \\ \faGithub \, \url{https://github.uio.no/FYS3150-G2-2023/Project-4}} % self-explanatory
 \date{\today}                             % self-explanatory
 \noaffiliation                            % ignore this, but keep it.
 
@@ -102,7 +102,7 @@
 \subfile{sections/results}
 
 % Conclusion
-\subfile{sections/conclusion}
+% \subfile{sections/conclusion}
 
 % Notes
 % \subfile{draft}
@@ -122,14 +122,12 @@
 
 \end{document}
 
-% Abstract: OK!
-% Introduction: Hyugen, Heisenberg
-% Methods: OK!
-% Results: OK
-% Conclusion: OK!
-% Appendices: 
+% Methods
+% P1: Theory, imag i = i, index i, j = \hat{i}, \hat{j}
+% P2:
+% 
 
 % Results
-% P7: OK!
-% P8: OK!
-% P9: OK
+% P7:
+% P8:
+% P9:

latex/sections/abstract.tex

@@ -3,18 +3,16 @@
 \begin{document}
 \begin{abstract}
 We have simulated the two-dimensional time-dependent Schrödinger equation, to study 
-variations of the double-slit experiment. To derive a discretized equation 
-we applied the Crank-Nicolson scheme in 2+1 dimensions. In addition, we have used 
-Dirichlet boundary conditions to express the equation in matrix form, and solve 
-it using the sparse matrix solver \verb|superlu|. Our implementation, and choice 
-of solver method, resulted in a deviation from conserved total probability on the 
-scale $10^{-14}$, for both the single and double slit setup. To illustrate the time 
-evolution of the probability function, we created colormap plots for time steps 
-$t=\{0, 0.001, 0.002\}$. We also included separate plots for each time step of 
-Re$(u_{\ivec, \jvec})$ and Im$(u_{\ivec, \jvec})$. In addition, we determined the 
-normalized particle detection probability $p(y \ | \ x=0.8, t=0.002)$, for single-, 
-double- and triple-slit.
+variations of the double-slit experiment. To solve the partial differential equations 
+we have applied the Crank-Nicolson scheme in 2+1 dimensions, to derive a discretized 
+equation. In addition, we have used Dirichlet boundary conditions to express the 
+equation in matrix form and solve it using the sparse matrix solver \verb|superlu|. 
+Our implementation, and choice of solver method, resulted in conserved total 
+probability $\sum_{\ivec, \jvec} p_{\ivec, \jvec}^{n}=1$ for both the single and 
+double slit setup. To illustrate the time evolution of the probability function, 
+we created colormap plots at time steps $t = [0, 0.001, 0.002]$. We also included 
+separate plots for each time step of Re$(u_{\ivec, \jvec})$ and Im$(u_{\ivec, \jvec})$.
+In addition, we determined the normalized particle detection probability $p(y \ | \ x=0.8, t=0.002)$, 
+for single-, double- and triple-slit.
 \end{abstract}
 \end{document}
-
-% $| \sum_{\ivec, \jvec} p_{\ivec, \jvec}^{n} - 1 | \approx $ 

latex/sections/appendices.tex

@@ -3,8 +3,7 @@
 \begin{document}
 \appendix
 \section{The Crank-Nicholson method}\label{ap:crank_nicolson}
-The Crank-Nicolson (CN) approach consider 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*}
@@ -21,17 +20,15 @@ The parameter $\theta$ is introduced for a general approach, however, for CN $\t
     \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] \\
 \end{align*}
 
-We need the first derivative in respect to both time and position, as well as the 
-second derivative in respect to position. Taylor expanding will result in a discretized 
-version.
+We need the first derivative in respect to both time and position, as well as the second derivative in respect to position. Taylor expanding will result in a discretized version, assume this is known...
 
-The Schrödinger equation contains $i$ on the left hand side, we rewrite it as
+Schrödinger contain $i$ at the lhs, factor it as
 \begin{align*}
     \frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} &= \frac{1}{2i} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \\
-    &= -\frac{i}{2} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \ ,
+    &= -\frac{i}{2} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] & \text{, where $\frac{1}{i} = -i$}
 \end{align*}
-where $1/i = -i$. Using Equation \eqref{eq:schrodinger_dimensionless}, which is 
-found in Section \ref{ssec:schrodinger}, we get
+
+Using Equation \eqref{eq:schrodinger_dimensionless}, we get
 \begin{align*}
     u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n} & -\frac{i \Delta t}{2} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \\
     &= -\frac{i \Delta t}{2} \bigg[ - \frac{u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1}}{2 \Delta x^{2}} \\
@@ -47,7 +44,8 @@ the left hand side, and the terms containing $n$ time step on the right hand sid
     &= 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} 
 \end{align*}
-Since we use an equal step size $h$ in both $x$ and $y$ direction, we can use 
+In addition, since we will use an equal step size $h$ in both $x$ and $y$ direction, 
+we can use 
 \begin{align*}
     \frac{i \Delta t}{2 \Delta h^{2}} = \frac{i \Delta t}{2 \Delta x^{2}} = \frac{i \Delta t}{2 \Delta y^{2}} \ ,
 \end{align*}
@@ -55,7 +53,7 @@ and define
 \begin{align*}
     r \equiv \frac{i \Delta t}{2 \Delta h^{2}}
 \end{align*}
-Now, the discretized Schrödinger equation can be rewritten as 
+Now, the discretized Schrödinger equation can be written as 
 \begin{align*}
     & u_{\ivec, \jvec}^{n+1} - r \big[ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1} \big] \\
     & - r \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} \\
@@ -65,7 +63,7 @@ Now, the discretized Schrödinger equation can be rewritten as
 
