From 2093e260f37a7b8bd66cfb115d7a0be00b3fdd2c Mon Sep 17 00:00:00 2001 From: Cory Date: Fri, 8 Dec 2023 10:30:30 +0100 Subject: [PATCH 01/32] Update Makefile --- Makefile | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/Makefile b/Makefile index 79ee877..288e1c4 100644 --- a/Makefile +++ b/Makefile @@ -29,14 +29,14 @@ SRC=./src INCLUDE=./include # Source files and object file locations -SRCFILES=utils.cpp testlib.cpp data_type.cpp IsingModel.cpp monte_carlo.cpp +SRCFILES=utils.cpp testlib.cpp SRCS=$(addprefix $(SRC)/, $(SRCS)) BINOBJS=$(addprefix $(BINOBJDIR)/, $(SRCFILES:.cpp=.o)) PROFOBJS=$(addprefix $(PROFOBJDIR)/, $(SRCFILES:.cpp=.o)) DEBUGOBJS=$(addprefix $(DEBUGOBJDIR)/, $(SRCFILES:.cpp=.o)) # Location for Binaries -EXEC=main test_suite phase_transition phase_transition_mpi time pd_estimate mcmc_progression +EXEC=main test_suite BINS=$(addprefix $(BINDIR)/, $(EXEC)) PROFBINS=$(PROFDIR)/phase_transition_mpi DEBUGBINS=$(addprefix $(DEBUGDIR)/, $(EXEC)) -- 2.20.1 From 4458ff2d126114cfb1f2193d5d889b9d198bde20 Mon Sep 17 00:00:00 2001 From: Cory Date: Tue, 12 Dec 2023 16:31:48 +0100 Subject: [PATCH 02/32] Create skeleton for the WaveSimulation class. This closes #4 --- include/WaveSimulation.hpp | 32 ++++++++++++++++++++++++++++++++ lib/WaveSimulation.cpp | 0 2 files changed, 32 insertions(+) create mode 100644 include/WaveSimulation.hpp create mode 100644 lib/WaveSimulation.cpp diff --git a/include/WaveSimulation.hpp b/include/WaveSimulation.hpp new file mode 100644 index 0000000..39e51b0 --- /dev/null +++ b/include/WaveSimulation.hpp @@ -0,0 +1,32 @@ +/** @file WaveSimulation.hpp + * + * @author Cory Alexander Balaton (coryab) + * @author Janita Ovidie Sandtrøen Willumsen (janitaws) + * + * @version 0.1 + * + * @brief The definition of the WaveSimulation class + * + * @bug No known bugs + * */ +#ifndef __WAVE_SIMULATION__ +#define __WAVE_SIMULATION__ + +#include + +class WaveSimulation { +protected: + int M; + arma::cx_mat U; + arma::cx_mat V; + arma::cx_mat A; + arma::cx_mat B; + double h; + double dt; +public: + virtual void solve() = 0; + void build_A(arma::cx_vec A_vec); + void build_B(arma::cx_vec B_vec); +}; + +#endif diff --git a/lib/WaveSimulation.cpp b/lib/WaveSimulation.cpp new file mode 100644 index 0000000..e69de29 -- 2.20.1 From 939aa3ad508252b2dcb63fd7bc385e37b71bcfb8 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Fri, 15 Dec 2023 16:17:33 +0100 Subject: [PATCH 03/32] Add draft for report, with a general setup of theoretical background and methods. Not ready for review! --- latex/draft.tex | 209 +++++++++++++++++++++++++++++++ latex/references.bib | 67 ++++++++++ latex/schrodinger_simulation.tex | 17 ++- latex/sections/appendices.tex | 41 ++++++ latex/sections/introduction.tex | 7 ++ latex/sections/methods.tex | 24 +++- 6 files changed, 363 insertions(+), 2 deletions(-) create mode 100644 latex/draft.tex diff --git a/latex/draft.tex b/latex/draft.tex new file mode 100644 index 0000000..50be835 --- /dev/null +++ b/latex/draft.tex @@ -0,0 +1,209 @@ +\documentclass[./schrodinger_simulation.tex]{subfiles} + +\begin{document} +\section{Theoretical background}\label{sec:theory} +% Introduction? +The nature of light has long been a subject of discussion, from the 1500s +and the invention of microscopes, through the 1700s where both a particle theory +and a wave theory. In the late 1600s, Christiaan Huygens proposed the wave theory of light, +which was challenged by Isaac Newton's particle theory. The particle theory made +was the leading theory in the beginning of 1800s, when Thomas Young demonstrated +the interference of light, through his double-slit experiment, the wave theory found +new hold. + +In the 1800s, the study of ideal black bodies done by Gustav R. Kirchhoff, lead to a +better understanding of heat radiation. Wilhelm Wien started working on determining the +spectral energy distribution, and Wien's law. The law did make sense for high frequencies, +however, there were inconsistencies when frequency were lower than a certain value. +Wiens law led to an exponential curve, which disagree with the law of conservation. + +Max Planck guessed a result which led to Plancks radiation law, which he later derived +using Boltzmanns statistical interpretation of the second law of thermodunamics. +Plancks findings gave rise to Einsteins quantum hypothesis, and later the wave-particle +duality \cite{britannica:1998:planck}. + +For small atoms classical mechanics are not able to explain the position of a particle, +and Heisenberg uncertainty priciple suggest that the particles have a wavelike behavior. +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. + +% Methods? +% Schrödinger +The Schrödinger equation has a general form +\begin{align} + i \hbar \frac{\partial}{\partial t} | \Psi \rangle &= \hat{H} | \Psi \rangle \ , + \label{eq:schrodinger_general} +\end{align} +where $i$ is the imaginary number, and $\hbar$ is Plancks constant. Here $| \Psi \rangle$ +is the quantum state and $\hat{H}$ is a Hamiltonian operator. +For two-dimensional position space, the quantum state can be expressed using the +time-dependent complex-valued wave function $\Psi (x, y, t)$. The square modulus of the wave function $|\psi|^{2}$, predicts probability of finding the particle at position $(x, y)$ at time t. +% Segue to Born +The modulus of the wave function, is related to the probability density function +\begin{align} + p(x, y \ | \ t) &= |\Psi(x, y, t)|^{2} = \Psi^{*}(x, y, t) \Psi(x, y, t) + \label{eq:born_rule} +\end{align} +using the Born rule, where $^{*}$ denotes the complex conjugated wave function. +When the potential is time-independent, the Schrödinger equation can be expressed as +\begin{align*} + i \hbar \frac{\partial}{\partial t} \Psi (x, y, t) &= - \frac{\hbar^{2}}{2m} \bigg( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \bigg) \Psi (x, y, t) \\ + & \quad + V(x, y, t) \Psi (x, y, t) \numberthis \ . + \label{eq:schrodinger_special} +\end{align*} +When we scale Schrödinger equation by the dimensionful variables, we are left with +a wave function $u$, potential $v$ and the dimensionless equation +\begin{align} + i \frac{\partial u}{\partial t} &= - \frac{\partial^{2} u}{\partial x^{2}} - \frac{\partial^{2} u}{\partial y^{2}} + v(x, y) u \ . + \label{eq:schrodinger_dimensionless} +\end{align} + +% Crank-Nicolson +To evaluate the position of a single particle, we have to consider partial differential +equations (PDE). To solve these numerically, we have to discretize Equation \eqref{eq:schrodinger_dimensionless}. +We use the Crank-Nicolson method (CN), which combines the forward and backward finite +difference method. % +% Include the general $\theta$-method +Using the $\theta$-rule \footnote{For $\theta \in [0, 1]$ we can derive Forward Euler using $\theta = 1$, Backward Euler using $\theta = 0$, and $\theta = 1/2$ gives Crank-Nicolson} +with $\theta = 1/2$, CN can be written as +\begin{align} + \frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} &= \frac{1}{2} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \ . + \label{eq:crank_nicolson} +\end{align} +To simplify notation and avoid confusion of indices with the imaginary number $i$, +we have used the notation $\ivec, \jvec$ in subscript to indicate the commonly named indices $i, j$ +in x- and y-direction. In addition, the superscript $n, n+1$ indicate position in time. +We use CN to derive the discretized Schrödinger equation +\begin{align*} + & u_{\ivec, \jvec}^{n+1} - \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1} \big] \\ + & - \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1} \big] + \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\ + &= u_{\ivec, \jvec}^{n} + \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n} \big] \\ + & \quad + \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n} \big] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \numberthis \ . + \label{eq:schrodinger_discretized} +\end{align*} +The full derivation of both Equation \eqref{eq:crank_nicolson} and Equation \eqref{eq:schrodinger_discretized} +can be found in Appendix \ref{ap:crank_nicolson} + +First, we Taylor +expand the wave equation $u$ around (position and time). + + +\section{Notes}\label{sec:notes} +\subsection*{Introduction - draft 2} +In classical mechanics we study the kinematics and dynamics of physical objects, +ignoring its intrinsic properties for simplicity. It allows us to describe the +forces acting on an object as well as the motion of the object. We can describe +a planets orbital movement \cite{britannica:2023:kepler}, calculate the ... necessary +to launch satellites into orbit, or simply figure out where a ball is going to land +when you throw it... However, when want to study an object at a microscopic level, +e.g. a single atom, classical mechanics falls short. + +Thomas Young first performed the double-slit experiment in 1801, to demonstrate +the principle of interference of light \cite{britannica:2023:young}, postulating +light as waves. In the study of blackbodies, scientists were not able to describe +the radiated intensity of increased frequencies using classical mechanichs, as they +contradicted the principle of conservation of energy \cite{britannica:1998:planck}. +Max Planck assumed that the radiated energy consist of discrete values, or quanta, +to describe the peak in radiated energy. + +Light as particles -> waves -> particles/packets +Erwin Schrödinger wanted to find a mathematical description of the wave characteristics +of matter, supporting the wave-particle idea. He postulated a function which varies +with position, where the function swuared can be interpreted as the probability +of finding an electron at a given position. + +\subsection*{Introduction - draft 1} +In classical mechanics, we study the motion of particles at a macroscopic ,using +physical laws such as Newtons second law. These laws and concepts were used to solve +the motion of the planets. + +However, when we zoom in on a single atom, at a microscopic level, classical mechanics +are no longer sufficient in describing the motion of particles. That is, elementary +particles do not behave in a way which can be described using classical mechanics. +Introducing quantum mechanics, where a quantum is the minimum amount of something, +such as an elementary particle of a given field. + +Beginning with the study of blackbodies and radiation, radiated intensity as a +function of frequency, contradicting the principle of conservation of energy. Max +Planck derived an equation Plancks Law, electromagnetic energy takes the form of +quanta (discrete packets now known as photons). Energy of the photon is equal to +plancks constant times the freq of light. + +Thomas Young first performed the double-slit experiment in 1801, to demonstrate +the principle of interference of light \cite{britannica:2023:young}, postulating +light as waves. + +In the 1860s James Clerk Maxwell predicted that light waves consisted of coupled +electric and magnetic fields. + +Erwin Schrödinger wanted to find a wave equation for matter when classical mechanics +fell short, the wave function. The wave function squared can be interpreted as the +probability of finding an electron at a given position. Schrödingers equation can +be solved for the hydrogen atom, wave equation of the energy levels for a hydrogen +atom. The time-dependent equation also shows how a quantum state evolves with time +\cite[p. 81]{wu:2023:quantum}. + +\subsection*{Research material} +Simulating the double-slit experiment requires solving partial differential equations, +when the number of spatial variables increases so does the complexity of the problem. + +Crank-Nicolson set out to find a numerical method to solve the diffusion equation, +with a higher stability than the explicit and implicit schemes. Combining both schemes, +they developed a finite difference method where a solution is stable for all dx and dt. + +To solve equations, such as Schrödingers, numerically, a discretized equation is +derived using difference methods of explicit and implicit scheme. + +Distribution of particle position. +The wave function can be interpreted in several ways, not unity in agreement, does it apply to one electron, or a system of electrons. Use to find probabilities of measurements. Probability (squared) of finding the electron at a given position, need to do measurements many times for statistical reasons... + +energy defined in quanta, photoelectric effect + +Applying the wave-particle duality to matter, allows us to analyze the behavior of elementary particles more accurately. + +Heisenberg uncertainty principle + +diffraction and interference, describing waves through double slit experiment (Thomas Young). Diffraction pattern where light interfere constructively result in bright spots, whereas destructive interference result in dark spots. + +Path difference: distance between the center of each slit times sine of the angle between the point on the screen and the slits. + +Progress in quantum mechanics without knowing how it works! +- Quantum is the minimum amount of something, elementary particle of a given field, smallest unit of excitation in a fundamental field (inject energy). +- Planck: father of quantum mechanics +Quantum mechanics (MIT open course ware): +- superposition principle: an object can simultaneously appear in two different places or have two different velocities \cite[p. 3]{wu:2023:quantum} +- quantum randomness: the outcome of a measurement is random when a particle is in a superposition of two positions. +- electron color: black or white, hardness: hard or soft +- Identical particles: an absolute in quantum mechanics, two particles can't be distinguished from each other. +- Wave-Particle duality: Electrons are waves, and can form standing waves around protons. +- Schrödinger equation: wave equation of the energy levels for a hydrogen atom. It also shows how a quantum state evolves with time \cite[p. 81]{wu:2023:quantum}. +\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) + V(x, y, t) \Psi (x, y, t) +\end{align} +- $\psi$ is a wave function and describes the quantum state of a particle at time t. The square modulus of the wave function $|\psi|^{2}$, predicts probability of finding the particle at position $(x, y)$ at time t. +- lhs: describe the rate of change of a wave function over time t. +- rhs: describe the variation of the wave function in space. +- eq: describe the relation between the lhs and rhs, formulation of wave-particle duality. + +Partial differential equations +- problems with many variables, constrained by boundary conditions and initial values. + +Diffusion equation: describe the density of a quantity as a function of time, where the flux density obeys the Gauss-Green theorem (div = 0). +Crank-Nicolson: combines the implicit and explicit methods in solving a pde, rewriting the pde as a set of mat-vec multiplications. +Originally the method was derived for the diffusion +equation, however, it can be used for the wave equation as well. + +Wave equation in two dimensions (lec. p. 322), discretize position and time, to obtain +\begin{align*} + u_{\ivec, \jvec}^{n+1} &- r [ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1}] - r [ u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1}] + \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\ + &= u_{\ivec, \jvec}^{n} + r [ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n}] + r [ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n}] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} +\end{align*} +Goal: Simulate the two-dimensional time-dependent schrödinger equation, to study a double-slit-in-a-box setup. +% Introduction +The double-slit experiment, interference using slits observe result on canvas. An experiment by Thomas Young to demonstrate the wave-particle duality of light. Electrons are waves as well as particles. + +When we consider an elementary particle, classical physics are not sufficient in describing the particles position in time. We introduce a probability density function for detecting the particle at a given position. + +We can study the quantum state of a particle in two dimensions, using Schrödinger's equation and the wave function. The kinetic energy are found from the partial derivative of the wave function in respect to the position varables. To solve this numerically, we use the crank-nicolson method to discretize position and time, which combines the explicit and implicit method in solving pdes. +\end{document} \ No newline at end of file diff --git a/latex/references.bib b/latex/references.bib index a8cc3b3..cb962f0 100644 --- a/latex/references.bib +++ b/latex/references.bib @@ -1,6 +1,73 @@ +# Double-slit and physics +@article{murray:2020:double_slits, + author = {Andrew Murray}, + journal = {Physics World}, + number = {2}, + title = {Double slits with single atoms}, + volume = {33}, + year = {2020} +} + +@misc{britannica:1999:light, + author = {The Editors of Encyclopaedia Britannica}, + title = {Light}, + publisher = {Britannica}, + url = {https://www.britannica.com/science/light}, + urldate = {2023-12-15}, +} + # Schrodinger and Crank-Nicolson +@book{wu:2023:quantum, + author = {Biao Wu and Ying Hu}, + title = {Quantum Mechanics: A Concise Introduction}, + publisher = {Springer Singapore}, + year = {2023}, + edition = {1}, +} +@misc{britannica:2023:young, + author = {Glenn Stark}, + title = {Young’s double-slit experiment}, + publisher = {Britannica}, + year = {2023}, + url = {https://www.britannica.com/science/light/Youngs-double-slit-experiment}, + note = {Last accessed 2023-12-12} +} +@misc{britannica:2023:kepler, + author = {The Editors of Encyclopaedia Britannica}, + title = {Kepler’s laws of planetary motion}, + publisher = {Britannica}, + url = {https://www.britannica.com/science/Keplers-laws-of-planetary-motion}, + note = {Last accessed 2023-12-13} +} + +@misc{britannica:1998:planck, + author = {The Editors of Encyclopaedia Britannica}, + title = {Planck’s radiation law}, + publisher = {Britannica}, + url = {https://www.britannica.com/science/Plancks-radiation-law}, + urldate = {2023-12-13} +} + +@misc{britannica:1998:newton, + author = {The Editors of Encyclopaedia Britannica}, + title = {Newton’s laws of motion}, + publisher = {Britannica}, + url = {https://www.britannica.com/science/Newtons-laws-of-motion}, + urldate = {2023-12-13} +} + +@article{crank:1947:numerical, + author = {John Crank and Phyllis Nicolson}, + title = {A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type}, + journal = {Mathematical proceedings of the Cambridge Philosophical Society}, + doi = {10.1017/S0305004100023197}, + year = {1947}, + volume = {43}, + number = {1}, + pages = {50--67} +} # Math and statistics @book{lindstrom:2016:kalkulus, diff --git a/latex/schrodinger_simulation.tex b/latex/schrodinger_simulation.tex index 445d00f..2c1a92e 100644 --- a/latex/schrodinger_simulation.tex +++ b/latex/schrodinger_simulation.tex @@ -46,6 +46,10 @@ % \def\biblio{\bibliographystyle{plain}\bibliography{../references/references}} \newcommand\numberthis{\addtocounter{equation}{1}\tag{\theequation}} +% Defines indices i and j to avoid confusion with imaginary i +\newcommand{\ivec}{\hat{\imath}} +\newcommand{\jvec}{\hat{\jmath}} + \usepackage{xr} \usepackage{subfiles} % \externaldocument[M-]{\subfix{main}} @@ -74,11 +78,14 @@ % Conclusion % \subfile{sections/conclusion} +% Notes +\subfile{draft} + \clearpage \newpage % Appendix -% \subfile{sections/appendices} +\subfile{sections/appendices} \clearpage \onecolumngrid @@ -89,4 +96,12 @@ \end{document} +% Methods +% P1: Theory, imag i = i, index i, j = \hat{i}, \hat{j} +% P2: +% +% Results +% P7: +% P8: +% P9: diff --git a/latex/sections/appendices.tex b/latex/sections/appendices.tex index db71d54..980e046 100644 --- a/latex/sections/appendices.tex +++ b/latex/sections/appendices.tex @@ -2,6 +2,47 @@ \begin{document} \appendix +\section{The Crank-Nicholson method}\label{ap:crank_nicolson} +The Crank-Nicolson \(CN\) approach considers both the forward difference, an explicit scheme, +\begin{equation*} + \frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} = F_{\ivec, \jvec}^{n} \ , +\end{equation*} +and the backward difference, an implicit scheme, +\begin{equation*} + \frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} = F_{\ivec, \jvec}^{n+1} \ . +\end{equation*} +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} \ . +\end{align*} +The parameter $\theta$ is introduced for a general approach, however, for CN $\theta = 1/2$. +\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] \\ +\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, assume this is known... + +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] & \text{, where $\frac{1}{i} = -i$} +\end{align*} + +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}} \\ + & \quad - \frac{u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1}}{2 \Delta y^{2}} + \frac{1}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\ + & \quad - \frac{u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n}}{2 \Delta x^{2}} \\ + & \quad - \frac{u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n}}{2 \Delta y^{2}} + \frac{1}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \bigg] \\ +\end{align*} +We rewrite the expression, +\begin{align*} + & u_{\ivec, \jvec}^{n+1} - \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1} \big] \\ + & - \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1} \big] + \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\ + &= u_{\ivec, \jvec}^{n} + \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n} \big] \\ + & \quad + \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n} \big] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \\ +\end{align*} \end{document} diff --git a/latex/sections/introduction.tex b/latex/sections/introduction.tex index 1f9346e..5de6f41 100644 --- a/latex/sections/introduction.tex +++ b/latex/sections/introduction.tex @@ -2,6 +2,13 @@ \begin{document} \section{Introduction}\label{sec:introduction} +% Light - particle or wave +% - double-slit, blackbodies radiation +% Position space +% - classical vs quantum mechanics \end{document} + +% crank-nicolson method! +% wave equation \ No newline at end of file diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 43bab2b..7a4946f 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -1,7 +1,29 @@ \documentclass[../schrodinger_simulation.tex]{subfiles} \begin{document} -\section{Methods}\label{sec:methods} +\section{Methods}\label{sec:methods} % +\subsection{Double-slit experiment}\label{ssec:double_slit} % +In the beginning of the 1800s, the general view was that light consisted of particles. +However, in 1801 Thomas Young demonstrated the principle of interference of light +\cite{britannica:2023:young}, while postulating light as waves rather than particles. +% Thomas Young first performed the double-slit experiment in 1801, to demonstrate +% the principle of interference of light \cite{britannica:2023:young}, postulating +% light as waves. In the study of blackbodies, scientists were not able to describe +% the radiated intensity of increased frequencies using classical mechanichs, as they +% contradicted the principle of conservation of energy \cite{britannica:1998:planck}. +% Max Planck assumed that the radiated energy consist of discrete values, or quanta, +% to describe the peak in radiated energy. +The double-slit experiment + +\subsection{Schrödinger equation}\label{ssec:schrodinger} % + +\subsection{Crank-Nicolson}\label{ssec:crank_nicolson} % + +% Might be better to move theory on double-slit here and have a subsection of light +% property etc. as a first section or in introduction? +\subsection{Implementation}\label{ssec:implementation} % + +\subsection{Tools}\label{ssec:tools} % \end{document} -- 2.20.1 From e02abe3ab768ab7ee9d14b005dd324cf758e3053 Mon Sep 17 00:00:00 2001 From: Cory Date: Sun, 17 Dec 2023 13:32:36 +0100 Subject: [PATCH 04/32] Update Makefile and .clangd --- .clangd | 2 +- Makefile | 45 ++++++++++++++++++++++++++++++++++----------- 2 files changed, 35 insertions(+), 12 deletions(-) diff --git a/.clangd b/.clangd index 3c44a56..46f80c6 100644 --- a/.clangd +++ b/.clangd @@ -1,2 +1,2 @@ CompileFlags: - Add: [-I../include] + Add: [-I../include, -std=c++11] diff --git a/Makefile b/Makefile index 288e1c4..c3b0dc3 100644 --- a/Makefile +++ b/Makefile @@ -1,5 +1,5 @@ # The compiler -CC=mpicxx +CC=g++ # Flags CFLAGS=-Wall -larmadillo -std=c++11 -O3 -fopenmp @@ -26,19 +26,26 @@ LATEXDIR=./latex # Source directories SRC=./src +LIB=./lib INCLUDE=./include # Source files and object file locations -SRCFILES=utils.cpp testlib.cpp -SRCS=$(addprefix $(SRC)/, $(SRCS)) -BINOBJS=$(addprefix $(BINOBJDIR)/, $(SRCFILES:.cpp=.o)) -PROFOBJS=$(addprefix $(PROFOBJDIR)/, $(SRCFILES:.cpp=.o)) -DEBUGOBJS=$(addprefix $(DEBUGOBJDIR)/, $(SRCFILES:.cpp=.o)) +#SRCFILES=utils.cpp testlib.cpp +#SRCS=$(addprefix $(SRC)/, $(SRCS)) +#BINOBJS=$(addprefix $(BINOBJDIR)/, $(SRCFILES:.cpp=.o)) +#PROFOBJS=$(addprefix $(PROFOBJDIR)/, $(SRCFILES:.cpp=.o)) +#DEBUGOBJS=$(addprefix $(DEBUGOBJDIR)/, $(SRCFILES:.cpp=.o)) + +# Lib files +LIBSRCS=$(notdir $(shell find $(LIB) -type f)) +LIBBINOBJS=$(addprefix $(BINOBJDIR)/, $(LIBSRCS:.cpp=.o)) +LIBPROFOBJS=$(addprefix $(PROFOBJDIR)/, $(LIBSRCS:.cpp=.o)) +LIBDEBUGOBJS=$(addprefix $(DEBUGOBJDIR)/, $(LIBSRCS:.cpp=.o)) # Location for Binaries -EXEC=main test_suite +EXEC=$(basename $(notdir $(shell find $(SRC) -type f))) BINS=$(addprefix $(BINDIR)/, $(EXEC)) -PROFBINS=$(PROFDIR)/phase_transition_mpi +PROFBINS=$(addprefix $(PROFDIR)/, $(EXEC)) DEBUGBINS=$(addprefix $(DEBUGDIR)/, $(EXEC)) # List phony targets @@ -56,23 +63,24 @@ latex: $(MAKE) -C $(LATEXDIR) # Rule for binaries -$(BINDIR)/%: $(BINOBJDIR)/%.o $(BINOBJS) +$(BINDIR)/%: $(BINOBJDIR)/%.o $(LIBBINOBJS) $(MKDIR) $(BINDIR) $(CC) $^ -o $@ $(CFLAGS) -I$(INCLUDE) # Rule for profiling binaries -$(PROFDIR)/%: $(PROFOBJDIR)/%.o $(PROFOBJS) +$(PROFDIR)/%: $(PROFOBJDIR)/%.o $(LIBPROFOBJS) $(MKDIR) $(PROFDIR) $(INSTRUMENT) $(CC) $^ -o $@ $(CFLAGS) $(PROFFLAGS) -I$(INCLUDE) # Rule for debug binaries -$(DEBUGDIR)/%: $(DEBUGOBJDIR)/%.o $(DEBUGOBJS) +$(DEBUGDIR)/%: $(DEBUGOBJDIR)/%.o $(LIBDEBUGOBJS) $(MKDIR) $(DEBUGDIR) $(CC) $^ -o $@ $(CFLAGS) $(DBGFLAGS) -I$(INCLUDE) # Rule for object files $(BINOBJDIR)/%.o: $(SRC)/%.cpp $(MKDIR) $(BINOBJDIR) + echo $(LIBBINOBJS) $(CC) -c $^ -o $@ $(CFLAGS) -I$(INCLUDE) # Rule for instrumented object files @@ -85,6 +93,21 @@ $(DEBUGOBJDIR)/%.o: $(SRC)/%.cpp $(MKDIR) $(DEBUGOBJDIR) $(CC) -c $^ -o $@ $(CFLAGS) $(DBGFLAGS) -I$(INCLUDE) +# Rule for object files +$(BINOBJDIR)/%.o: $(LIB)/%.cpp + $(MKDIR) $(BINOBJDIR) + $(CC) -c $^ -o $@ $(CFLAGS) -I$(INCLUDE) + +# Rule for instrumented object files +$(PROFOBJDIR)/%.o: $(LIB)/%.cpp + $(MKDIR) $(PROFOBJDIR) + $(INSTRUMENT) $(CC) -c $^ -o $@ $(CFLAGS) $(PROFFLAGS) -I$(INCLUDE) + +# Rule for debug object files +$(DEBUGOBJDIR)/%.o: $(LIB)/%.cpp + $(MKDIR) $(DEBUGOBJDIR) + $(CC) -c $^ -o $@ $(CFLAGS) $(DBGFLAGS) -I$(INCLUDE) + # Cleaning clean: objclean binclean latexclean -- 2.20.1 From eff6291e0b12b2fe11c68c25cf1c3db03e4452c6 Mon Sep 17 00:00:00 2001 From: Cory Date: Sun, 17 Dec 2023 13:33:06 +0100 Subject: [PATCH 05/32] Minimum, messy code --- include/WaveSimulation.hpp | 29 ++++-- include/constants.hpp | 17 ++++ include/literals.hpp | 19 ++++ include/utils.hpp | 10 ++- lib/WaveSimulation.cpp | 180 +++++++++++++++++++++++++++++++++++++ lib/literals.cpp | 17 ++++ lib/utils.cpp | 43 +++++++++ src/main.cpp | 120 +++++++++++++++++++++++++ src/test_suite.cpp | 4 + 9 files changed, 427 insertions(+), 12 deletions(-) create mode 100644 include/constants.hpp create mode 100644 include/literals.hpp create mode 100644 lib/literals.cpp diff --git a/include/WaveSimulation.hpp b/include/WaveSimulation.hpp index 39e51b0..0ef9acf 100644 --- a/include/WaveSimulation.hpp +++ b/include/WaveSimulation.hpp @@ -12,21 +12,34 @@ #ifndef __WAVE_SIMULATION__ #define __WAVE_SIMULATION__ +#include "constants.hpp" +#include "literals.hpp" + #include +#include class WaveSimulation { protected: - int M; - arma::cx_mat U; - arma::cx_mat V; - arma::cx_mat A; - arma::cx_mat B; + uint32_t M; + arma::sp_cx_mat B; double h; double dt; + double T; + public: - virtual void solve() = 0; - void build_A(arma::cx_vec A_vec); - void build_B(arma::cx_vec B_vec); + int32_t N; + arma::cx_mat V; + arma::cx_mat U; + arma::sp_cx_mat A; + WaveSimulation(double h, double dt, double T); + virtual void solve(std::ofstream& ofile); + void build_A(); + void build_B(); + void initialize_U(double x_c, double y_c, double sigma_x, double sigma_y, + double p_x, double p_y); + void write_U(std::ofstream &ofile); + void step(); + void build_V(double thickness, double pos_x, double aperture_sparation, double aperture, uint32_t slits); }; #endif diff --git a/include/constants.hpp b/include/constants.hpp new file mode 100644 index 0000000..4373d64 --- /dev/null +++ b/include/constants.hpp @@ -0,0 +1,17 @@ +/** @file constants.hpp + * + * @author Cory Alexander Balaton (coryab) + * @author Janita Ovidie Sandtrøen Willumsen (janitaws) + * + * @version 1.0 + * + * @brief Library of constants + * + * @bug No known bugs + * */ +#ifndef __CONST__ +#define __CONST__ + +#define I std::complex{0., 1.} + +#endif diff --git a/include/literals.hpp b/include/literals.hpp new file mode 100644 index 0000000..3a29d48 --- /dev/null +++ b/include/literals.hpp @@ -0,0 +1,19 @@ +/** @file literals.hpp + * + * @author Cory Alexander Balaton (coryab) + * @author Janita Ovidie Sandtrøen Willumsen (janitaws) + * + * @version 1.0 + * + * @brief Useful literals + * + * @bug No known bugs + * */ +#ifndef __LITERALS__ +#define __LITERALS__ + +#include + +std::complex operator ""_i(long double magnitude); + +#endif diff --git a/include/utils.hpp b/include/utils.hpp index 0618908..83f98a8 100644 --- a/include/utils.hpp +++ b/include/utils.hpp @@ -117,10 +117,10 @@ std::string dirname(const std::string &path); /** @brief Take 2 strings and concatenate them and make sure there is a * directory separator (/) between them. * - * @details This function doesn't care whether or not the values given as - * parameters are valid path strings. It is the responsibility of the user to make - * sure that the values given are valid path strings. - * The function only guarantees that the output string is a valid path string. + * @details This function doesn't care whether or not the values given as + * parameters are valid path strings. It is the responsibility of the user to + * make sure that the values given are valid path strings. The function only + * guarantees that the output string is a valid path string. * * @param left The left hand side of the result string * @param right The right hand side of the result string @@ -129,6 +129,8 @@ std::string dirname(const std::string &path); * */ std::string concatpath(const std::string &left, const std::string &right); +// A function that prints the structure of a sparse matrix to screen. +void print_sp_matrix_structure(const arma::sp_cx_mat &A); } // namespace utils #endif diff --git a/lib/WaveSimulation.cpp b/lib/WaveSimulation.cpp index e69de29..056cb05 100644 --- a/lib/WaveSimulation.cpp +++ b/lib/WaveSimulation.cpp @@ -0,0 +1,180 @@ +/** @file WaveSimulation.cpp + * + * @author Cory Alexander Balaton (coryab) + * @author Janita Ovidie Sandtrøen Willumsen (janitaws) + * + * @version 0.1 + * + * @brief Implementation of the WaveSimulation class. + * + * @bug No known bugs + * */ +#include "WaveSimulation.hpp" + +#include "utils.hpp" + +#include +#include +#include +#include + +WaveSimulation::WaveSimulation(double h, double dt, double T) +{ + this->dt = dt; + this->h = h; + this->T = T; + this->M = 1. / h; + this->N = M - 2; + this->V.set_size(this->N, this->N); + this->V.fill(0.); + this->U.set_size(this->N, this->N); + this->U.fill(0.); +} + +void WaveSimulation::solve(std::ofstream &ofile) +{ + ofile << this->N << '\n'; + uint32_t iterations = this->T / this->dt; + for (size_t i = 0; i < iterations; i++) { + this->write_U(ofile); + this->step(); + } +} + +void WaveSimulation::step() +{ + DEBUG("Inside step"); + arma::cx_vec tmp = this->B * this->U.as_col(); + arma::spsolve(this->U, this->A, tmp); +} + +void WaveSimulation::build_A() +{ + // Create the diagonal + arma::cx_vec diagonal(this->N * this->N); + + // Set diagonal values + std::complex r = (1._i * this->dt) / (2 * h * h); + for (size_t i = 0; i < diagonal.size(); i++) { + diagonal(i) = 1. + 4. * r + (1._i * this->dt / 2.) * this->V(i); + } + + // Create the submatrix + arma::cx_mat sub_matrix(this->N, this->N, arma::fill::zeros); + sub_matrix.diag(-1).fill(-r); + sub_matrix.diag(1).fill(-r); + + // Set the size of A + this->A.set_size(this->N * this->N, this->N * this->N); + + // Fill in the values in the submatrix diagonal + for (size_t i = 0; i < this->A.n_cols; i += this->N) { + this->A.submat(i, i, i + this->N - 1, i + this->N - 1) = sub_matrix; + } + + // Fill the last sub/sup-diagonals + this->A.diag() = diagonal; + this->A.diag(-this->N).fill(-r); + this->A.diag(this->N).fill(-r); +} + +void WaveSimulation::build_B() +{ + std::complex r = (1._i * this->dt) / (2 * h * h); + + // Create the diagonal + arma::cx_vec diagonal(this->N * this->N); + + for (size_t i = 0; i < diagonal.size(); i++) { + diagonal(i) = 1. - 4. * r - (1._i * this->dt / 2.) * this->V(i); + } + + // Create the submatrix + arma::cx_mat sub_matrix(this->N, this->N, arma::fill::zeros); + sub_matrix.diag(-1).fill(r); + sub_matrix.diag(1).fill(r); + + // Set the size of B + this->B.set_size(this->N * this->N, this->N * this->N); + + // Fill in the values in the submatrix diagonal + for (size_t i = 0; i < this->B.n_cols; i += this->N) { + this->B.submat(i, i, i + this->N - 1, i + this->N - 1) = sub_matrix; + } + + // Fill the last sub/sup-diagonals + this->B.diag() = diagonal; + this->B.diag(-this->N).fill(r); + this->B.diag(this->N).fill(r); +} + +void WaveSimulation::initialize_U(double x_c, double y_c, double sigma_x, + double sigma_y, double p_x, double p_y) +{ + double x, y, diff_x, diff_y; + std::complex sum = 0.; + for (size_t j = 0; j < this->U.n_cols; j++) { + x = j * h; + diff_x = x - x_c; + for (size_t i = 0; i < this->U.n_rows; i++) { + y = i * h; + diff_y = y - y_c; + this->U(i, j) = + std::exp(-(diff_x * diff_x) / (2. * sigma_x * sigma_x) + - (diff_y * diff_y) / (2. * sigma_y * sigma_y) + + p_x * x * 1._i + p_y * y * 1._i); + sum += this->U(i, j) * std::conj(this->U(i, j)); + } + } + if (std::abs(sum.imag()) > 1e-7) { + abort(); + } + double norm = 1. / std::sqrt(sum.real()); + + this->U.for_each([norm](std::complex &el) { el *= norm; }); +} + +void WaveSimulation::write_U(std::ofstream &ofile) +{ + this->U.for_each( + [&ofile](std::complex el) { ofile << el << '\t'; }); + ofile << '\n'; +} + +void WaveSimulation::build_V(double thickness, double pos_x, + double aperture_separation, double aperture, + uint32_t slits) +{ + uint32_t mid_y = this->N / 2 - (this->N % 2 == 0); + arma::cx_vec res; + + if (slits % 2 == 0) { + res = arma::cx_vec(aperture_separation/this->h,arma::fill::value(1e10)); + for (size_t i=0; i < slits; i+=2) { + res = arma::join_cols(res, arma::cx_vec(aperture/this->h,arma::fill::zeros)); + res = arma::join_cols(arma::cx_vec(aperture/this->h,arma::fill::zeros), res); + res = arma::join_cols(res, arma::cx_vec(aperture_separation/this->h,arma::fill::value(1e10))); + res = arma::join_cols(arma::cx_vec(aperture_separation/this->h,arma::fill::value(1e10)), res); + } + } + else { + res = arma::cx_vec(aperture/this->h,arma::fill::value(0)); + for (size_t i=0; i < slits-1; i+=2) { + res = arma::join_cols(res, arma::cx_vec(aperture_separation/this->h,arma::fill::value(1e10))); + res = arma::join_cols(arma::cx_vec(aperture_separation/this->h,arma::fill::value(1e10)), res); + res = arma::join_cols(res, arma::cx_vec(aperture/this->h,arma::fill::zeros)); + res = arma::join_cols(arma::cx_vec(aperture/this->h,arma::fill::zeros), res); + } + } + if (res.size() > this->N) { + abort(); + } + uint32_t fill = (this->N - res.size()) / 2; + res = arma::join_cols(arma::cx_vec(fill, arma::fill::value(1e10)), res); + res = arma::join_cols(res, arma::cx_vec(fill + ((this->N - res.size()) % 2), arma::fill::value(1e10))); + + uint32_t start = pos_x/this->h - thickness/this->h/2; + for (size_t i=0; i < thickness/this->h; i++) { + this->V.col(start+i) = res; + } +} diff --git a/lib/literals.cpp b/lib/literals.cpp new file mode 100644 index 0000000..57daf51 --- /dev/null +++ b/lib/literals.cpp @@ -0,0 +1,17 @@ +/** @file literals.cpp + * + * @author Cory Alexander Balaton (coryab) + * @author Janita Ovidie Sandtrøen Willumsen (janitaws) + * + * @version 1.0 + * + * @brief The implementation of the literals. + * + * @bug No known bugs + * */ +#include "literals.hpp" + +std::complex operator""_i(long double magnitude) +{ + return std::complex(0.,magnitude); +} diff --git a/lib/utils.cpp b/lib/utils.cpp index 554ee31..5dabba2 100644 --- a/lib/utils.cpp +++ b/lib/utils.cpp @@ -70,4 +70,47 @@ std::string concatpath(const std::string &left, const std::string &right) } } +void print_sp_matrix_structure(const arma::sp_cx_mat &A) +{ + using namespace std; + using namespace arma; + + // Declare a C-style 2D array of strings. + string S[A.n_rows][A.n_cols]; + + // Initialise all the strings to " ". + for (int i = 0; i < A.n_rows; i++) { + for (int j = 0; j < A.n_cols; j++) { + S[i][j] = " "; + } + } + + // Next, we want to set the string to a dot at each non-zero element. + // To do this we use the special loop iterator from the sp_cx_mat class + // to help us loop over only the non-zero matrix elements. + sp_cx_mat::const_iterator it = A.begin(); + sp_cx_mat::const_iterator it_end = A.end(); + + int nnz = 0; + for (; it != it_end; ++it) { + S[it.row()][it.col()] = "•"; + nnz++; + } + + // Finally, print the matrix to screen. + cout << endl; + for (int i = 0; i < A.n_rows; i++) { + cout << "| "; + for (int j = 0; j < A.n_cols; j++) { + cout << S[i][j] << " "; + } + cout << "|\n"; + } + + cout << endl; + cout << "matrix size: " << A.n_rows << "x" << A.n_cols << endl; + cout << "non-zero elements: " << nnz << endl; + cout << endl; +} + } // namespace utils diff --git a/src/main.cpp b/src/main.cpp index e69de29..079eae5 100644 --- a/src/main.cpp +++ b/src/main.cpp @@ -0,0 +1,120 @@ +/** @file main.cpp + * + * @author Cory Alexander Balaton (coryab) + * @author Janita Ovidie Sandtrøen Willumsen (janitaws) + * + * @version 1.0 + * + * @brief Implementation of the testing library + * + * @bug No known bugs + * */ +#include "WaveSimulation.hpp" +#include "utils.hpp" + +void probability_deviation() +{ + WaveSimulation sim_narrow(.005, 2.5e-5, .008); + sim_narrow.initialize_U(.25, .5, .05, .05, 200., 0.); + sim_narrow.build_A(); + sim_narrow.build_B(); + + WaveSimulation sim_wide(.005, 2.5e-5, .008); + sim_wide.build_V(.02, .5, .05, .05, 2); + sim_wide.initialize_U(.25, .5, .05, .10, 200., 0.); + sim_wide.build_A(); + sim_wide.build_B(); + + std::ofstream ofile; + utils::mkpath("data"); + ofile.open("data/probability_deviation.txt"); + + double sum_narrow, sum_wide; + + for (size_t i = 0; i < 320; i++) { + sum_narrow = 0; + sum_wide = 0; + for (size_t j = 0; j < sim_narrow.U.n_elem; j++) { + sum_narrow += (sim_narrow.U(j) * std::conj(sim_narrow.U(j))).real(); + sum_wide += (sim_wide.U(j) * std::conj(sim_wide.U(j))).real(); + } + + sim_narrow.step(); + sim_wide.step(); + ofile << i << ',' << sum_narrow << ',' << sum_wide << '\n'; + } + + ofile.close(); +} + +void color_map() +{ + WaveSimulation sim(.005, 2.5e-5, .008); + sim.build_V(.02, .5, .05, .05, 2); + sim.initialize_U(.25, .5, .05, .10, 200., 0.); + sim.build_A(); + sim.build_B(); + + std::ofstream ofile; + ofile.open("data/color_map.txt"); + ofile << sim.N << '\n'; + sim.write_U(ofile); + for (size_t i = 0; i < 40; i++) { + sim.step(); + } + sim.write_U(ofile); + for (size_t i = 0; i < 40; i++) { + sim.step(); + } + sim.write_U(ofile); + ofile.close(); +} + +void detector_screen() +{ + WaveSimulation sim(.005, 2.5e-5, .008); + sim.build_V(.02, .5, .05, .05, 1); + sim.initialize_U(.25, .5, .05, .10, 200., 0.); + sim.build_A(); + sim.build_B(); + + std::ofstream ofile; + ofile.open("data/screen/single_slit.txt"); + ofile << sim.N << '\n'; + for (size_t i = 0; i < 80; i++) { + sim.step(); + } + sim.write_U(ofile); + ofile.close(); + + sim.build_V(.02, .5, .05, .05, 2); + sim.initialize_U(.25, .5, .05, .10, 200., 0.); + + ofile.open("data/screen/double_slit.txt"); + ofile << sim.N << '\n'; + for (size_t i = 0; i < 80; i++) { + sim.step(); + } + sim.write_U(ofile); + ofile.close(); + + sim.build_V(.02, .5, .05, .05, 3); + sim.initialize_U(.25, .5, .05, .10, 200., 0.); + + ofile.open("data/screen/triple_slit.txt"); + ofile << sim.N << '\n'; + for (size_t i = 0; i < 80; i++) { + sim.step(); + } + sim.write_U(ofile); + ofile.close(); +} + +int