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23
latex/references.bib
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@ -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},

2
latex/schrodinger_simulation.tex
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@ -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.

2
latex/sections/abstract.tex
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@ -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.

4
latex/sections/appendices.tex
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@ -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 =

23
latex/sections/conclusion.tex
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@ -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
in the barrier, increased the number of areas of both high and low probability of
particle detection.
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}

22
latex/sections/introduction.tex
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@ -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
proposed by Max Planck in the radiation law, which he later derived using Boltzmanns
statistical interpretation of the second law of thermodynamics. Plancks findings
gave rise to the quantum hypothesis, and later Einsteins wave-particle duality \cite{britannica:1998:planck}.
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 1800s, when
Thomas Young demonstrated the interference of light, through his double-slit experiment
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. Kirchhoffs study of ideal blackbodies,
showed that light exibits both wavelike and particle-like characteristics. The
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 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}

44
latex/sections/methods.tex
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@ -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.
% Add something about kinetic and potential energy, to introduce the potential V
When the potential is time-independent, and the particle is non-relativistic,
the Schrödinger equation can be expressed as
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
\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
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 $\theta = 0$},
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.
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 $h$,
and step size in t-direction $\Delta t$, such that
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 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}

28
latex/sections/results.tex
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@ -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
with the barrier resulting in a change in kinetic energy. The result is more prone
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 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
the probability function is a product of $u_{\ivec, \jvec}$ and its conjugate $u_{\ivec, \jvec}^{*}$,
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
@ -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.
When we use a triple-slit barrier, we observe an increase in interference which
result in narrow peaks. In addition, the probability of detecting a particle at
the ends of the screen increase with number of slits.
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}