 
 \section{Matrix structure}\label{ap:matrix_structure}
-For a $u$ vector of length $(M-2) = 3$, the matrices $A$ and $B$ have size 
+For $u$ vector of length $(M-2) = 3$, the matrices $A$ and $B$ have size 
 $(M-2)^{2} \times (M-2)^{2} = 9 \times 9$ given by 
 \begin{align*}
     A = 
@@ -96,65 +94,4 @@ $(M-2)^{2} \times (M-2)^{2} = 9 \times 9$ given by
     0   &  0   &  0   &  0    &   0   &  r    &  0   &  r    &  b_{8}  \\
     \end{bmatrix}
 \end{align*}
-
-
-\section{Figures}\label{ap:figures}
-We created colormap plots of the real and imaginary part of $u_{\ivec, \jvec}$, 
-in Figure \ref{fig:colormap_real_imag}.
-\begin{figure*}
-    \centering
-    \begin{subfigure}[b]{0.3\textwidth}
-        \centering
-        \includegraphics[width=\textwidth]{images/color_map_0_real.pdf}
-        \caption{Re$(u_{\ivec, \jvec})$ at time $t=0$.}
-        \label{fig:colormap_0_real}
-    \end{subfigure}
-    \hfill
-    \begin{subfigure}[b]{0.3\textwidth}
-        \centering
-        \includegraphics[width=\textwidth]{images/color_map_1_real.pdf}
-        \caption{Re$(u_{\ivec, \jvec})$ at time $t=0.001$.}
-        \label{fig:colormap_1_real}
-    \end{subfigure}
-    \hfill
-    \begin{subfigure}[b]{0.3\textwidth}
-        \centering
-        \includegraphics[width=\textwidth]{images/color_map_2_real.pdf}
-        \caption{Re$(u_{\ivec, \jvec})$ at time $t=0.002$.}
-        \label{fig:colormap_2_real}
-    \end{subfigure}
-    \newline
-    \begin{subfigure}[b]{0.3\textwidth}
-        \centering
-        \includegraphics[width=\textwidth]{images/color_map_0_imag.pdf}
-        \caption{Im$(u_{\ivec, \jvec})$ at time $t=0$.}
-        \label{fig:colormap_0_imag}
-    \end{subfigure}
-    \hfill
-    \begin{subfigure}[b]{0.3\textwidth}
-        \centering
-        \includegraphics[width=\textwidth]{images/color_map_1_imag.pdf}
-        \caption{Im$(u_{\ivec, \jvec})$ at time $t=0.001$.}
-        \label{fig:colormap_1_imag}
-    \end{subfigure}
-    \hfill
-    \begin{subfigure}[b]{0.3\textwidth}
-        \centering
-        \includegraphics[width=\textwidth]{images/color_map_2_imag.pdf}
-        \caption{Im$(u_{\ivec, \jvec})$ at time $t=0.002$.}
-        \label{fig:colormap_2_imag}
-    \end{subfigure}
-    \caption{The time evolution of Re($u_{\ivec, \jvec}$) and Im($u_{\ivec, \jvec}$), 
-    for time steps $t=[0, 0.001, 0.002]$.}
-    \label{fig:colormap_real_imag}
-\end{figure*}
-% \begin{figure*}
-%     \centering
-%     \includegraphics[width=0.9\textwidth]{images/color_map_all.pdf}
-%     \caption{The time evolution of the probability function $p_{\ivec, \jvec}^{n}$ (top row), 
-%     Re($u_{\ivec, \jvec}^{n}$) (middle row), and Im($u_{\ivec, \jvec}^{n}$) (bottom row). 
-%     Time step $t=0$ in the left column, $t=0.001$ in the middle column, and $t=0.002$ 
-%     in the right column.}
-%     \label{fig:colormap_all}
-% \end{figure*}
 \end{document}