main() +{ + probability_deviation(); + color_map(); + detector_screen(); + + return 0; +} diff --git a/src/test_suite.cpp b/src/test_suite.cpp index e69de29..905869d 100644 --- a/src/test_suite.cpp +++ b/src/test_suite.cpp @@ -0,0 +1,4 @@ +int main() +{ + return 0; +} -- 2.20.1 From cce5d0a0fddc8048a7fb9a4610d9117e5b16f22f Mon Sep 17 00:00:00 2001 From: Cory Date: Sun, 17 Dec 2023 13:40:01 +0100 Subject: [PATCH 06/32] Minor fix --- data/color_map.txt | 4 + data/probability_deviation.txt | 320 +++++++++++++++++++++++++++++++++ python_scripts/heat_map.py | 43 +++++ python_scripts/plot_v.py | 27 +++ src/main.cpp | 1 + 5 files changed, 395 insertions(+) create mode 100644 data/color_map.txt create mode 100644 data/probability_deviation.txt create mode 100644 python_scripts/heat_map.py create mode 100644 python_scripts/plot_v.py diff --git a/data/color_map.txt b/data/color_map.txt new file mode 100644 index 0000000..3b0e419 --- /dev/null +++ b/data/color_map.txt @@ -0,0 +1,4 @@ +39204 +(5.54049e-13,0) (7.10524e-13,0) (9.08916e-13,0) (1.1598e-12,0) (1.47624e-12,0) (1.87432e-12,0) (2.37381e-12,0) (2.99891e-12,0) (3.77914e-12,0) (4.75049e-12,0) (5.95659e-12,0) (7.45025e-12,0) (9.29519e-12,0) (1.15681e-11,0) (1.43607e-11,0) (1.77831e-11,0) (2.1966e-11,0) (2.70652e-11,0) (3.32647e-11,0) (4.07823e-11,0) (4.98739e-11,0) (6.084e-11,0) (7.4032e-11,0) (8.98595e-11,0) (1.08798e-10,0) (1.314e-10,0) (1.58301e-10,0) (1.90233e-10,0) (2.28035e-10,0) (2.72666e-10,0) (3.25219e-10,0) (3.86931e-10,0) (4.59205e-10,0) (5.43618e-10,0) (6.41942e-10,0) (7.56156e-10,0) (8.88467e-10,0) (1.04132e-09,0) (1.21743e-09,0) 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(-6.41119e-09,1.00662e-08) (-8.49371e-09,3.79079e-09) (-7.2432e-09,-1.129e-09) (-4.34418e-09,-3.70527e-09) (-1.285e-09,-4.04818e-09) (1.00994e-09,-2.92358e-09) (2.21352e-09,-1.28694e-09) (2.42416e-09,6.9189e-11) (1.93485e-09,7.07401e-10) (1.05003e-09,5.88419e-10) diff --git a/data/probability_deviation.txt b/data/probability_deviation.txt new file mode 100644 index 0000000..d545f12 --- /dev/null +++ b/data/probability_deviation.txt @@ -0,0 +1,320 @@ +0,1,1 +1,1,1 +2,1,1 +3,1,1 +4,1,1 +5,1,1 +6,1,1 +7,1,1 +8,1,1 +9,1,1 +10,1,1 +11,1,1 +12,1,1 +13,1,1 +14,1,1 +15,1,1 +16,1,1 +17,1,1 +18,1,1 +19,1,1 +20,1,1 +21,1,1 +22,1,1 +23,1,1 +24,1,1 +25,1,1 +26,1,1 +27,1,1 +28,1,1 +29,1,1 +30,1,1 +31,1,1 +32,1,1 +33,1,1 +34,1,1 +35,1,1 +36,1,1 +37,1,1 +38,1,1 +39,1,1 +40,1,1 +41,1,1 +42,1,1 +43,1,1 +44,1,1 +45,1,1 +46,1,1 +47,1,1 +48,1,1 +49,1,1 +50,1,1 +51,1,1 +52,1,1 +53,1,1 +54,1,1 +55,1,1 +56,1,1 +57,1,1 +58,1,1 +59,1,1 +60,1,1 +61,1,1 +62,1,1 +63,1,1 +64,1,1 +65,1,1 +66,1,1 +67,1,1 +68,1,1 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+294,1,1 +295,1,1 +296,1,1 +297,1,1 +298,1,1 +299,1,1 +300,1,1 +301,1,1 +302,1,1 +303,1,1 +304,1,1 +305,1,1 +306,1,1 +307,1,1 +308,1,1 +309,1,1 +310,1,1 +311,1,1 +312,1,1 +313,1,1 +314,1,1 +315,1,1 +316,1,1 +317,1,1 +318,1,1 +319,1,1 diff --git a/python_scripts/heat_map.py b/python_scripts/heat_map.py new file mode 100644 index 0000000..8700bff --- /dev/null +++ b/python_scripts/heat_map.py @@ -0,0 +1,43 @@ +import numpy as np +import matplotlib.pyplot as plt +import matplotlib +from matplotlib.animation import FuncAnimation +import ast + +wave_arr = [] +fig = plt.figure() +ax = plt.gca() +img = ax.imshow([[]]) + +def plot(): + with open("test.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) + wave_arr.append(arr.T) + # print(arr) + + # plt.imshow(arr, cmap="hot", interpolation="nearest") + # plt.show() + + +def animation(i): + norm = matplotlib.cm.colors.Normalize(vmin=0, vmax=np.max(wave_arr[i])) + img.set_norm(norm) + img.set_data(wave_arr[i]) + + +if __name__ == "__main__": + plot() + + norm = matplotlib.cm.colors.Normalize(vmin=0, vmax=np.max(wave_arr[0])) + img = ax.imshow(wave_arr[0], extent=[0,1,0,1], cmap=plt.get_cmap("viridis"), norm=norm) + anim = FuncAnimation(fig, animation, interval=1, frames=np.arange(0,len(wave_arr)), repeat=True, blit=0) + # plt.show() + + anim.save("./animation.mp4", writer="ffmpeg", bitrate=10000, fps=15) diff --git a/python_scripts/plot_v.py b/python_scripts/plot_v.py new file mode 100644 index 0000000..3e48134 --- /dev/null +++ b/python_scripts/plot_v.py @@ -0,0 +1,27 @@ +import numpy as np +import matplotlib.pyplot as plt +import matplotlib +from matplotlib.animation import FuncAnimation +import ast + + +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() diff --git a/src/main.cpp b/src/main.cpp index 079eae5..c6891b4 100644 --- a/src/main.cpp +++ b/src/main.cpp @@ -79,6 +79,7 @@ void detector_screen() sim.build_B(); std::ofstream ofile; + utils::mkpath("data/screen"); ofile.open("data/screen/single_slit.txt"); ofile << sim.N << '\n'; for (size_t i = 0; i < 80; i++) { -- 2.20.1 From b0ce8ef73057db0c57bb48af011ff8a59ce23a43 Mon Sep 17 00:00:00 2001 From: Cory Date: Sun, 17 Dec 2023 13:40:17 +0100 Subject: [PATCH 07/32] Add screen data --- data/screen/double_slit.txt | 2 ++ data/screen/single_slit.txt | 2 ++ data/screen/triple_slit.txt | 2 ++ 3 files changed, 6 insertions(+) create mode 100644 data/screen/double_slit.txt create mode 100644 data/screen/single_slit.txt create mode 100644 data/screen/triple_slit.txt diff --git a/data/screen/double_slit.txt b/data/screen/double_slit.txt new file mode 100644 index 0000000..78e6d97 --- /dev/null +++ b/data/screen/double_slit.txt @@ -0,0 +1,2 @@ +39204 +(-1.04461e-09,2.01638e-09) (-9.00354e-10,3.24732e-09) (-2.82911e-09,6.56239e-09) (-2.63599e-09,6.52117e-09) (-3.37847e-09,1.18304e-08) (-5.82484e-09,1.1571e-08) (-2.9779e-09,1.61989e-08) (-7.46749e-09,2.10235e-08) (-5.60068e-09,2.12628e-08) (-3.51068e-09,3.09278e-08) (-7.71267e-09,3.6318e-08) (-2.97356e-09,4.07418e-08) (8.76734e-10,5.545e-08) (-1.00864e-09,6.88659e-08) (2.31811e-09,8.07691e-08) (1.00756e-08,1.0137e-07) (1.24455e-08,1.30323e-07) (1.04126e-08,1.57805e-07) (1.36592e-08,1.85326e-07) (2.23708e-08,2.2027e-07) (3.3942e-08,2.65035e-07) (4.59457e-08,3.20245e-07) (5.89983e-08,3.84029e-07) (7.60503e-08,4.58807e-07) (9.66299e-08,5.47163e-07) (1.22407e-07,6.50754e-07) 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(-6.26083e-29,-9.14729e-29) (2.08045e-30,-7.1775e-29) (2.84655e-29,-3.65161e-29) (2.86271e-29,-7.97214e-30) (1.77044e-29,6.85521e-30) (6.37889e-30,1.02568e-29) (-7.00086e-31,7.62391e-30) (-3.25576e-30,3.56025e-30) (-2.95976e-30,5.2366e-31) (-1.63652e-30,-9.00201e-31) (-4.52552e-31,-1.12933e-30) (1.55531e-31,-8.08381e-31) (2.25536e-31,-3.82553e-31) -- 2.20.1 From f27cca4baf3c185c69d02cf7aaac695de47230ca Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Tue, 19 Dec 2023 14:50:06 +0100 Subject: [PATCH 08/32] Finish first draft of introduction and method section. --- latex/draft.tex | 389 ++++++++++++++++++++++++------- latex/references.bib | 70 +++++- latex/schrodinger_simulation.tex | 32 ++- latex/sections/appendices.tex | 2 +- latex/sections/introduction.tex | 53 ++++- latex/sections/methods.tex | 237 +++++++++++++++++-- latex/sections/results.tex | 11 + 7 files changed, 688 insertions(+), 106 deletions(-) diff --git a/latex/draft.tex b/latex/draft.tex index 50be835..55ce765 100644 --- a/latex/draft.tex +++ b/latex/draft.tex @@ -2,103 +2,48 @@ \begin{document} \section{Theoretical background}\label{sec:theory} +Scientists have made use of light in devices leading to... + +The use of light have facilitated many discoveries through history. The invention +of the compound microscope, which used lenses to focus light, led to the first +observation of animal cells. And the telescope, which led to a greater understanding +of the universe. + +The nature of light has long been a subject of interest and discussion. Around year +1600 the first microscope was invented, which made use of light through lenses. +The observation made using the microscope led to a greater understanding of the +world at a microscopic scale. However, the study of light itself have led to (something). + +Through the 1600s, the view shifted from particle to wave. % Introduction? -The nature of light has long been a subject of discussion, from the 1500s +The nature of light has long been a subject of interest and discussion, from the 1500s and the invention of microscopes, through the 1700s where both a particle theory and a wave theory. In the late 1600s, Christiaan Huygens proposed the wave theory of light, -which was challenged by Isaac Newton's particle theory. The particle theory made -was the leading theory in the beginning of 1800s, when Thomas Young demonstrated -the interference of light, through his double-slit experiment, the wave theory found +which was challenged by Isaac Newton's particle theory. 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 wave theory found new hold. In the 1800s, the study of ideal black bodies done by Gustav R. Kirchhoff, lead to a better understanding of heat radiation. Wilhelm Wien started working on determining the spectral energy distribution, and Wien's law. The law did make sense for high frequencies, -however, there were inconsistencies when frequency were lower than a certain value. +however, as there were inconsistencies when the frequency were lower than a threshold value. Wiens law led to an exponential curve, which disagree with the law of conservation. -Max Planck guessed a result which led to Plancks radiation law, which he later derived -using Boltzmanns statistical interpretation of the second law of thermodunamics. -Plancks findings gave rise to Einsteins quantum hypothesis, and later the wave-particle +Max Planck guessed a solution to the problem, which led to Planck's radiation law. +He later derived the radiation law 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}. -For small atoms classical mechanics are not able to explain the position of a particle, +For small atoms classical mechanics are not sufficient in describing the position of a particle, and Heisenberg uncertainty priciple suggest that the particles have a wavelike behavior. 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. % Methods? -% Schrödinger -The Schrödinger equation has a general form -\begin{align} - i \hbar \frac{\partial}{\partial t} | \Psi \rangle &= \hat{H} | \Psi \rangle \ , - \label{eq:schrodinger_general} -\end{align} -where $i$ is the imaginary number, and $\hbar$ is Plancks constant. Here $| \Psi \rangle$ -is the quantum state and $\hat{H}$ is a Hamiltonian operator. -For two-dimensional position space, the quantum state can be expressed using the -time-dependent complex-valued wave function $\Psi (x, y, t)$. The square modulus of the wave function $|\psi|^{2}$, predicts probability of finding the particle at position $(x, y)$ at time t. -% Segue to Born -The modulus of the wave function, is related to the probability density function -\begin{align} - p(x, y \ | \ t) &= |\Psi(x, y, t)|^{2} = \Psi^{*}(x, y, t) \Psi(x, y, t) - \label{eq:born_rule} -\end{align} -using the Born rule, where $^{*}$ denotes the complex conjugated wave function. -When the potential is time-independent, the Schrödinger equation can be expressed as -\begin{align*} - i \hbar \frac{\partial}{\partial t} \Psi (x, y, t) &= - \frac{\hbar^{2}}{2m} \bigg( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \bigg) \Psi (x, y, t) \\ - & \quad + V(x, y, t) \Psi (x, y, t) \numberthis \ . - \label{eq:schrodinger_special} -\end{align*} -When we scale Schrödinger equation by the dimensionful variables, we are left with -a wave function $u$, potential $v$ and the dimensionless equation -\begin{align} - i \frac{\partial u}{\partial t} &= - \frac{\partial^{2} u}{\partial x^{2}} - \frac{\partial^{2} u}{\partial y^{2}} + v(x, y) u \ . - \label{eq:schrodinger_dimensionless} -\end{align} - -% Crank-Nicolson -To evaluate the position of a single particle, we have to consider partial differential -equations (PDE). To solve these numerically, we have to discretize Equation \eqref{eq:schrodinger_dimensionless}. -We use the Crank-Nicolson method (CN), which combines the forward and backward finite -difference method. % -% Include the general $\theta$-method -Using the $\theta$-rule \footnote{For $\theta \in [0, 1]$ we can derive Forward Euler using $\theta = 1$, Backward Euler using $\theta = 0$, and $\theta = 1/2$ gives Crank-Nicolson} -with $\theta = 1/2$, CN can be written as -\begin{align} - \frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} &= \frac{1}{2} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \ . - \label{eq:crank_nicolson} -\end{align} -To simplify notation and avoid confusion of indices with the imaginary number $i$, -we have used the notation $\ivec, \jvec$ in subscript to indicate the commonly named indices $i, j$ -in x- and y-direction. In addition, the superscript $n, n+1$ indicate position in time. -We use CN to derive the discretized Schrödinger equation -\begin{align*} - & u_{\ivec, \jvec}^{n+1} - \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1} \big] \\ - & - \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1} \big] + \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\ - &= u_{\ivec, \jvec}^{n} + \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n} \big] \\ - & \quad + \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n} \big] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \numberthis \ . - \label{eq:schrodinger_discretized} -\end{align*} -The full derivation of both Equation \eqref{eq:crank_nicolson} and Equation \eqref{eq:schrodinger_discretized} -can be found in Appendix \ref{ap:crank_nicolson} - -First, we Taylor -expand the wave equation $u$ around (position and time). - - -\section{Notes}\label{sec:notes} -\subsection*{Introduction - draft 2} -In classical mechanics we study the kinematics and dynamics of physical objects, -ignoring its intrinsic properties for simplicity. It allows us to describe the -forces acting on an object as well as the motion of the object. We can describe -a planets orbital movement \cite{britannica:2023:kepler}, calculate the ... necessary -to launch satellites into orbit, or simply figure out where a ball is going to land -when you throw it... However, when want to study an object at a microscopic level, -e.g. a single atom, classical mechanics falls short. - +% Double-slit experiment Thomas Young first performed the double-slit experiment in 1801, to demonstrate the principle of interference of light \cite{britannica:2023:young}, postulating light as waves. In the study of blackbodies, scientists were not able to describe @@ -107,6 +52,292 @@ contradicted the principle of conservation of energy \cite{britannica:1998:planc Max Planck assumed that the radiated energy consist of discrete values, or quanta, to describe the peak in radiated energy. +% Reference: https://tex.stackexchange.com/questions/469109/how-to-change-arrowheads-to-lie-on-a-plane +\begin{tikzpicture}[scale=1.25,every node/.append style={transform shape}] + \foreach \x in {-1,-0.75,...,0} { + \draw (\x,-1) -- (\x,1); + } + \draw[fill=black!10] (0.5,-2,-1) -- (0.5,-2,1) -- (0.5,2,1) -- (0.5,2,-1) -- (0.5,-2,-1); + \fill (0.5,0,0) circle (0.05); + \foreach \r in {0.25,0.5,...,1.75} { + \draw (0.5,0) ++(-60:\r) arc (-60:60:\r); + } + \draw[fill=black!10] (2,-2,-1) -- (2,-2,1) -- (2,2,1) -- (2,2,-1) -- (2,-2,-1); + %\fill (2,0.5) circle (0.05) (2,-0.5) circle (0.05); + \foreach \r in {0.25,0.5,...,2} { + \draw (2,0.5) ++(-60:\r) arc (-60:60:\r); + \draw (2,-0.5) ++(-60:\r) arc (-60:60:\r); + } + \draw[fill=black!10] (4,-2,-1) -- (4,-2,1) -- (4,2,1) -- (4,2,-1) -- (4,-2,-1); +% LABELLING + \begin{scope}[canvas is yz plane at x=2,rotate=-90] + \node[circle,inner sep=0.5mm,fill,label=above:{S${}_1$}] at (0,0.5){}; + \node[circle,inner sep=0.5mm,fill,label=below:{S${}_2$}] at (0,-0.5) {}; + \draw[dash pattern=on 1.5pt off 1pt,thin] (-0.5,0.5) -- (0,0.5) + (-0.5,-0.5) -- (0,-0.5); + \pgflowlevelsynccm + \draw[|<->|] (-0.5,0.5) -- (-0.5,-0.5) node[midway,right=-0.1cm] {d}; + \end{scope} + \begin{scope}[canvas is yz plane at x=0.5,rotate=-90] + \node[below left=-0.1cm] at (0,0) {S${}_0$}; + \end{scope} + \begin{scope}[xshift=4cm,yshift=2cm,rotate=-90,canvas is xy plane at z=0] + \fill[white] (0,0) rectangle (4,4); + \begin{axis}[ + width=5.575cm, + xmin=-0.5, + xmax=0.5, + ticks=none + ] + \addplot [samples=1000,blue + ] + {(cos(deg(5*pi*sin(deg(x)))))^(2)*((sin(deg(4*pi*sin(deg(x)))))/(4*pi*sin(deg(x))))^(2)}; + \end{axis} + \end{scope} + \draw[thin,densely dashed,blue] (2,0) -- (6.9,0); + \draw[thin,densely dashed,blue] (2,0) -- +(15:2.5); + \draw[thin,densely dashed,blue] (2,0) -- +(-15:2.5); + \draw[thin,densely dashed,blue] (2,0) -- +(32:2.5); + \draw[thin,densely dashed,blue] (2,0) -- +(-32:2.5); +\end{tikzpicture} + +% Author: Izaak Neutelings (June 2020) +% Inspiration: +% https://courses.physics.ucsd.edu/2011/Summer/session1/physics2c/diffraction.pdf +% https://tex.stackexchange.com/questions/201830/periodic-shading-in-tikz +% \begin{tikzpicture}[ +% nodal/.style={mylightgreen,dashed,very thin}, +% declare function={ +% %xnode(\n,\dn,\lam,\f) = sqrt( (\n^2+(\n+\dn)^2)*\lambd^2/2 - (\n^2-(\n+\dn)^2)^2*\lambd^4/(4*\a^2) - \a^2/4 ); +% xnode(\n,\dn,\lam,\f) = \lam/\f*sqrt( \n^2*(\f^2-\dn^2)+\n*\dn*(\f^2-\dn^2)+\dn^2*\f^2/2-(\f^4+\dn^4)/4 ); +% ynode(\n,\dn,\lam,\a) = (2*\n*\dn+\dn^2)*\lam/(2*\f); +% intensity(\y,\lam,\a,\L) = cos(180*\a*\y/(2*\lam*sqrt(\L*\L+\y*\y)))^2; +% } +% ] + +% \def\L{3.8} % distance between walls +% \def\H{5.4} % total wall height +% \def\h{2.8} % plane wave height +% \def\t{0.15} % wall thickness +% \def\a{1.15} % slit distance +% \def\d{0.20} % slit size +% \def\N{21} % number of waves +% \def\lambd{0.20} % wavelength +% \def\R{\N*\lambd} % wave radius +% \def\Nlines{3} % number of nodal lines +% \def\A{1.6} % amplitude +% %\def\r{0.06} % point source radius +% %\def\nmax{10} +% \def\nsamples{100} +% \def\ang{62} + +% \begin{scope} +% \clip (-\t/2,-\H/2) rectangle (\L,\H/2); +% %\clip (-\t/2,0.7*\a) -- (0.6*\L,\H/2) -- (\L,\H/2) -- +% % (\L,-\H/2) -- (0.6*\L,-\H/2) -- (-\t/2,-0.7*\a) -- cycle; + +% % NODAL LINES +% \draw[nodal] +% (0.08*\N*\lambd,0) -- (1.06*\R,0); +% \coordinate (NP0) at (\L,0); % to avoid "Dimension too large error" +% \foreach \dn [evaluate={ +% \f=\a/\lambd; +% \nmin=2.5+0.2*\dn; %0.501*(-\dn+\f) +% \nmax=10; %(NP0) +% \c=int(\dn<\f); +% \y=\L/sqrt((\a/(\lambd*\dn))^2-1); +% }] in {1,...,\Nlines}{ +% \coordinate (NP+\dn) at (\L,\y); % to avoid "Dimension too large error" +% \coordinate (NP-\dn) at (\L,-\y); % to avoid "Dimension too large error" +% \ifnum\c=1 +% \draw[nodal,variable=\n,samples=\nsamples,smooth] +% plot[domain=\nmin:\nmax] ({xnode(\n,\dn,\lambd,\f)},{ynode(\n,\dn,\lambd,\f)}) +% -- (NP+\dn); +% \draw[nodal,variable=\n,samples=\nsamples,smooth] +% plot[domain=\nmin:\nmax] ({xnode(\n,\dn,\lambd,\f)},{-ynode(\n,\dn,\lambd,\f)}) +% -- (NP-\dn); +% \fi +% } + +% % WAVES +% \foreach \i [evaluate={\R=\i*\lambd;}] in {1,...,\N}{ +% \ifodd\i +% \draw[myblue,line width=0.8] (0,\a/2)++(\ang:\R) arc (\ang:-\ang:\R); +% \draw[myred,line width=0.8] (0,-\a/2)++(\ang:\R) arc (\ang:-\ang:\R); +% \else +% \draw[myblue!80,line width=0.1] (0,\a/2)++(\ang:\R) arc (\ang:-\ang:\R); +% \draw[myred!80,line width=0.1] (0,-\a/2)++(\ang:\R) arc (\ang:-\ang:\R); +% \fi +% } +% \end{scope} + +% % PLANE WAVES +% \foreach \i [evaluate={\x=-\i*\lambd;}] in {0,...,5}{ +% \ifodd\i +% \draw[myblue,line width=0.8] (\x,-\h/2) -- (\x,\h/2); +% \else +% \draw[myblue,line width=0.1] (\x,-\h/2) -- (\x,\h/2); +% \fi +% } + +% % WALL +% \fill[wall] +% (\t/2,\a/2-\d/2) rectangle (-\t/2,-\a/2+\d/2) +% (\t/2,\a/2+\d/2) rectangle (-\t/2,\H/2) +% (\t/2,-\a/2-\d/2) rectangle (-\t/2,-\H/2) +% (\L,-\H/2) rectangle (\L+\t,\H/2); + +% % SHADES +% \begin{scope}[shift={(1.08*\L,0)}] +% \def\yz{\L/sqrt((\a/\lambd)^2-1)} % m = +- 1/2 +% \def\yZ{\L/sqrt((\a/\lambd/2)^2-1)} % m = +- 1 +% \clip (0,-\H/2) rectangle (1.1*\A,\H/2); +% \fill[white] (0,-\H/2) rectangle++ (\A,\H); % to fill seams +% \foreach \i [evaluate={\n=0.5*\i;\yn=\L/sqrt((\a/(2*\lambd*\n))^2-1); +% }] in {1,...,\Nlines}{ +% \ifodd\i % if even +% \fill[myshadow] (0,{-\yn-0.1}) rectangle++ (\A,0.2); % to fill seams +% \fill[myshadow] (0,{ \yn-0.1}) rectangle++ (\A,0.2); % to fill seams +% \fi +% } +% \path[left color=myshadow,right color=myshadow,middle color=white,shading angle={180}] +% (0,{-\yz}) rectangle (\A,{\yz}); +% \foreach \i [evaluate={ +% \n=0.5*\i; +% \m=0.5*(\i+1); +% \yn=\L/sqrt((\a/(2*\lambd*\n))^2-1); +% \ym=\L/sqrt((\a/(2*\lambd*\m))^2-1); +% \dang=mod(\i,2)*180; +% }] in {1,...,\Nlines}{ +% \path[left color=myshadow,right color=white,shading angle={\dang}] +% (0,\yn) rectangle (\A,\ym); +% \path[left color=myshadow,right color=white,shading angle={180+\dang}] +% (0,-\yn) rectangle (\A,-\ym); +% } +% \end{scope} + +% % INTENSITY +% \begin{scope}[shift={(1.1*\L+1.1*\A,0)}] +% \draw[->,thick] (-0.08*\A,0) -- (1.3*\A,0) node[right=-2] {$\expval{I}$}; % I axis +% \draw[->,thick] (0,-0.52*\H) -- (0,0.54*\H) node[right] {$y$}; % y axis +% \draw[nodal] (NP0) --++ (0.15*\L+2.1*\A,0); % green nodal lines +% \foreach \i [evaluate={\y=\L/sqrt((\a/(\lambd*\i))^2-1)}] in {1,...,\Nlines}{ % green nodal lines +% \draw[nodal] (NP+\i) --++ ({0.15*\L+1.1*\A+\A*intensity(\y,\lambd,\a,\L)},0); +% \draw[nodal] (NP-\i) --++ ({0.15*\L+1.1*\A+\A*intensity(\y,\lambd,\a,\L)},0); +% } +% \draw[myred,thick,variable=\y,samples=\nsamples,smooth,domain=-\H/2:\H/2] +% plot({\A*intensity(\y,\lambd,\a,\L)},\y); +% \foreach \i [evaluate={ % ticks +% \modd=\i; %int(\i); +% \meven=int(\i-1); +% \y=\L/sqrt((\a/(\lambd*\i))^2-1); +% }] in {1,...,\Nlines}{ +% \ifodd\i +% \tick{0,-\y}{180} node[right=0,scale=0.85] {$m=-\frac{\modd}{2}$}; +% \tick{0,\y}{180} node[right=0,scale=0.85] {$m=+\frac{\modd}{2}$}; +% \else +% \tick{0,-\y}{180} node[right=0,scale=0.85] {$m=-\meven$}; +% \tick{0,\y}{180} node[right=0,scale=0.85] {$m=+\meven$}; +% \fi +% } +% \end{scope} + +% \end{tikzpicture} + +% Schrödinger +% ----------- +% The wave equation, Schrödinger equation has a general form +% \begin{align} +% i \hbar \frac{\partial}{\partial t} | \Psi \rangle &= \hat{H} | \Psi \rangle \ , +% \label{eq:schrodinger_general} +% \end{align} +% where $i$ is the imaginary number, and $\hbar$ is Plancks constant. Here $| \Psi \rangle$ +% is the quantum state and $\hat{H}$ is a Hamiltonian operator. % which represent the energy of the system +% For two-dimensional position space, the quantum state can be expressed using the +% time-dependent complex-valued wave function $\Psi (x, y, t)$. The square modulus of the wave function $|\psi|^{2}$, predicts probability of finding the particle at position $(x, y)$ at time t. +% % Segue to Born +% The modulus of the wave function, is related to the probability density function +% \begin{align} +% p(x, y \ | \ t) &= |\Psi(x, y, t)|^{2} = \Psi^{*}(x, y, t) \Psi(x, y, t) +% \label{eq:born_rule} +% \end{align} +% using the Born rule, where $^{*}$ denotes the complex conjugated wave function. +% When the potential is time-independent, the Schrödinger equation can be expressed as +% \begin{align*} +% i \hbar \frac{\partial}{\partial t} \Psi (x, y, t) &= - \frac{\hbar^{2}}{2m} \bigg( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \bigg) \Psi (x, y, t) \\ +% & \quad + V(x, y, t) \Psi (x, y, t) \numberthis \ . +% \label{eq:schrodinger_special} +% \end{align*} +% When we scale Schrödinger equation by the dimensionful variables, we are left with +% a wave function $u$, potential $v$ and the dimensionless equation +% \begin{align} +% i \frac{\partial u}{\partial t} &= - \frac{\partial^{2} u}{\partial x^{2}} - \frac{\partial^{2} u}{\partial y^{2}} + v(x, y) u \ . +% \label{eq:schrodinger_dimensionless} +% \end{align} + +% Crank-Nicolson +% -------------- +% To evaluate the position of a single particle, we have to consider partial differential +% equations (PDE). To solve these numerically, we have to discretize Equation \eqref{eq:schrodinger_dimensionless}. +% We use the Crank-Nicolson method (CN), which combines the forward and backward finite +% difference method. +% Include the general $\theta$-method +% Using the $\theta$-rule \footnote{For $\theta \in [0, 1]$ we can derive Forward Euler using $\theta = 1$, Backward Euler using $\theta = 0$, and $\theta = 1/2$ gives Crank-Nicolson} +% with $\theta = 1/2$, CN can be written as +% \begin{align} +% \frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} &= \frac{1}{2} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \ . +% \label{eq:crank_nicolson} +% \end{align} +% To simplify notation and avoid confusion of indices with the imaginary number $i$, +% we have used the notation $\ivec, \jvec$ in subscript to indicate the commonly named indices $i, j$ +% in x- and y-direction. In addition, the superscript $n, n+1$ indicate position in time. +% We use CN to derive the discretized Schrödinger equation +% \begin{align*} +% & u_{\ivec, \jvec}^{n+1} - \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1} \big] \\ +% & - \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1} \big] + \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\ +% &= u_{\ivec, \jvec}^{n} + \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n} \big] \\ +% & \quad + \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n} \big] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \numberthis \ . +% \label{eq:schrodinger_discretized} +% \end{align*} +% The full derivation of both Equation \eqref{eq:crank_nicolson} and Equation \eqref{eq:schrodinger_discretized} +% can be found in Appendix \ref{ap:crank_nicolson}. + +% Implementation +The implementation of CN is simplified usein Dirichlet boundary conditions, which +can be found in Table \ref{tab:dirichlet}. In addition, we use Gaussian wave packets +for the initial wave function +\begin{align} + u(x, y, t=0) = e^{- \frac{(x-x_{c})^{2}}{2 \sigma_{x}^{2}} - \frac{(y-y_{c})^{2}}{2 \sigma_{y}^{2}} + ip_{x}x + ip_{y}y} +\end{align} +\begin{algorithm}[H] + \caption{Crank-Nicolson scheme} + \label{algo:cn_scheme} + \begin{algorithmic} + \Procedure{Crank-Nicolson}{$args$} + \State Insert pseudo code $var \leftarrow \text{ some text}$ + \EndProcedure + \end{algorithmic} +\end{algorithm} + + +\section{Notes}\label{sec:notes} +\subsection*{Introduction - draft 2} +In classical mechanics we study the kinematics and dynamics of physical objects, +ignoring their intrinsic properties for simplicity. It allows us to describe the +forces acting on an object as well as the motion of the object. We can describe +a planets orbital movement \cite{britannica:2023:kepler}, calculate the ... necessary +to launch satellites into orbit, or simply figure out where a ball is going to land +when you throw it... However, when want to study an object at a microscopic level, +e.g. a single atom, classical mechanics falls short. + +% Thomas Young first performed the double-slit experiment in 1801, to demonstrate +% the principle of interference of light \cite{britannica:2023:young}, postulating +% light as waves. In the study of blackbodies, scientists were not able to describe +% the radiated intensity of increased frequencies using classical mechanichs, as they +% contradicted the principle of conservation of energy \cite{britannica:1998:planck}. +% Max Planck assumed that the radiated energy consist of discrete values, or quanta, +% to