latex/references.bib

@@ -187,6 +187,29 @@
   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

@@ -83,7 +83,7 @@
 \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://github.uio.no/FYS3150-G2-2023/Project-4}} % self-explanatory
+\author{Cory Alexander Balaton \& Janita Ovidie Sandtrøen Willumsen \\ \faGithub \, \url{https://github.uio.no/FYS3150-G2-2023/Project-5}} % self-explanatory
 \date{\today}                             % self-explanatory
 \noaffiliation                            % ignore this, but keep it.
 

latex/sections/abstract.tex

@@ -10,7 +10,7 @@ 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 
+$t=\{0, 0.001, 0.002\}$. 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.

latex/sections/appendices.tex

@@ -47,7 +47,7 @@ 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 will use an equal step size $h$ in both $x$ and $y$ direction, we can use 
+Since we 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*}
@@ -65,7 +65,7 @@ Now, the discretized Schrödinger equation can be rewritten as
 
 
 \section{Matrix structure}\label{ap:matrix_structure}
-For $u$ vector of length $(M-2) = 3$, the matrices $A$ and $B$ have size 
+For a $u$ 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 = 

latex/sections/conclusion.tex

@@ -2,19 +2,6 @@
 
 \begin{document}
 \section{Conclusion}\label{sec:conclusion}
-% 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.
-% Rewrite this section to differ from the abstract
 We simulated the two-dimensional time-dependent Schrödinger equation, and studied 
 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 
@@ -25,13 +12,13 @@ deviated from $1.0$ by a factor of $10^{-14}$ for both the single and double sli
 setup. % Add something about computational accuracy?
 
 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 
+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 a visible diffraction 
+to show the components of the complex values. This resulted in visible diffraction 
 patterns for the double-slit experiment.
 
 In addition, we studied the normalized particle detection probability $p(y \ | \ x=0.8, t=0.002)$, 
-for single-, double- and triple-slit. We found that increasing the number of slits 
+for single-, double- and triple-slit setups. We found that increasing the number of slits 
-in the barrier, increased the number of areas of both high and low probability of 
+in the barrier, resulted in an increased number of areas of both high and low probability 
-particle detection. 
+for particle detection. It also increased the variance of particle detection probability.
 \end{document}

latex/sections/introduction.tex

@@ -6,27 +6,27 @@
 The nature of light has long been a subject of interest and discussion. In classical 
 mechanics, we study the kinematics and dynamics of physical objects, while ignoring 
 their intrinsic properties for simplicity. Elementary particles, such as photons 
-and electrons, does not abide by the laws of classical mechanics. A solution was 
+and electrons, do not abide by the laws of classical mechanics. A solution was 
-proposed by Max Planck in the radiation law, which he later derived using Boltzmanns 
+proposed by Max Planck with the radiation law, which he later derived using Boltzmann's 
-statistical interpretation of the second law of thermodynamics. Plancks findings 
+statistical interpretation of the second law of thermodynamics. Planck's findings 
-gave rise to the quantum hypothesis, and later Einsteins wave-particle duality \cite{britannica:1998:planck}.
+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 1800s, when 
+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 
+Thomas Young demonstrated the interference of light, through his double-slit experiment,  
 while postulating light as waves \cite{young:1804:double_slit}. The study of 
-interference of light, and Gustav R. Kirchhoffs study of ideal blackbodies, 
+interference of light, and Gustav R. Kirchhoff's study of ideal blackbodies, 
-showed that light exibits both wavelike and particle-like characteristics. The 
+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 a particle, resulting in the wave equation.
+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 walls such as single- and triple-slit. To solve the equation, 
+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 algorithms and tools used in the implementation. 
+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}

latex/sections/methods.tex

@@ -3,11 +3,8 @@
 \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.
 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 
+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 
@@ -17,21 +14,20 @@ has a general form
     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 Plancks constant. $\hat{H}$ is 
+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 $| \Psi \rangle$ 
 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 
+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 
 \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. 
+where $\Psi^{*}$ denotes the complex conjugate of the wave function. When the potential 
-% Add something about kinetic and potential energy, to introduce the potential V
+is time-independent, and the particle is non-relativistic, the Schrödinger equation 
-When the potential is time-independent, and the particle is non-relativistic, 
+can be expressed as 
-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 \ .
@@ -39,9 +35,9 @@ the Schrödinger equation can be expressed as
 \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 $V = V(x, y)$
+the potential is time-independent, resulting in $V = V(x, y)$.
 