latex/sections/conclusion.tex

@@ -2,23 +2,24 @@
 
 \begin{document}
 \section{Conclusion}\label{sec:conclusion}
-We simulated the two-dimensional time-dependent Schrödinger equation, and studied 
+% Rewrite this section to differ from the abstract
+We have simulated the two-dimensional time-dependent Schrödinger equation, to study 
 variations of the double-slit experiment. To solve the partial differential equations 
-we applied the Crank-Nicolson scheme in 2+1 dimensions, and derived a discretized 
-equation. We used Dirichlet boundary conditions to simplify the equation, 
-expressed the equation in matrix form and solved it using the sparse matrix solver 
-\verb|superlu|. The total probability $\sum_{\ivec, \jvec} p_{\ivec, \jvec}^{n}$ 
-deviated from $1.0$ by a factor of $10^{-14}$ for both the single and double slit 
-setup. % Add something about computational accuracy?
+we have applied the Crank-Nicolson scheme in 2+1 dimensions, and derived a discretized 
+equation. In addition, we have used Dirichlet boundary conditions to express the 
+equation in matrix form and solve it using the sparse matrix solver \verb|superlu|. 
+Our implementation, and choice of solver method, resulted in conserved total 
+probability $\sum_{\ivec, \jvec} p_{\ivec, \jvec}^{n}=1$ for both the single and 
+double slit setup. 
 
-We illustrated the time evolution of the probability function $p_{\ivec, \jvec}^{n} = u_{\ivec, \jvec}^{n*} u_{\ivec, \jvec}^{n}$, 
-using colormap plots for time steps $t = \{0, 0.001, 0.002\}$. In addition, we included 
-separate plots for each time step of Re$(u_{\ivec, \jvec})$ and Im$(u_{\ivec, \jvec})$, 
-to show the components of the complex values. This resulted in visible diffraction 
-patterns for the double-slit experiment.
+To illustrate the time evolution of the probability function $p_{\ivec, \jvec}^{n} = u_{\ivec, \jvec}^{n*} u_{\ivec, \jvec}^{n}$, 
+we created colormap plots at time steps $t = [0, 0.001, 0.002]$. Since we are working 
+with complex numbers, we included separate plots for each time step of Re$(u_{\ivec, \jvec})$ 
+and Im$(u_{\ivec, \jvec})$.
+% We observed something...
+
+In addition, we determined the normalized particle detection probability $p(y \ | \ x=0.8, t=0.002)$, 
+for single-, double- and triple-slit.
+% We observed something here as well...
 
-In addition, we studied the normalized particle detection probability $p(y \ | \ x=0.8, t=0.002)$, 
-for single-, double- and triple-slit setups. We found that increasing the number of slits 
-in the barrier, resulted in an increased number of areas of both high and low probability 
-for particle detection. It also increased the variance of particle detection probability.
 \end{document}

latex/sections/introduction.tex

@@ -2,70 +2,42 @@
 
 \begin{document}
 \section{Introduction}\label{sec:introduction}
-% Light: wave particle
-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, do not abide by the laws of classical mechanics. A solution was 
-proposed by Max Planck with the radiation law, which he later derived using Boltzmann's 
-statistical interpretation of the second law of thermodynamics. Planck's findings 
-gave rise to the quantum hypothesis, and later Einstein's wave-particle duality \cite{britannica:1998:planck}.
-
-The particle theory was the leading theory in the beginning of the 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. Kirchhoff's study of ideal blackbodies, 
-showed that light exhibits 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 particles, 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 barriers with single-slit and triple-slits. 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 methods 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}
-
+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 
+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)} 
@@ -83,7 +55,11 @@ discuss our findings. Lastly, we will conclude our findings in Section \ref{sec:
 % 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

@@ -3,81 +3,70 @@
 \begin{document}
 \section{Methods}\label{sec:methods} % 
 \subsection{The Schrödinger equation}\label{ssec:schrodinger} %
-Erwin Schrödinger wanted to find a mathematical description of the wave characteristics 
-of matter, which supported 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 
+% 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 unit, and $\hbar$ is the reduced Planck's constant. $\hat{H}$ is 
-a Hamiltonian operator, which represents the energy for the system, and $| \Psi \rangle$ 
+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 the Born rule, the square modulus of the wave function is proportional to the 
-probability density of detecting a particle at position $(x, y)$ at time $t$. The 
-relation is given by 
+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 conjugate of the wave function. When the potential 
-is time-independent, and the particle is non-relativistic, the Schrödinger equation 
-can be expressed as 
+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 are expressions of the kinetic energy, and the potental $V$ 
-encodes 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 the Schrödinger equation by the dimensionful variables, we are left with 
-the wave function $u$ and the potential $v$. The dimensionless equation is given by 
+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} % 
-As a result of working in position space, the Born rule is given by 
+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) \ ,
+    p(x, y \ | \ t) &= |u(x, y, t)|^{2} = u^{*}(x, y, t) u(x, y, t) \ .
     \label{eq:born_rule_scaled}
 \end{align}
-where we assume a normalized wave function $u(x, y, t)$. We will initialize the wave 
-function, using a Gaussian wavepacket, given by 
-\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*}
-$x_{c}$ and $y_{c}$ are the coordinates of the center of the wavepacket, $\sigma_{x}$ 
-and $\sigma_{y}$ are the width of the wavepacket. The wave packet momenta are 
-given by $p_{x}$ and $p_{y}$.
+
 
 
 \subsection{The Crank-Nicolson scheme}\label{ssec:crank_nicolson} %
-When we evaluate a particle's position, we have to consider partial differential 
-equations (PDEs). To solve these numerically, we have to discretize Equation \eqref{eq:schrodinger_dimensionless}. 
-We use the $\theta$-rule \footnote{We can derive Forward Euler using $\theta = 1$, and Backward Euler using $\theta = 0$, in the $\theta$-rule.}, 
-to combine the forward (explicit) and backward (implicit) finite difference methods. 
+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 the notation, and avoid any confusion of the indices with the imaginary unit $i$, 
+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 derive the Crank-Nicolson scheme (CN) by using $\theta = 1/2$, given by
+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_scheme}
+    \label{eq:crank_nicolson_method}
 \end{align} %
-Using CN, we derive the discretized Schrödinger equation, given by 
+Using CN, we derive the discretized Schrödinger equation given by 
 \begin{align*}
     & u_{\ivec, \jvec}^{n+1} - r \big[ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1} \big] \\
     & - r \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} \\
@@ -87,35 +76,34 @@ Using CN, we derive the discretized Schrödinger equation, given by
 \end{align*} %
 where $r$ is defined as
 \begin{align*}
-    r \equiv \frac{i \Delta t}{2 \Delta h^{2}} \ .
+    r \equiv \frac{i \Delta t}{2 \Delta h^{2}}
 \end{align*} %
-The full derivation of both Equation \eqref{eq:crank_nicolson_scheme} and Equation \eqref{eq:schrodinger_discretized} 
+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 the interference of light \cite{britannica:2023:young}, while postulating 
-light as waves rather than particles. The double-slit experiment results in a diffraction 
+principle of interference of light \cite{britannica:2023:young}, while postulating 
+light as waves rather than particles. The double-slit experiment result in a diffraction 
 pattern on a detector screen, where constructive interference of light result in 
-bright spots, and destructive interference result in dark spots. An illustration 
-of Thomas Young's setup can be found in Figure \ref{fig:youngs_double_slit}.
+bright spots, and destructive interference result in dark spots as showed in Figure 
+\ref{fig:youngs_double_slit}.
 \begin{figure}
     \centering 
     \includegraphics[width=\linewidth]{images/youngs_double_slit.pdf}
     \caption{The setup of Thomas Young's double slit experiment, where $S_{0}$ denotes 
-    the light source, $S_{1}$ and $S_{2}$ denotes the slits in the barrier \cite[p. 4]{mit:2004:physics}.}
+    the light source, $S_{1}$ and $S_{2}$ denotes the slits in the wall.}
     \label{fig:youngs_double_slit}
 \end{figure}
 
-After the wave passes through the barrier, the pattern observed is determined by 
-the path difference given by 
-\begin{align}
+After the wave passes through the two slits, the pattern observed is determined by 
+the path difference determined by 
+\begin{align*}
     \delta = d \sin (\theta) = m \lambda \ ,
-    \label{eq:interference}
-\end{align}
+\end{align*}
 where $\lambda$ is the wavelength and $m$ is called the order number. $d$ is the 
-distance between the center of the two slits, while assuming that the distance between 
+distance between the center of the two slits, assuming that the distance between 
 the wall and the detector screen $L >> \delta$ \cite[p. 6]{mit:2004:physics}. In 
 this case, we observe constructive interference when 
 \begin{align*}
@@ -123,31 +111,23 @@ this case, we observe constructive interference when
 \end{align*} 
 and destructive interference when
 \begin{align*}
-    \delta = (m + \frac{1}{2}) \lambda && m = 0, \pm 1, \pm 2 \dots \ .
+    \delta = (m + \frac{1}{2}) \lambda && m = 0, \pm 1, \pm 2 \dots \ ,
 \end{align*} 
+
 % Something about Heisenberg uncertainty principle 
 