-When we scale Schrödinger equation by the dimensionful variables, we are left with 
+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 
 \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 \ .
@@ -63,9 +59,9 @@ given by $p_{x}$ and $p_{y}$.
 
 
 \subsection{The Crank-Nicolson scheme}\label{ssec:crank_nicolson} %
-When we evaluate a particles position, we have to consider partial differential 
+When we evaluate a particle's position, we have to consider partial differential 
-equations (PDE). To solve these numerically, we have to discretize Equation \eqref{eq:schrodinger_dimensionless}. 
+equations (PDEs). 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 $\theta = 0$}, 
+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. 
 The result is a linear combination of the explicit and implicit scheme, given by 
 \begin{align}
@@ -99,7 +95,7 @@ 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 
+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 
 pattern on a detector screen, where constructive interference of light result in 
 bright spots, and destructive interference result in dark spots. An illustration 
@@ -133,14 +129,14 @@ and destructive interference when
 
 
 \subsection{Implementation}\label{ssec:implementation} % 
-In this experiment, we set up the grid with an equal step size in x- and y-direction $h$, 
+In this experiment, we set up the grid with an equal step size in the x- and y-direction $h$, 
-and step size in t-direction $\Delta t$, such that
+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 simplified the indices such that 
+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} \ ,
@@ -152,7 +148,7 @@ conditions, given by
     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*}
-which allowed us to express Equation \eqref{eq:schrodinger_discretized} as a matrix 
+which allows us to express Equation \eqref{eq:schrodinger_discretized} as a matrix 
 equation 
 \begin{align}
     A u^{n+1} = B u^{n} \ .
@@ -221,12 +217,12 @@ with the following pattern
        \end{matrix}
     \end{bmatrix} \ .
 \end{align*}
-To fill the matrices $A$ and $B$, we used 
+To fill the matrices $A$ and $B$, we use 
 \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 filled matrices can be found in Appendix \ref{ap:matrix_structure}.
+An example of a pair 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}. 
@@ -265,7 +261,7 @@ and for the simulations, we used the parameter settings in Table \ref{tab:sim_se
         \hline
     \end{tabular}
     \caption{Simulation settings used in the double slit experiment. Setting 1 is 
-    used when the barrier is switched off and setting 2 is used when the barrier 
+    applied when the barrier is switched off and setting 2 is applied when the barrier 
     switched on.}
     \label{tab:sim_settings}
 \end{table}
@@ -279,7 +275,7 @@ was correct, we computed the deviation from $1.0$ given by
 
 \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|NumPy| \cite{harris:2020:numpy}, \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}
 

latex/sections/results.tex

@@ -11,7 +11,7 @@ memory usage compared to \verb|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 number of computations using a solver for dense matrices. 
+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 $1.0$.
 \begin{figure}
@@ -24,11 +24,11 @@ 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 is showed in Figure \ref{fig:deviation}. The wave interacts  
+to no barrier. The result can found in Figure \ref{fig:deviation}. When the wave interacts  
-with the barrier resulting in a change in kinetic energy. The result is more prone 
+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 limitation of a computer, some computational 
+In addition, we have to consider the limitations of the computer, therefore some computational 
 error is to be expected.
 
 
@@ -39,9 +39,9 @@ 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 
+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}.
+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}
@@ -65,8 +65,8 @@ 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}, found in Appendix \ref{ap:figures}. Since 
+separately in Figure \ref{fig:colormap_real_imag}. Since the probability function 
-the probability function is a product of $u_{\ivec, \jvec}$ and its conjugate $u_{\ivec, \jvec}^{*}$, 
+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 
@@ -91,11 +91,11 @@ better visualize the diffraction pattern.
 
 \subsection{Particle detection}\label{ssec:particle_detection}
 % Problem 9: Plot detection probability for single-, double- and triple-slit
-We simulation the wave equation using setting 2 in Table \ref{tab:sim_settings}, 
+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 is found in Figure \ref{fig:particle_detection_single}, 
+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}
@@ -121,8 +121,8 @@ respectively.
 \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 are both constructive and destructive interference. 
+result in more peaks, as there is both constructive and destructive interference. 
-When we use a triple-slit barrier, we observe an increase in interference which 
+When we used a triple-slit barrier, we observed an increase in interference which 
-result in narrow peaks. In addition, the probability of detecting a particle at 
+resulted in narrow peaks. In addition, the probability of detecting a particle at 
-the ends of the screen increase with number of slits.
+the ends of the screen increased with the number of slits.
 \end{document}