-
 \subsection{Implementation}\label{ssec:implementation} % 
-In this experiment, we set up the grid with an equal step size in the x- and y-direction $h$, 
-and step size in the t-direction $\Delta t$, 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*}
-In addition, we simplify the 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 results in a matrix $U^{n}$ that contains elements $u_{\ivec, \jvec}^{n}$, and 
-a matrix $V$ that contains elements $v_{\ivec, \jvec}$. We used Dirichlet boundary 
-conditions, given by 
-\begin{align*}
-    u(x=0, y, t) &= 0  & u(x=1, y, t) &= 0 \\
-    u(x, y=0, t) &= 0 & u(x, y=1, t) &= 0 \ ,
-\end{align*}
+% 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*}
+We use Dirichlet boundary conditions, as given in Table \ref{tab:boundary_conditions}, 
 which allows us to express Equation \eqref{eq:schrodinger_discretized} as a matrix 
 equation 
 \begin{align}
@@ -157,8 +137,8 @@ Here, both $u^{n+1}$ and $u^{n}$ are column vectors containing the internal poin
 of the $xy$ grid at time step $n+1$ and $n$, respectively. Since we have $M$ points 
 in $x$- and $y$-direction, we have $M-2$ internal points. Both $u$ vectors have 
 length $(M-2)^{2}$, and the matrices $A$ and $B$ have size $(M-2)^{2} \times (M-2)^{2}$. 
-The matrices are sparse and can be decomposed as submatrices of size $(M-2) \times (M-2)$, 
-with the following pattern 
+The matrices can be decomposed as submatrices of size $(M-2) \times (M-2)$, with 
+the following pattern 
 \begin{align*}
     A, B = 
     \begin{bmatrix}
@@ -215,18 +195,46 @@ with the following pattern
            \bullet & \bullet & \bullet \\
            \phantom{\bullet} & \bullet & \bullet 
        \end{matrix}
-    \end{bmatrix} \ .
+    \end{bmatrix}
 \end{align*}
-To fill the matrices $A$ and $B$, we use 
+To fill the matrices $A$ and $B$, we used 
 \begin{align*}
     a_{k} &= 1 + 4r + \frac{i \Delta t}{2} v_{\ivec, \jvec} \\
     b_{k} &= 1 - 4r - \frac{i \Delta t}{2} v_{\ivec, \jvec} \ .
 \end{align*} 
-An example of a pair of filled matrices can be found in Appendix \ref{ap:matrix_structure}.
+An example of filled matrices can be found in Appendix \ref{ap:matrix_structure}.
 
-For the general setup of the barrier, we used the values in Table \ref{tab:barrier_setup}, 
-and for the simulations, we used the parameter settings in Table \ref{tab:sim_settings}. 
-% Insert Heisenberg uncertainty here? Or refer to it?
+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 r} % @{\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 r} % @{\extracolsep{\fill}}
@@ -239,14 +247,15 @@ and for the simulations, we used the parameter settings in Table \ref{tab:sim_se
         Slit aperture & $0.05$  \\
         \hline
     \end{tabular}
-    \caption{Barrier parameters and values.}
-    \label{tab:barrier_setup}
+    \caption{Wall setup.}
+    \label{tab:wall_setup}
 \end{table}
+
 \begin{table}[H]
     \centering
     \begin{tabular}{l r r} % @{\extracolsep{\fill}}
         \hline
-        Parameter & Setting 1 & Setting 2 \\
+        Simulation & $1$ & $2$ \\
         \hline 
         $h$ & $0.005$ & $0.005$ \\
         $\Delta t$ & $2.5 \times 10^{-5}$ & $2.5 \times 10^{-5}$ \\
@@ -260,39 +269,12 @@ and for the simulations, we used the parameter settings in Table \ref{tab:sim_se
         $v_{0}$ & $0$ & $1 \times10^{10}$ \\
         \hline
     \end{tabular}
-    \caption{Simulation settings used in the double slit experiment. Setting 1 is 
-    applied when the barrier is switched off and setting 2 is applied when the barrier 
-    switched on.}
-    \label{tab:sim_settings}
+    \caption{Wall setup.}
+    \label{tab:sim_setup}
 \end{table}
 
-To check if the total probability is conserved over time, and that the implementation 
-was correct, we computed the deviation from $1.0$ given by
-\begin{align*}
-    s^{n} &= |1.0 - \sum_{\ivec , \jvec} p_{\ivec , \jvec}^{n}| \\
-    &= |1.0 - \sum_{\ivec , \jvec} u_{\ivec , \jvec}^{n*} u_{\ivec , \jvec}^{n}| \ .
-\end{align*}
-
 \subsection{Tools}\label{ssec:tools} %
 The double-slit experiment is implemented in C++. We use the Python library 
-\verb|NumPy| \cite{harris:2020:numpy}, \verb|matplotlib| \cite{hunter:2007:matplotlib} to produce all the plots, and 
+\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}
-
-
-
-% \begin{table}[H]
-%     \centering
-%     \begin{tabular}{l r} % @{\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}

latex/sections/results.tex

@@ -4,125 +4,32 @@
 \section{Results}\label{sec:results}
 \subsection{Deviation}\label{ssec:deviation}
 % Problem 3: Discuss approaches to solve Au^{n+1} = b, dealing with sparse matrix...
-We used the \verb|superlu| solver, which is a solver for sparse matrices. It is 
-generally used to solve nonsymmetric, sparse matrices. However, as the alternative 
-solver \verb|lapack| converts a sparse matrix to a dense matrix, it will increase 
-memory usage compared to \verb|superlu|.
+We used the superlu solver, which is a dedicated solver for sparse matrices. It is 
+generally used to solve nonsymmetric, sparse matrices. However, as the lapack solver 
+converts the sparse matrix to a dense matrix, it will increase memory usage compared 
+to superlu.
 % Problem 7: Consequenses of solver choice, in regards to accuracy of probability conserved
 %            Add plot of deviation for both single- and double-slit
-Since we used a solver for sparse matrices, we decrease the number of computations performed
-compared to the number of computations using a solver for dense matrices. 
-We checked if the total probability was conserved over time, by plotting the deviation 
-from $1.0$.
+Since we use a solver for sparse matrices, we decrease number of computations performed 
+compared to solver using dense matrix. We check if the total probability is conserved 
+over time, by plotting the deviation $s$ as
+\begin{align*}
+    s^{n} = 1 - \sum_{\ivec , \jvec} p_{\ivec , \jvec}^{n} = 1 - \sum_{\ivec , \jvec} u_{\ivec , \jvec}^{n*} u_{\ivec , \jvec}^{n} \ .
+\end{align*}
+The deviation as a function of time is plotted in Figure \ref{fig:deviation}.
 \begin{figure}
     \centering
     \includegraphics[width=\linewidth]{images/probability_deviation.pdf}
-    \caption{Deviation of total probability, for time $t \in [0, T]$ where $T=0.008$.}
+    \caption{Deviation for $t \in [0, T]$ where $T=0.008$.}
     \label{fig:deviation}
 \end{figure}
-We simulated the wave equation with the barrier switched off, using setting 1 in 
-Table \ref{tab:sim_settings} found in Section \ref{ssec:implementation}. When the 
-barrier was switched on, we used setting 2 in \ref{tab:sim_settings}. We observed 
-a larger deviation of total probability for a barrier with double slits compared 
-to no barrier. The result can found in Figure \ref{fig:deviation}. When the wave interacts  
-with the barrier, it results in a larger change in kinetic energy. The result is more prone 
-to computational errors, than if the wave propagates without interacting with a 
-barrier. No interaction results in a more stable deviation from the total probability. 
-In addition, we have to consider the limitations of the computer, therefore some computational 
-error is to be expected.
-
 
 \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
-We studied the time evolution of the probability function, using setting 2 in 
-Table \ref{tab:sim_settings}, found in Section \ref{ssec:implementation}. To visualize 
-the time evolution, we created colormap plots for different time steps. Figure \ref{fig:colormap_0_prob}, 
-Figure \ref{fig:colormap_1_prob}, and Figure \ref{fig:colormap_2_prob} show the 
-results for time steps $t=\{0, 0.001, 0.002\}$, respectively. In addition, we created 
-separate plots for the real and imaginary part of $u_{\ivec, \jvec}$, for the same 
-time steps. The results can be found in Appendix \ref{ap:figures}, in Figure \ref{fig:colormap_real_imag}.
-\begin{figure}
-    \centering
-    \includegraphics[width=\linewidth]{images/color_map_0_prob.pdf}
-    \caption{The probability function $p_{\ivec, \jvec}^{n}$, at time $t=0$.}
-    \label{fig:colormap_0_prob}
-\end{figure}
-\begin{figure}
-    \centering
-    \includegraphics[width=\linewidth]{images/color_map_1_prob.pdf}
-    \caption{The probability function $p_{\ivec, \jvec}^{n}$, at time $t=0.001$.}
-    \label{fig:colormap_1_prob}
-\end{figure}
-\begin{figure}
-    \centering
-    \includegraphics[width=\linewidth]{images/color_map_2_prob.pdf}
-    \caption{The probability function $p_{\ivec, \jvec}^{n}$, at time $t=0.002$.}
-    \label{fig:colormap_2_prob}
-\end{figure}
-At time step $t=0.001$, Figure \ref{fig:colormap_1_prob}, when the wave interacts 
-with the double slit barrier, we observe a clear diffraction pattern in the 
-probability function. At time step $t=0$ (Figure \ref{fig:colormap_0_prob}) and 
-$t=0.002$ (Figure \ref{fig:colormap_2_prob}), the diffraction pattern is not as 
-clear. It is, however, more visible when we observe the real and imaginary part 
-separately in Figure \ref{fig:colormap_real_imag}. Since the probability function 
-is a product of $u_{\ivec, \jvec}$ and its conjugate $u_{\ivec, \jvec}^{*}$, 
-initialized by a Gaussian wavepacket, the result is a sum of the real and imaginary part. 
-% This can be found using Euler's formula, and the diffraction pattern is determined by interference given by \eqref{eq:interference}
-In Figure \ref{fig:colormap_2_prob}, the probability function result in positive 
-areas at both sides of the barries. Some of the probability function is reflected 
-by the barrier, while the the rest spread out after passing the barrier. This is 
-a consequence of the wave-particle duality. 
-
-To compare the probability function $p_{\ivec, \jvec}$ for all the time steps in 
-a colormap plot, with the real and imaginary part of $u_{\ivec, \jvec}$, we created 
-Figure \ref{fig:colormap_all}. Where we excluded all x- and y-ticks, and labels, to 
-better visualize the diffraction pattern.
-\begin{figure}
-    \centering
-    \includegraphics[width=\linewidth]{images/color_map_all.pdf}
-    \caption{The time evolution of the probability function $p_{\ivec, \jvec}^{n}$ (top row), 
-    Re($u_{\ivec, \jvec}$) (middle row), and Im($u_{\ivec, \jvec}$) (bottom row). 
-    Time step $t=0$ in the left column, $t=0.001$ in the middle column, and $t=0.002$ 
-    in the right column.}
-    \label{fig:colormap_all}
-\end{figure}
-
 
 \subsection{Particle detection}\label{ssec:particle_detection}
 % Problem 9: Plot detection probability for single-, double- and triple-slit
-We simulated the wave equation using setting 2 in Table \ref{tab:sim_settings}, 
-and assumed a detector screen located at $x=0.8$. To visualize the pattern of constructive 
-and destructive interference, we plotted the probability of particle detection, 
-along the screen, at time $t=0.002$. We adjusted the parameters to include single-, double-, and triple-slit 
-barriers. The results are found in Figure \ref{fig:particle_detection_single}, 
-Figure \ref{fig:particle_detection_double}, and Figure \ref{fig:particle_detection_triple}, 
-respectively.
-\begin{figure}
-    \centering
-    \includegraphics[width=\linewidth]{images/single_slit_detector.pdf}
-    \caption{Probability of particle detection along a detector screen at time $t=0.002$, 
-    when using a single-slit barrier.}
-    \label{fig:particle_detection_single}
-\end{figure}
-\begin{figure}
-    \centering
-    \includegraphics[width=\linewidth]{images/double_slit_detector.pdf}
-    \caption{Probability of particle detection along a detector screen at time $t=0.002$, 
-    when using a double-slit barrier.}
-    \label{fig:particle_detection_double}
-\end{figure}
-\begin{figure}
-    \centering
-    \includegraphics[width=\linewidth]{images/triple_slit_detector.pdf}
-    \caption{Probability of particle detection along a detector screen at time $t=0.002$, 
-    when using a triple-slit barrier.}
-    \label{fig:particle_detection_triple}
-\end{figure}
-When the barrier has a single slit, there is no destructive interference and we 
-observe a single peak in the probability of particle detection. Adding another slit 
-result in more peaks, as there is both constructive and destructive interference. 
-When we used a triple-slit barrier, we observed an increase in interference which 
-resulted in narrow peaks. In addition, the probability of detecting a particle at 
-the ends of the screen increased with the number of slits.
+
+
 \end{document}

python_scripts/colormap.py

@@ -19,7 +19,6 @@ plt.rcParams.update(params)
 
 
 def plot():
-    ticks = [0, 0.25, 0.5, 0.75, 1.0]
     with open("data/color_map.txt") as f:
         lines = f.readlines()
         size = int(lines[0])
@@ -39,19 +38,16 @@ def plot():
                 np.multiply(arr, arr.conj()).real,
                 interpolation="nearest",
                 cmap=sns.color_palette("mako", as_cmap=True),
-                extent=[0, 1.0, 0, 1.0]
             )
             color_map2 = ax2.imshow(
                 arr.real,
                 interpolation="nearest",
                 cmap=sns.color_palette("mako", as_cmap=True),
-                extent=[0, 1.0, 0, 1.0]
             )
             color_map3 = ax3.imshow(
                 arr.imag,
                 interpolation="nearest",
                 cmap=sns.color_palette("mako", as_cmap=True),
-                extent=[0, 1.0, 0, 1.0]
             )
 
             # Create color bar
@@ -64,26 +60,10 @@ def plot():
             ax2.grid(False)
             ax3.grid(False)
 
-            # Set custom ticks
-            ax1.set_xticks(ticks)
-            ax1.set_yticks(ticks)
-            ax2.set_xticks(ticks)
-            ax2.set_yticks(ticks)
-            ax3.set_xticks(ticks)
-            ax3.set_yticks(ticks)
-
-            # Set labels
-            ax1.set_xlabel("x-axis")
-            ax1.set_ylabel("y-axis")
-            ax2.set_xlabel("x-axis")
-            ax2.set_ylabel("y-axis")
-            ax3.set_xlabel("x-axis")
-            ax3.set_ylabel("y-axis")
-
             # Save the figures
-            fig1.savefig(f"latex/images/color_map_{i}_prob.pdf", bbox_inches="tight")
-            fig2.savefig(f"latex/images/color_map_{i}_real.pdf", bbox_inches="tight")
-            fig3.savefig(f"latex/images/color_map_{i}_imag.pdf", bbox_inches="tight")
+            fig1.savefig(f"latex/images/color_map_{i}_prob.pdf")
+            fig2.savefig(f"latex/images/color_map_{i}_real.pdf")
+            fig3.savefig(f"latex/images/color_map_{i}_imag.pdf")
 
             # Close figures
             plt.close(fig1)

python_scripts/colormap_all.py

@@ -1,62 +0,0 @@
-import ast
-
-import matplotlib.pyplot as plt
-import numpy as np
-import seaborn as sns
-
-sns.set_theme()
-params = {
-    "font.family": "Serif",
-    "font.serif": "Roman",
-    "text.usetex": True,
-    "xtick.bottom": False,
-    "xtick.labelbottom": False,
-    "ytick.left": False,
-    "ytick.labelleft": False,
-    "axes.grid": False,
-    # "figure.autolayout": True
-    # "figure.constrained_layout.use": False,
-    # "figure.constrained_layout.h_pad":  0.04167,
-    # "figure.constrained_layout.w_pad":  0.04167
-    # "figure.subplot.wspace": 0,
-    # "figure.subplot.hspace": 0
-}
-plt.rcParams.update(params)
-
-
-def plot():
-    fig, axes = plt.subplots(figsize=(6,6), nrows=3, ncols=3)
-    with open("data/color_map.txt") as f:
-        lines = f.readlines()
-        size = int(lines[0])
-        for i, line in enumerate(lines[1:]):
-            arr = line.strip().split("\t")
-            arr = np.asarray(list(map(lambda x: complex(*ast.literal_eval(x)), arr)))
-            # Reshape and transpose array
-            arr = arr.reshape(size, size).T
-
-            # Plot color maps
-            axes[0, i].imshow(
-                np.multiply(arr, arr.conj()).real,
-                interpolation="nearest",
-                cmap=sns.color_palette("mako", as_cmap=True)
-            )
-            axes[1, i].imshow(
-                arr.real,
-                interpolation="nearest",
-                cmap=sns.color_palette("mako", as_cmap=True)
-            )
-            axes[2, i].imshow(
-                arr.imag,
-                interpolation="nearest",
-                cmap=sns.color_palette("mako", as_cmap=True)
-            )
-
-    # Create tight subplots and save figure
-    fig.subplots_adjust(wspace=0, hspace=0)
-    fig.savefig(f"latex/images/color_map_all.pdf", bbox_inches="tight")
-
-
-if __name__ == "__main__":
-    plot()
-

python_scripts/detector.py

@@ -27,7 +27,6 @@ def plot():
         "latex/images/double_slit_detector.pdf",
         "latex/images/triple_slit_detector.pdf",
     ]
-    colors = sns.color_palette("mako", n_colors=2)
     for file, output in zip(files, outputs):
         with open(file) as f:
             lines = f.readlines();
@@ -44,10 +43,8 @@ def plot():
             slice *= norm
             slice = np.asarray([i * i.conjugate() for i in slice])
 
-            ax.plot(x, slice, color=colors[0])
-            ax.set_xlabel("Detector screen (y-axis)")
-            ax.set_ylabel("Detection probability")
-            fig.savefig(output, bbox_inches="tight")
+            ax.plot(x, slice)
+            plt.savefig(output)
             plt.close(fig)
 
 

python_scripts/deviation.py

@@ -21,17 +21,9 @@ def plot():
         "data/probability_deviation_no_slits.txt",
         "data/probability_deviation_slits.txt",
     ]
-    labels = [
-        "none",
-        "double-slit"
-    ]
-    colors = sns.color_palette("mako", n_colors=2)
-
-    fig, ax = plt.subplots() 
-
-    for file, label, color in zip(files, labels, colors):
+    for file in files:
         with open(file) as f:
-            lines = f.readlines()
+            lines = f.readlines();
             x = []
             arr = []
             for line in lines:
@@ -39,13 +31,10 @@ def plot():
                 x.append(float(tmp[0]))
                 arr.append(float(tmp[1]))
 
-            ax.plot(x, arr, label=label, color=color)
-        
-        ax.legend(title="Barrier")
-        ax.set_xlabel("Time (t)")
-        ax.set_ylabel("Deviation")
-    fig.savefig("latex/images/probability_deviation.pdf", bbox_inches="tight")
+            plt.plot(x,arr)
 
+        plt.savefig("latex/images/probability_deviation.pdf")
 
 if __name__ == "__main__":
-    plot()
+    plot()
+

python_scripts/plot_v.py

@@ -0,0 +1,40 @@
+import numpy as np
+import matplotlib.pyplot as plt
+import matplotlib
+from matplotlib.animation import FuncAnimation
+import ast
+
+import seaborn as sns 
+params = {
+    "font.family": "Serif",
+    "font.serif": "Roman",
+    "text.usetex": True,
+    "axes.titlesize": "large",
+    "axes.labelsize": "large",
+    "xtick.labelsize": "large",
+    "ytick.labelsize": "large",
+    "legend.fontsize": "medium",
+}
+plt.rcParams.update(params)
+
+
+def plot():
+    with open("v.txt") as f:
+        lines = f.readlines();
+        size = int(lines[0])
+        for line in lines[1:]:
+            arr = line.strip().split("\t")
+            arr = np.asarray(list(map(lambda x: ((a := complex(*ast.literal_eval(x)))*a.conjugate()).real, arr)))
+
+            # print(sum(arr))
+            arr = arr.reshape(size,size)
+            # print(arr)
+
+            plt.imshow(arr.T, cmap="hot", interpolation="nearest")
+            plt.show()
+
+
+
+
+if __name__ == "__main__":
+    plot()