describe the peak in radiated energy. + Light as particles -> waves -> particles/packets Erwin Schrödinger wanted to find a mathematical description of the wave characteristics of matter, supporting the wave-particle idea. He postulated a function which varies diff --git a/latex/references.bib b/latex/references.bib index cb962f0..78cee6f 100644 --- a/latex/references.bib +++ b/latex/references.bib @@ -9,11 +9,71 @@ } @misc{britannica:1999:light, - author = {The Editors of Encyclopaedia Britannica}, - title = {Light}, - publisher = {Britannica}, - url = {https://www.britannica.com/science/light}, - urldate = {2023-12-15}, + author = {The Editors of Encyclopaedia Britannica}, + title = {Light}, + publisher = {Britannica}, + url = {https://www.britannica.com/science/light}, + urldate = {2023-12-15}, +} + +@article{sinha:2010:multi_order_inference, + author = {Urbasi Sinha and Christophe Couteau and Thomas Jennewein and Raymond Laflamme and Gregor Weihs}, + journal = {Science}, + title = {Ruling Out Multi-Order Interference in Quantum Mechanics}, + volume = {329}, + number = {5990}, + pages = {418--421}, + year = {2010}, + doi = {10.1126/science.1190545}, + url = {https://www.science.org/doi/abs/10.1126/science.1190545}, + urldate = {2023-12-16} +} + +@article{young:1804:double_slit, + author = {Thomas Young}, + title = {I. The Bakerian Lecture. Experiments and calculations relative to physical optics}, + journal = {Philosophical Transactions of the Royal Society of London}, + volume = {94}, + pages = {1--16}, + year = {1804}, + doi = {10.1098/rstl.1804.0001}, + url = {https://royalsocietypublishing.org/doi/abs/10.1098/rstl.1804.0001}, + urldate = {2023-12-16}, +} + +@book{benacquista:2018:classical_mechanics, + author = {Matthew J. Benacquista and Joseph D. Romano}, + title = {Classical Mechanics}, + publisher = {Springer International Publishing : Imprint: Springer}, + year = {2018}, + edition = {1} +} + +@book{griffiths:2018:quantum_mechanics, + author = {David J. Griffiths and Darrell F. Schroeter}, + title = {Introduction to quantum mechanics}, + publisher = {Cambridge University Press}, + year = {2018}, + edition = {3} +} + +@book{springer:2018:compendium_quantum_physics, + author = {Daniel Greenberger and Klaus Hentschel and Friedel Weinert}, + title = {Compendium of Quantum Physics}, + publisher = {Springer Berlin, Heidelberg}, + year = {2009}, + edition = {1} +} + +@article{key, + author = {Asher Peres and Daniel R. Terno}, + title = {Quantum Information and Relativity Theory}, + journal = {Reviews of Modern Physics}, + number = {1}, + volume = {76}, + doi = {10.1103/revmodphys.76.93}, + year = {2004}, + url = {http://dx.doi.org/10.1103/RevModPhys.76.93} } # Schrodinger and Crank-Nicolson diff --git a/latex/schrodinger_simulation.tex b/latex/schrodinger_simulation.tex index 2c1a92e..3341bb6 100644 --- a/latex/schrodinger_simulation.tex +++ b/latex/schrodinger_simulation.tex @@ -20,6 +20,7 @@ %% I recommend downloading TeXMaker, because it includes a large library of the most common packages. \usepackage{physics,amssymb} % mathematical symbols (physics imports amsmath) +\usepackage[no-test-for-array]{nicematrix} \usepackage{graphicx} % include graphics such as plots \graphicspath{.images/} @@ -33,6 +34,29 @@ \usepackage{algorithm} \usepackage[noend]{algpseudocode} \usepackage{tikz} +% \usepackage{pgfplots} +% \pgfplotsset{compat=1.18} +% \usetikzlibrary{3d} +\usepackage{xcolor} +% \usepackage{etoolbox} %ifthen +% \usetikzlibrary{calc} +% \usetikzlibrary{arrows,arrows.meta} +% \usetikzlibrary{decorations.markings} +% \usetikzlibrary{angles,quotes} +% \usetikzlibrary{fadings} +% \tikzset{>=latex} +% \colorlet{wall}{blue!30!black} +% \colorlet{myblue}{blue!70!black} +% \colorlet{myred}{red!70!black} +% \colorlet{mydarkred}{red!50!black} +% \colorlet{mylightgreen}{green!60!black!70} +% \colorlet{mygreen}{green!60!black} +% \colorlet{myredgrey}{red!50!black!80} +% \colorlet{myshadow}{blue!30!black!90} +% \tikzstyle{wave}=[myblue,thick] +% \tikzstyle{mydashed}=[black!70,dashed,thin] +% \tikzstyle{mymeas}=[{Latex[length=3,width=2]}-{Latex[length=3,width=2]},thin] +% \tikzstyle{mysmallarr}=[-{Latex[length=3,width=2]}] % \usetikzlibrary{quantikz} % defines the color of hyperref objects % Blending two colors: blue!80!black = 80% blue and 20% black @@ -44,7 +68,9 @@ % Biblio stuff % \def\biblio{\bibliographystyle{plain}\bibliography{../references/references}} + \newcommand\numberthis{\addtocounter{equation}{1}\tag{\theequation}} +\newcommand{\rvline}{\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}} % Defines indices i and j to avoid confusion with imaginary i \newcommand{\ivec}{\hat{\imath}} @@ -67,10 +93,10 @@ \maketitle % Introduction -% \subfile{sections/introduction} +\subfile{sections/introduction} % Methods -% \subfile{sections/methods} +\subfile{sections/methods} % Results % \subfile{sections/results} @@ -79,7 +105,7 @@ % \subfile{sections/conclusion} % Notes -\subfile{draft} +% \subfile{draft} \clearpage \newpage diff --git a/latex/sections/appendices.tex b/latex/sections/appendices.tex index 980e046..b14a0e9 100644 --- a/latex/sections/appendices.tex +++ b/latex/sections/appendices.tex @@ -3,7 +3,7 @@ \begin{document} \appendix \section{The Crank-Nicholson method}\label{ap:crank_nicolson} -The Crank-Nicolson \(CN\) approach considers both the forward difference, an explicit scheme, +The Crank-Nicolson (CN) approach consider both the forward difference, an explicit scheme, \begin{equation*} \frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} = F_{\ivec, \jvec}^{n} \ , \end{equation*} diff --git a/latex/sections/introduction.tex b/latex/sections/introduction.tex index 5de6f41..47f43f2 100644 --- a/latex/sections/introduction.tex +++ b/latex/sections/introduction.tex @@ -2,12 +2,63 @@ \begin{document} \section{Introduction}\label{sec:introduction} +The nature of light has long been a subject of interest and discussion. % +% Important part of human behavior is observing and understanding our surroundings. +% Many big discoveries have been made through observations, verified by mathematical +% explanations. Classical physics is based on calculation predicting something we +% verify by observation etc. But what happens when we move down to the microscopic +% scale, can we still predict the position of a microscopic ball, also called an atom? +In classical mechanics, we study the kinematics and dynamics of physical objects, +while ignoring their intrinsic properties for simplicity. Newton's second law can be +applied to an object to describe its trajectory. % It allows us to describe the +% forces acting on an object as well as the motion of the object. We can describe +% a planets orbital movement \cite{britannica:2023:kepler}, calculate the ... necessary +% to launch satellites into orbit, or simply figure out where a ball is going to land +% when you throw it... However, when want to study an object at a microscopic level, +% e.g. a single atom, classical mechanics falls short. + +Elementary particles such as electrons, does not abide by the laws of classical mechanics. +For several years, scientists did not agree on whether light was a particle or a +wave. Through the study of interference of light, and radiation of ideal blackbodies, +it has been shown that light has both wavelike and particle-like characteristics. +This is known as the wave-particle duality, and was showed by Albert Einstein in +1905. % +% Thomas Young studied the interference of light, and found that light to showed +% wavelike characteristics \cite{young:1804:double_slit}. This did not agree with +% Newtons particle-theory + +Erwin Schrödinger wanted to find a mathematical description of the wave characteristics +of matter, supporting the wave-particle idea. He postulated a wave function which varies +with position, where the function squared can be interpreted as the probability +of finding an electron at a given position. This resulted in the Schrödinger equation, +a wave eqution of the energy levels for a hydrogen atom. It also shows how a quantum +state evolves with time \cite[p. 81]{wu:2023:quantum}. + +We will simulate the time-dependent Schrödinger equation in two dimensions, to +study the light wave interference in the double-slit experiment. In addition, we +will include variations of walls such as single- and triple-slit. To solve the equation, +we will apply the Crank-Nicolson method in 2+1 dimensions. + +% However, according to the Heisenberg uncertainty principle, we can't find dx and/or +% dp = 0. dx = sqrt{Var(x)} "spread in position", dp = hat{\Psi}(p) = sqrt{Var(p)} +% dx \cdot dp \geq \frac{\hbar}{2} +% Fourier transform \Psi(x) \doublearrow hat{\Psi}(p) +% \Psi(x) = \int_{infty}^{infty} (alt sum) hat{\Psi}(p) \cos(px) dp sum of different wave forms % Light - particle or wave % - double-slit, blackbodies radiation - +% The nature of light. % Position space % - classical vs quantum mechanics +% Intuitions of the behavior of physical objects, predictable. Not like the quantum +% when we scale down to the atom, our intuition are not as reliable and prediction +% are not as perfect. +% Instead of finding the path of a ball, we find all the possible paths a ball can take. +% The world is not one-dimensional, and modelling it require partial diff eqs +In Section \ref{sec:methods}, we will present the theoretical background for +this experiment, as well as the algorithms and tools used in the implementation. +Continuing with Section \ref{sec:results}, we will present our results and +discuss our findings. Lastly, we will conclude our findings in Section \ref{sec:conclusion}. \end{document} % crank-nicolson method! diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 7a4946f..68455e0 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -2,28 +2,231 @@ \begin{document} \section{Methods}\label{sec:methods} % +\subsection{The Schrödinger equation}\label{ssec:schrodinger} % +% Add something that takes Planck to Schrödinger +% In classical mechanics, we have Newton laws and conservation of energy. In quantum +% mechanics, we have Schrödinger equation. +The Schrödinger equation has a general form +\begin{align} + i \hbar \frac{\partial}{\partial t} | \Psi \rangle &= \hat{H} | \Psi \rangle \ , + \label{eq:schrodinger_general} +\end{align} +where $i$ is the imaginary number, and $\hbar$ is Plancks constant. $\hat{H}$ is +a Hamiltonian operator, which represent the energy for the system, and $| \Psi \rangle$ +is the quantum state. In two-dimensional position space, the quantum state can +be expressed using the time-dependent complex-valued wave function $\Psi (x, y, t)$. +Using Born rule, the square modulus of the wave function is proportional to the +probability density of finding a particle at position $(x, y)$ at time t. The relation +is given by +\begin{align} + p(x, y \ | \ t) &= |\Psi(x, y, t)|^{2} = \Psi^{*}(x, y, t) \Psi(x, y, t) \ , + \label{eq:born_rule} +\end{align} +where $\Psi^{*}$ denotes the complex conjugated wave function. +% Add something about kinetic and potential energy, to introduce the potential V +When the potential is time-independent, the Schrödinger equation can be expressed as +\begin{align*} + i \hbar \frac{\partial}{\partial t} \Psi (x, y, t) &= - \frac{\hbar^{2}}{2m} \bigg( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} \bigg) \Psi (x, y, t) \\ + & \quad + V(x, y, t) \Psi (x, y, t) \numberthis \ . + \label{eq:schrodinger_special} +\end{align*} +The partial derivatives (...) gives the kinetic energy, and the potental $V$ is +the external environment. In this experiment we will only consider the case where +the potential is time-independent, resulting in $V = V(x, y)$ +When we scale Schrödinger equation by the dimensionful variables, we are left with +a wave function $u$, potential $v$ and the dimensionless equation +\begin{align} + i \frac{\partial u}{\partial t} &= - \frac{\partial^{2} u}{\partial x^{2}} - \frac{\partial^{2} u}{\partial y^{2}} + v(x, y) u \ . + \label{eq:schrodinger_dimensionless} +\end{align} % +This gives us the Born rule +\begin{align} + p(x, y \ | \ t) &= |u(x, y, t)|^{2} = u^{*}(x, y, t) u(x, y, t) \ . + \label{eq:born_rule_scaled} +\end{align} -\subsection{Double-slit experiment}\label{ssec:double_slit} % -In the beginning of the 1800s, the general view was that light consisted of particles. -However, in 1801 Thomas Young demonstrated the principle of interference of light -\cite{britannica:2023:young}, while postulating light as waves rather than particles. -% Thomas Young first performed the double-slit experiment in 1801, to demonstrate -% the principle of interference of light \cite{britannica:2023:young}, postulating -% light as waves. In the study of blackbodies, scientists were not able to describe -% the radiated intensity of increased frequencies using classical mechanichs, as they -% contradicted the principle of conservation of energy \cite{britannica:1998:planck}. -% Max Planck assumed that the radiated energy consist of discrete values, or quanta, -% to describe the peak in radiated energy. -The double-slit experiment -\subsection{Schrödinger equation}\label{ssec:schrodinger} % -\subsection{Crank-Nicolson}\label{ssec:crank_nicolson} % +\subsection{The Crank-Nicolson scheme}\label{ssec:crank_nicolson} % +When we evaluate a particle in position space, we have to consider partial differential +equations (PDE). To solve these numerically, we have to discretize Equation \eqref{eq:schrodinger_dimensionless}. +We use the $\theta$-rule \footnote{Using the $\theta$-rule, we can derive Forward Euler using $\theta = 1$, and Backward Euler using $\theta = 0$}, +to combine the forward (explicit) and backward (implicit) finite difference method. +The result is a linear combination of the explicit and implicit scheme, given by +\begin{align} + \frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} &= \theta F_{\ivec, \jvec}^{n+1} + (1 - \theta) F_{\ivec, \jvec}^{n} \ , + \label{eq:theta_rule} +\end{align} % +where $\theta \in [0, 1]$. +To simplify notation and avoid confusion of indices with the imaginary number $i$, +we have used the notation $\ivec, \jvec$ in subscript to indicate the commonly named indices $i, j$ +in x- and y-direction. In addition, the superscript $n, n+1$ indicate position in time. +We get the Crank-Nicolson method (CN) when $\theta = 1/2$ gives Crank-Nicolson +\begin{align} + \frac{u_{\ivec, \jvec}^{n+1} - u_{\ivec, \jvec}^{n}}{\Delta t} &= \frac{1}{2} \bigg[ F_{\ivec, \jvec}^{n+1} + F_{\ivec, \jvec}^{n} \bigg] \ . + \label{eq:crank_nicolson_method} +\end{align} % +Using CN, we derive the discretized Schrödinger equation given by +\begin{align*} + & u_{\ivec, \jvec}^{n+1} - \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1} \big] \\ + & - \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1} \big] + \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\ + &= u_{\ivec, \jvec}^{n} + \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n} \big] \\ + & \quad + \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n} \big] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \numberthis \ . + \label{eq:schrodinger_discretized} +\end{align*} % +The full derivation of both Equation \eqref{eq:crank_nicolson_method} and Equation \eqref{eq:schrodinger_discretized} +can be found in Appendix \ref{ap:crank_nicolson}. + + +\subsection{The double-slit experiment}\label{ssec:double_slit} % +Thomas Young first performed the double-slit experiment in 1801 to demonstrate the +principle of interference of light \cite{britannica:2023:young}, while postulating +light as waves rather than particles. The double-slit experiment gives a diffraction +pattern, where constructive interference of light result in bright spots, and destructive +interference result in dark spots. +% Something about Heisenberg uncertainty principle -% Might be better to move theory on double-slit here and have a subsection of light -% property etc. as a first section or in introduction? \subsection{Implementation}\label{ssec:implementation} % +% Add tables of parameters used, initial conditions, notation etc. +A, B are sparse csc matrix +- theory of csc matrix? +% \begin{equation*} +% \begin{pNiceArray}{ccc|ccc} +% \bullet & \bullet & & \bullet & & & & & \\ +% \bullet & \bullet & \bullet & & \bullet & & & & \\ +% & \bullet & \bullet & & & \bullet & & & +% \end{pNiceArray} +% \end{equation*} + +\begin{equation*} + \begin{bmatrix} + \begin{matrix} + \bullet & \bullet & \phantom{\bullet} \\ + \bullet & \bullet & \bullet \\ + \phantom{\bullet} & \bullet & \bullet + \end{matrix} + & \rvline & + \begin{matrix} + \bullet & \phantom{\bullet} & \phantom{\bullet} \\ + \phantom{\bullet} & \bullet & \phantom{\bullet} \\ + \phantom{\bullet} & \phantom{\bullet} & \bullet + \end{matrix} + & \rvline & + \begin{matrix} + \phantom{\bullet} & \phantom{\bullet} & \phantom{\bullet} \\ + \phantom{\bullet} & \phantom{\bullet} & \phantom{\bullet} \\ + \phantom{\bullet} & \phantom{\bullet} & \phantom{\bullet} + \end{matrix} \\ + \hline + \begin{matrix} + \bullet & \phantom{\bullet} & \phantom{\bullet} \\ + \phantom{\bullet} & \bullet & \phantom{\bullet} \\ + \phantom{\bullet} & \phantom{\bullet} & \bullet + \end{matrix} + & \rvline & + \begin{matrix} + \bullet & \bullet & \phantom{\bullet} \\ + \bullet & \bullet & \bullet \\ + \phantom{\bullet} & \bullet & \bullet + \end{matrix} + & \rvline & + \begin{matrix} + \bullet & \phantom{\bullet} & \phantom{\bullet} \\ + \phantom{\bullet} & \bullet & \phantom{\bullet} \\ + \phantom{\bullet} & \phantom{\bullet} & \bullet + \end{matrix} \\ + \hline + \begin{matrix} + \phantom{\bullet} & \phantom{\bullet} & \phantom{\bullet} \\ + \phantom{\bullet} & \phantom{\bullet} & \phantom{\bullet} \\ + \phantom{\bullet} & \phantom{\bullet} & \phantom{\bullet} + \end{matrix} + & \rvline & + \begin{matrix} + \bullet & \phantom{\bullet} & \phantom{\bullet} \\ + \phantom{\bullet} & \bullet & \phantom{\bullet} \\ + \phantom{\bullet} & \phantom{\bullet} & \bullet + \end{matrix} + & \rvline & + \begin{matrix} + \bullet & \bullet & \phantom{\bullet} \\ + \bullet & \bullet & \bullet \\ + \phantom{\bullet} & \bullet & \bullet + \end{matrix} + \end{bmatrix} +\end{equation*} + +Notations: +In addition, we use an equal step size in x- and y-direction, $h$ such that +\begin{align*} + x \in [0, 1] && x \rightarrow x_{\ivec} = \ivec h && \ivec = 0, 1, \dots, M-1 \\ + y \in [0, 1] && y \rightarrow y_{\jvec} = \jvec h && \jvec = 0, 1, \dots, M-1 \\ + t \in [0, T] && t \rightarrow t_{n} = n \Delta t && n = 0, 1, \dots, N_{t}-1 +\end{align*} +And simplify indices such that +\begin{align*} + u(x, y, t) \rightarrow u(\ivec h, \jvec h, n \Delta t) \equiv u_{\ivec, \jvec}^{n} \\ + v(x, y) \rightarrow u(\ivec h, \jvec h) \equiv v_{\ivec, \jvec} +\end{align*} +which gives a matrix $U^{n}$ that contains elements $u_{\ivec, \jvec}^{n}$, and +a matrix $V$ that contains elements $v_{\ivec, \jvec}$. +\begin{table}[H] + \centering + \begin{tabular}{l l l} % @{\extracolsep{\fill}} + \hline + Position & Value \\ + \hline + $u(x=0, y, t)$ & $0$ \\ + $u(x=1, y, t)$ & $0$ \\ + $u(x, y=0, t)$ & $0$ \\ + $u(x, y=1, t)$ & $0$ \\ + \hline + \end{tabular} + \caption{Boundary conditions in the xy-plane, also known as Dirichlet boundary conditions.} + \label{tab:boundary_conditions} +\end{table} + +For the general setup of the wall, we used +\begin{table}[H] + \centering + \begin{tabular}{l l} % @{\extracolsep{\fill}} + \hline + Parameter & Value \\ + \hline + Wall thickness & $0.02$ \\ + Wall position & $0.5$ \\ + Separator length & $0.05$ \\ + Slit aperture & $0.05$ \\ + \hline + \end{tabular} + \caption{Wall setup.} + \label{tab:wall_setup} +\end{table} + +\begin{table}[H] + \centering + \begin{tabular}{l l l} % @{\extracolsep{\fill}} + \hline + Simulation & $1$ & $2$ \\ + \hline + $h$ & $0.005$ & $0.005$ \\ + $\Delta t$ & $2.5 \times 10^{-5}$ & $2.5 \times 10^{-5}$ \\ + $T$ & $0.008$ & $0.002$ \\ + $x_{c}$ & $0.25$ & $0.25$ \\ + $\sigma_{x}$ & $0.05$ & $0.05$ \\ + $p_{x}$ & $200$ & $200$ \\ + $y_{c}$ & $0.5$ & $0.5$ \\ + $\sigma_{y}$ & $0.05$ & $0.20$ \\ + $p_{y}$ & $0$ & $0$ \\ + $v_{0}$ & $0$ & $1 \times10^{10}$ \\ + \hline + \end{tabular} + \caption{Wall setup.} + \label{tab:sim_setup} +\end{table} \subsection{Tools}\label{ssec:tools} % - +The double-slit experiment is implemented in C++. We use the Python library +\verb|matplotlib| \cite{hunter:2007:matplotlib} to produce all the plots, and +\verb|seaborn| \cite{waskom:2021:seaborn} to set the theme in the figures. \end{document} diff --git a/latex/sections/results.tex b/latex/sections/results.tex index c0c702d..d482e1c 100644 --- a/latex/sections/results.tex +++ b/latex/sections/results.tex @@ -2,6 +2,17 @@ \begin{document} \section{Results}\label{sec:results} +\subsection{Deviation}\label{ssec:deviation} +% Problem 3: Discuss approaches to solve Au^{n+1} = b, dealing with sparse matrix... +% Problem 7: Consequenses of solver choice, in regards to accuracy of probability conserved +% Add plot of deviation for both single- and double-slit + +\subsection{Time evolution}\label{ssec:time_evolution} +% Problem 8: Colormap, include plot of both Re and Im for different time steps +% Account for color scale + +\subsection{Particle detection}\label{ssec:particle_detection} +% Problem 9: Plot detection probability for single-, double- and triple-slit \end{document} -- 2.20.1 From aebc76f389bde1b0040339774b0ee9dc23bd9c51 Mon Sep 17 00:00:00 2001 From: Cory Date: Tue, 19 Dec 2023 16:08:35 +0100 Subject: [PATCH 09/32] Data generated by the program --- data/color_map.txt | 8 +- data/probability_deviation.txt | 640 ++++++++++++++++----------------- data/screen/double_slit.txt | 4 +- data/screen/single_slit.txt | 4 +- data/screen/triple_slit.txt | 4 +- 5 files changed, 330 insertions(+), 330 deletions(-) diff --git a/data/color_map.txt b/data/color_map.txt index 3b0e419..79e2f46 100644 --- a/data/color_map.txt +++ b/data/color_map.txt @@ -1,4 +1,4 @@ -39204 -(5.54049e-13,0) (7.10524e-13,0) (9.08916e-13,0) (1.1598e-12,0) (1.47624e-12,0) (1.87432e-12,0) (2.37381e-12,0) (2.99891e-12,0) (3.77914e-12,0) (4.75049e-12,0) (5.95659e-12,0) (7.45025e-12,0) (9.29519e-12,0) (1.15681e-11,0) (1.43607e-11,0) (1.77831e-11,0) (2.1966e-11,0) (2.70652e-11,0) (3.32647e-11,0) (4.07823e-11,0) (4.98739e-11,0) (6.084e-11,0) (7.4032e-11,0) 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(1.41632e-06,1.01476e-05) (5.08159e-06,1.01607e-05) (8.86618e-06,8.9283e-06) (1.23638e-05,6.39363e-06) (1.51487e-05,2.62819e-06) (1.68148e-05,-2.16151e-06) (1.70174e-05,-7.63832e-06) (1.55166e-05,-1.33522e-05) (1.22152e-05,-1.87731e-05) (7.18657e-06,-2.33379e-05) (6.86707e-07,-2.65089e-05) (-6.85597e-06,-2.78359e-05) (-1.48733e-05,-2.70151e-05) (-2.27114e-05,-2.39341e-05) (-2.96997e-05,-1.86962e-05) (-3.52258e-05,-1.16173e-05) (-3.88067e-05,-3.19663e-06) (-4.01441e-05,5.93709e-06) (-3.91573e-05,1.50959e-05) (-3.59896e-05,2.36094e-05) (-3.09862e-05,3.08978e-05) (-2.46499e-05,3.65313e-05) (-1.75791e-05,4.02688e-05) (-1.04002e-05,4.20728e-05) (-3.70037e-06,4.20994e-05) (2.02892e-06,4.06681e-05) (6.43261e-06,3.8216e-05) (9.314e-06,3.52441e-05) (1.06373e-05,3.22631e-05) (1.05128e-05,2.97435e-05) (9.16952e-06,2.80769e-05) (6.91852e-06,2.75499e-05) (4.11377e-06,2.83311e-05) (1.11399e-06,3.04715e-05) (-1.7504e-06,3.39152e-05) (-4.20298e-06,3.85181e-05) (-6.03828e-06,4.40706e-05) (-7.13008e-06,5.03223e-05) (-7.43171e-06,5.70043e-05) (-6.96981e-06,6.38503e-05) (-5.83326e-06,7.06115e-05) (-4.15903e-06,7.70678e-05) (-2.11648e-06,8.30337e-05) (1.08246e-07,8.83603e-05) (2.32706e-06,9.2934e-05) (4.36298e-06,9.66722e-05) (6.06111e-06,9.95186e-05) (7.2974e-06,0.000101437) (7.98506e-06,0.000102409) (8.07878e-06,0.000102424) (7.57675e-06,0.000101486) (6.52069e-06,9.96063e-05) (4.99367e-06,9.68079e-05) (3.11593e-06,9.31296e-05) (1.03842e-06,8.86296e-05) (-1.06585e-06,8.33913e-05) (-3.01269e-06,7.75283e-05) (-4.61963e-06,7.11879e-05) (-5.72044e-06,6.45525e-05) (-6.18055e-06,5.78383e-05) (-5.91242e-06,5.12888e-05) (-4.88939e-06,4.51642e-05) (-3.15654e-06,3.97268e-05) (-8.36801e-07,3.52204e-05) (1.86939e-06,3.18487e-05) (4.69224e-06,2.97505e-05) (7.30924e-06,2.89785e-05) (9.37039e-06,2.94799e-05) (1.05306e-05,3.10872e-05) (1.04872e-05,3.3517e-05) (9.01826e-06,3.63828e-05) (6.01867e-06,3.92212e-05) (1.527e-06,4.15296e-05) (-4.25992e-06,4.28151e-05) (-1.09895e-05,4.26483e-05) (-1.81746e-05,4.07154e-05) (-2.52342e-05,3.68642e-05) (-3.15482e-05,3.11335e-05) (-3.65239e-05,2.3763e-05) (-3.9663e-05,1.51795e-05) (-4.06224e-05,5.95844e-06) (-3.92586e-05,-3.23405e-06) (-3.56499e-05,-1.17133e-05) (-3.00908e-05,-1.8852e-05) (-2.30601e-05,-2.41499e-05) (-1.51666e-05,-2.72873e-05) (-7.07962e-06,-2.81558e-05) (5.45827e-07,-2.68613e-05) (7.13729e-06,-2.37018e-05) (1.22594e-05,-1.91236e-05) (1.56478e-05,-1.36634e-05) (1.722e-05,-7.88677e-06) (1.70658e-05,-2.32983e-06) (1.54203e-05,2.54874e-06) (1.26268e-05,6.40173e-06) (9.09418e-06,9.01277e-06) (5.2542e-06,1.03023e-05) (1.5218e-06,1.0322e-05) (-1.74038e-06,9.23978e-06) (-4.25169e-06,7.31721e-06) (-5.83614e-06,4.87901e-06) (-6.43606e-06,2.27707e-06) (-6.11407e-06,-1.49763e-07) (-5.04118e-06,-2.11598e-06) (-3.47045e-06,-3.42433e-06) (-1.69806e-06,-3.98922e-06) (-1.72367e-08,-3.84338e-06) (1.32747e-06,-3.12508e-06) (2.1767e-06,-2.04734e-06) (2.4762e-06,-8.54632e-07) (2.27589e-06,2.24153e-07) (1.70872e-06,1.01655e-06) (9.54545e-07,1.4326e-06) (1.97901e-07,1.4713e-06) (-4.10498e-07,1.20738e-06) (-7.79964e-07,7.63317e-07) (-8.89001e-07,2.74609e-07) (-7.77306e-07,-1.42615e-07) (-5.24842e-07,-4.14114e-07) (-2.25056e-07,-5.1613e-07) (4.01307e-08,-4.70142e-07) (2.17521e-07,-3.27506e-07) (2.8901e-07,-1.49787e-07) (2.67575e-07,9.18678e-09) (1.86089e-07,1.15137e-07) (8.3548e-08,1.57132e-07) (-6.90883e-09,1.44152e-07) (-6.52778e-08,9.70992e-08) (-8.63586e-08,3.96465e-08) (-7.67916e-08,-9.1439e-09) (-4.95694e-08,-3.87582e-08) (-1.8347e-08,-4.75289e-08) (6.69472e-09,-4.02573e-08) (2.06503e-08,-2.46818e-08) (2.35343e-08,-8.22465e-09) (1.85697e-08,4.0792e-09) (1.00978e-08,1.02249e-08) (1.8869e-09,1.07445e-08) (-3.77598e-09,7.62918e-09) (-6.20898e-09,3.20638e-09) (-5.92778e-09,-6.70182e-10) (-4.0334e-09,-2.99502e-09) (-1.66363e-09,-3.60392e-09) (3.47825e-10,-2.92941e-09) (1.58243e-09,-1.67028e-09) (1.97761e-09,-4.89393e-10) (1.68824e-09,1.93391e-10) (9.50789e-10,2.87447e-10) diff --git a/data/probability_deviation.txt b/data/probability_deviation.txt index d545f12..f9609b8 100644 --- a/data/probability_deviation.txt +++ b/data/probability_deviation.txt @@ -1,320 +1,320 @@ -0,1,1 -1,1,1 -2,1,1 -3,1,1 -4,1,1 -5,1,1 -6,1,1 -7,1,1 -8,1,1 -9,1,1 -10,1,1 -11,1,1 -12,1,1 -13,1,1 -14,1,1 -15,1,1 -16,1,1 -17,1,1 -18,1,1 -19,1,1 -20,1,1 -21,1,1 -22,1,1 -23,1,1 -24,1,1 -25,1,1 -26,1,1 -27,1,1 -28,1,1 -29,1,1 -30,1,1 -31,1,1 -32,1,1 -33,1,1 -34,1,1 -35,1,1 -36,1,1 -37,1,1 -38,1,1 -39,1,1 -40,1,1 -41,1,1 -42,1,1 -43,1,1 -44,1,1 -45,1,1 -46,1,1 -47,1,1 -48,1,1 -49,1,1 -50,1,1 -51,1,1 -52,1,1 -53,1,1 -54,1,1 -55,1,1 -56,1,1 -57,1,1 -58,1,1 -59,1,1 -60,1,1 -61,1,1 -62,1,1 -63,1,1 -64,1,1 -65,1,1 -66,1,1 -67,1,1 -68,1,1 -69,1,1 -70,1,1 -71,1,1 -72,1,1 -73,1,1 -74,1,1 -75,1,1 -76,1,1 -77,1,1 -78,1,1 -79,1,1 -80,1,1 -81,1,1 -82,1,1 -83,1,1 -84,1,1 -85,1,1 -86,1,1 -87,1,1 -88,1,1 -89,1,1 -90,1,1 -91,1,1 -92,1,1 -93,1,1 -94,1,1 -95,1,1 -96,1,1 -97,1,1 -98,1,1 -99,1,1 -100,1,1 -101,1,1 -102,1,1 -103,1,1 -104,1,1 -105,1,1 -106,1,1 -107,1,1 -108,1,1 -109,1,1 -110,1,1 -111,1,1 -112,1,1 -113,1,1 -114,1,1 -115,1,1 -116,1,1 -117,1,1 -118,1,1 -119,1,1 -120,1,1 -121,1,1 -122,1,1 -123,1,1 -124,1,1 -125,1,1 -126,1,1 -127,1,1 -128,1,1 -129,1,1 -130,1,1 -131,1,1 -132,1,1 -133,1,1 -134,1,1 -135,1,1 -136,1,1 -137,1,1 -138,1,1 -139,1,1 -140,1,1 -141,1,1 -142,1,1 -143,1,1 -144,1,1 -145,1,1 -146,1,1 -147,1,1 -148,1,1 -149,1,1 -150,1,1 -151,1,1 -152,1,1 -153,1,1 -154,1,1 -155,1,1 -156,1,1 -157,1,1 -158,1,1 -159,1,1 -160,1,1 -161,1,1 -162,1,1 -163,1,1 -164,1,1 -165,1,1 -166,1,1 -167,1,1 -168,1,1 -169,1,1 -170,1,1 -171,1,1 -172,1,1 -173,1,1 -174,1,1 -175,1,1 -176,1,1 -177,1,1 -178,1,1 -179,1,1 -180,1,1 -181,1,1 -182,1,1 -183,1,1 -184,1,1 -185,1,1 -186,1,1 -187,1,1 -188,1,1 -189,1,1 -190,1,1 -191,1,1 -192,1,1 -193,1,1 -194,1,1 -195,1,1 -196,1,1 -197,1,1 -198,1,1 -199,1,1 -200,1,1 -201,1,1 -202,1,1 -203,1,1 -204,1,1 -205,1,1 -206,1,1 -207,1,1 -208,1,1 -209,1,1 -210,1,1 -211,1,1 -212,1,1 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(1.28861e-06,-2.85067e-06) (2.05062e-06,-1.84025e-06) (2.30327e-06,-7.33468e-07) (2.09609e-06,2.58957e-07) (1.55486e-06,9.79484e-07) (8.47822e-07,1.34773e-06) (1.46314e-07,1.36605e-06) (-4.11005e-07,1.10681e-06) (-7.42277e-07,6.85835e-07) (-8.30478e-07,2.29754e-07) (-7.15344e-07,-1.54159e-07) (-4.73302e-07,-3.98696e-07) (-1.92144e-07,-4.8419e-07) (5.23267e-08,-4.32997e-07) (2.1205e-07,-2.94636e-07) (2.72129e-07,-1.27155e-07) (2.46513e-07,1.9603e-08) (1.67093e-07,1.14752e-07) (7.06128e-08,1.4949e-07) (-1.24575e-08,1.33481e-07) (-6.44307e-08,8.70964e-08) (-8.15578e-08,3.27566e-08) (-7.08794e-08,-1.19055e-08) (-4.48577e-08,-3.75643e-08) (-1.60226e-08,-4.32955e-08) (6.73352e-09,-3.4216e-08) (1.95445e-08,-1.81131e-08) (2.29023e-08,-2.38001e-09) (1.97865e-08,7.95392e-09) (1.36018e-08,1.09237e-08) (6.71775e-09,7.34521e-09) -- 2.20.1 From f1a4cf4a24a768a30721465c8c6f939b3b68b4f9 Mon Sep 17 00:00:00 2001 From: Cory Date: Tue, 19 Dec 2023 16:08:58 +0100 Subject: [PATCH 10/32] Make some changes --- include/WaveSimulation.hpp | 20 +++++---- lib/WaveSimulation.cpp | 83 +++++++++++++++++++++++++++----------- src/main.cpp | 69 ++++++++++++++++--------------- 3 files changed, 107 insertions(+), 65 deletions(-) diff --git a/include/WaveSimulation.hpp b/include/WaveSimulation.hpp index 0ef9acf..ae9ba7e 100644 --- a/include/WaveSimulation.hpp +++ b/include/WaveSimulation.hpp @@ -26,20 +26,26 @@ protected: double dt; double T; + void build_A(); + void build_B(); + void initialize_U(double x_c, double y_c, double sigma_x, double sigma_y, + double p_x, double p_y); + void build_V(double thickness, double pos_x, double aperture_separation, + double aperture, uint32_t slits); public: int32_t N; arma::cx_mat V; arma::cx_mat U; arma::sp_cx_mat A; - WaveSimulation(double h, double dt, double T); - virtual void solve(std::ofstream& ofile); - void build_A(); - void build_B(); - void initialize_U(double x_c, double y_c, double sigma_x, double sigma_y, - double p_x, double p_y); + WaveSimulation(double h, double dt, double T, double x_c, double y_c, + double sigma_x, double sigma_y, double p_x, double p_y, + double thickness, double pos_x, double ap_sep, double ap, + uint32_t slits); + WaveSimulation(double h, double dt, double T, double x_c, double y_c, + double sigma_x, double sigma_y, double p_x, double p_y); + virtual void solve(std::ofstream &ofile); void write_U(std::ofstream &ofile); void step(); - void build_V(double thickness, double pos_x, double aperture_sparation, double aperture, uint32_t slits); }; #endif diff --git a/lib/WaveSimulation.cpp b/lib/WaveSimulation.cpp index 056cb05..7be2fd5 100644 --- a/lib/WaveSimulation.cpp +++ b/lib/WaveSimulation.cpp @@ -18,17 +18,36 @@ #include #include -WaveSimulation::WaveSimulation(double h, double dt, double T) +WaveSimulation::WaveSimulation(double h, double dt, double T, double x_c, + double y_c, double sigma_x, double sigma_y, + double p_x, double p_y, double thickness, + double pos_x, double ap_sep, double ap, + uint32_t slits) { this->dt = dt; this->h = h; this->T = T; this->M = 1. / h; this->N = M - 2; - this->V.set_size(this->N, this->N); - this->V.fill(0.); - this->U.set_size(this->N, this->N); - this->U.fill(0.); + this->build_V(thickness, pos_x, ap_sep, ap, slits); + this->initialize_U(x_c, y_c, sigma_x, sigma_y, p_x, p_y); + this->build_A(); + this->build_B(); +} + +WaveSimulation::WaveSimulation(double h, double dt, double T, double x_c, double y_c, + double sigma_x, double sigma_y, double p_x, double p_y) +{ + this->dt = dt; + this->h = h; + this->T = T; + this->M = 1. / h; + this->N = M - 2; + this->build_V(0.,0.,0.,0., 0); + this->initialize_U(x_c, y_c, sigma_x, sigma_y, p_x, p_y); + this->build_A(); + this->build_B(); + } void WaveSimulation::solve(std::ofstream &ofile) @@ -43,7 +62,6 @@ void WaveSimulation::solve(std::ofstream &ofile) void WaveSimulation::step() { - DEBUG("Inside step"); arma::cx_vec tmp = this->B * this->U.as_col(); arma::spsolve(this->U, this->A, tmp); } @@ -111,6 +129,7 @@ void WaveSimulation::build_B() void WaveSimulation::initialize_U(double x_c, double y_c, double sigma_x, double sigma_y, double p_x, double p_y) { + this->U.set_size(this->N, this->N); double x, y, diff_x, diff_y; std::complex sum = 0.; for (size_t j = 0; j < this->U.n_cols; j++) { @@ -145,25 +164,42 @@ void WaveSimulation::build_V(double thickness, double pos_x, double aperture_separation, double aperture, uint32_t slits) { - uint32_t mid_y = this->N / 2 - (this->N % 2 == 0); + this->V.set_size(this->N, this->N); + this->V.fill(0.); + if (slits == 0) { + return; + } arma::cx_vec res; if (slits % 2 == 0) { - res = arma::cx_vec(aperture_separation/this->h,arma::fill::value(1e10)); - for (size_t i=0; i < slits; i+=2) { - res = arma::join_cols(res, arma::cx_vec(aperture/this->h,arma::fill::zeros)); - res = arma::join_cols(arma::cx_vec(aperture/this->h,arma::fill::zeros), res); - res = arma::join_cols(res, arma::cx_vec(aperture_separation/this->h,arma::fill::value(1e10))); - res = arma::join_cols(arma::cx_vec(aperture_separation/this->h,arma::fill::value(1e10)), res); + res = arma::cx_vec(aperture_separation / this->h, + arma::fill::value(1e10)); + for (size_t i = 0; i < slits; i += 2) { + res = arma::join_cols( + res, arma::cx_vec(aperture / this->h, arma::fill::zeros)); + res = arma::join_cols( + arma::cx_vec(aperture / this->h, arma::fill::zeros), res); + res = + arma::join_cols(res, arma::cx_vec(aperture_separation / this->h, + arma::fill::value(1e10))); + res = arma::join_cols(arma::cx_vec(aperture_separation / this->h, + arma::fill::value(1e10)), + res); } } else { - res = arma::cx_vec(aperture/this->h,arma::fill::value(0)); - for (size_t i=0; i < slits-1; i+=2) { - res = arma::join_cols(res, arma::cx_vec(aperture_separation/this->h,arma::fill::value(1e10))); - res = arma::join_cols(arma::cx_vec(aperture_separation/this->h,arma::fill::value(1e10)), res); - res = arma::join_cols(res, arma::cx_vec(aperture/this->h,arma::fill::zeros)); - res = arma::join_cols(arma::cx_vec(aperture/this->h,arma::fill::zeros), res); + res = arma::cx_vec(aperture / this->h, arma::fill::value(0)); + for (size_t i = 0; i < slits - 1; i += 2) { + res = + arma::join_cols(res, arma::cx_vec(aperture_separation / this->h, + arma::fill::value(1e10))); + res = arma::join_cols(arma::cx_vec(aperture_separation / this->h, + arma::fill::value(1e10)), + res); + res = arma::join_cols( + res, arma::cx_vec(aperture / this->h, arma::fill::zeros)); + res = arma::join_cols( + arma::cx_vec(aperture / this->h, arma::fill::zeros), res); } } if (res.size() > this->N) { @@ -171,10 +207,11 @@ void WaveSimulation::build_V(double thickness, double pos_x, } uint32_t fill = (this->N - res.size()) / 2; res = arma::join_cols(arma::cx_vec(fill, arma::fill::value(1e10)), res); - res = arma::join_cols(res, arma::cx_vec(fill + ((this->N - res.size()) % 2), arma::fill::value(1e10))); + res = arma::join_cols(res, arma::cx_vec(fill + ((this->N - res.size()) % 2), + arma::fill::value(1e10))); - uint32_t start = pos_x/this->h - thickness/this->h/2; - for (size_t i=0; i < thickness/this->h; i++) { - this->V.col(start+i) = res; + uint32_t start = pos_x / this->h - thickness / this->h / 2; + for (size_t i = 0; i < thickness / this->h; i++) { + this->V.col(start + i) = res; } } diff --git a/src/main.cpp b/src/main.cpp index c6891b4..8c2cb6a 100644 --- a/src/main.cpp +++ b/src/main.cpp @@ -11,19 +11,14 @@ * */ #include "WaveSimulation.hpp" #include "utils.hpp" +#include void probability_deviation() { - WaveSimulation sim_narrow(.005, 2.5e-5, .008); - sim_narrow.initialize_U(.25, .5, .05, .05, 200., 0.); - sim_narrow.build_A(); - sim_narrow.build_B(); + WaveSimulation sim_narrow(.005, 2.5e-5, .008, .25, .5, .05, .05, 200., 0.); - WaveSimulation sim_wide(.005, 2.5e-5, .008); - sim_wide.build_V(.02, .5, .05, .05, 2); - sim_wide.initialize_U(.25, .5, .05, .10, 200., 0.); - sim_wide.build_A(); - sim_wide.build_B(); + WaveSimulation sim_wide(.005, 2.5e-5, .008, .25, .5, .05, .10, 200., 0., + 0.02, .5, .05, .05, 2); std::ofstream ofile; utils::mkpath("data"); @@ -41,7 +36,8 @@ void probability_deviation() sim_narrow.step(); sim_wide.step(); - ofile << i << ',' << sum_narrow << ',' << sum_wide << '\n'; + ofile << i << ',' << utils::scientific_format(sum_narrow) << ',' + << utils::scientific_format(sum_wide) << '\n'; } ofile.close(); @@ -49,11 +45,8 @@ void probability_deviation() void color_map() { - WaveSimulation sim(.005, 2.5e-5, .008); - sim.build_V(.02, .5, .05, .05, 2); - sim.initialize_U(.25, .5, .05, .10, 200., 0.); - sim.build_A(); - sim.build_B(); + WaveSimulation sim(.005, 2.5e-5, .002, .25, .5, .05, .20, 200., 0., 0.02, + .5, .05, .05, 2); std::ofstream ofile; ofile.open("data/color_map.txt"); @@ -72,50 +65,56 @@ void color_map() void detector_screen() { - WaveSimulation sim(.005, 2.5e-5, .008); - sim.build_V(.02, .5, .05, .05, 1); - sim.initialize_U(.25, .5, .05, .10, 200., 0.); - sim.build_A(); - sim.build_B(); + WaveSimulation *sim = new WaveSimulation( + .005, 2.5e-5, .002, .25, .5, .05, .20, 200., 0., 0.02, .5, .05, .05, 1); std::ofstream ofile; utils::mkpath("data/screen"); ofile.open("data/screen/single_slit.txt"); - ofile << sim.N << '\n'; + ofile << sim->N << '\n'; for (size_t i = 0; i < 80; i++) { - sim.step(); + sim->step(); } - sim.write_U(ofile); + sim->write_U(ofile); ofile.close(); - sim.build_V(.02, .5, .05, .05, 2); - sim.initialize_U(.25, .5, .05, .10, 200., 0.); + delete sim; + sim = new WaveSimulation(.005, 2.5e-5, .002, .25, .5, .05, .20, 200., 0., + 0.02, .5, .05, .05, 2); ofile.open("data/screen/double_slit.txt"); - ofile << sim.N << '\n'; + ofile << sim->N << '\n'; for (size_t i = 0; i < 80; i++) { - sim.step(); + sim->step(); } - sim.write_U(ofile); + sim->write_U(ofile); ofile.close(); - sim.build_V(.02, .5, .05, .05, 3); - sim.initialize_U(.25, .5, .05, .10, 200., 0.); + delete sim; + sim = new WaveSimulation(.005, 2.5e-5, .002, .25, .5, .05, .20, 200., 0., + 0.02, .5, .05, .05, 3); ofile.open("data/screen/triple_slit.txt"); - ofile << sim.N << '\n'; + ofile << sim->N << '\n'; for (size_t i = 0; i < 80; i++) { - sim.step(); + sim->step(); } - sim.write_U(ofile); + sim->write_U(ofile); ofile.close(); } int main() { - probability_deviation(); - color_map(); + //probability_deviation(); + //color_map(); detector_screen(); + + //std::ofstream ofile; + //ofile.open("test.txt"); + //WaveSimulation sim(.005, 2.5e-5, .008, .25, .5, .05, .10, 200., 0., + //0.02, .5, .05, .05, 2); + //sim.solve(ofile); + return 0; } -- 2.20.1 From 2b68e98dd115adf753aad51a83340208e1c1f443 Mon Sep 17 00:00:00 2001 From: Cory Date: Tue, 19 Dec 2023 16:09:28 +0100 Subject: [PATCH 11/32] Create scripts for plotting --- python_scripts/colormap.py | 26 +++++++++++++++++++++ python_scripts/detector.py | 41 +++++++++++++++++++++++++++++++++ python_scripts/normalization.py | 25 ++++++++++++++++++++ 3 files changed, 92 insertions(+) create mode 100644 python_scripts/colormap.py create mode 100644 python_scripts/detector.py create mode 100644 python_scripts/normalization.py diff --git a/python_scripts/colormap.py b/python_scripts/colormap.py new file mode 100644 index 0000000..37ef2da --- /dev/null +++ b/python_scripts/colormap.py @@ -0,0 +1,26 @@ +import numpy as np +import matplotlib.pyplot as plt +import ast +import seaborn as sns + +sns.set_theme() + +def plot(): + with open("data/color_map.txt") as f: + lines = f.readlines(); + size = int(lines[0]) + for i, line in enumerate(lines[1:]): + fig, ax = plt.subplots() + arr = line.strip().split("\t") + arr = np.asarray(list(map(lambda x: ((a := complex(*ast.literal_eval(x)))*a.conjugate()).real, arr))) + + arr = arr.reshape(size,size) + + ax.imshow(arr.T, interpolation="nearest") + plt.savefig(f"latex/images/color_map_{i}.pdf") + plt.close(fig) + + +if __name__ == "__main__": + plot() + diff --git a/python_scripts/detector.py b/python_scripts/detector.py new file mode 100644 index 0000000..6795735 --- /dev/null +++ b/python_scripts/detector.py @@ -0,0 +1,41 @@ +import numpy as np +import matplotlib.pyplot as plt +import ast +import seaborn as sns + +sns.set_theme() + +def plot(): + files = [ + "data/screen/single_slit.txt", + "data/screen/double_slit.txt", + "data/screen/triple_slit.txt", + ] + outputs = [ + "latex/images/single_slit_detector.pdf", + "latex/images/double_slit_detector.pdf", + "latex/images/triple_slit_detector.pdf", + ] + for file, output in zip(files, outputs): + with open(file) as f: + lines = f.readlines(); + size = int(lines[0]) + column = int(.8 * size) + fig, ax = plt.subplots() + arr = lines[1].strip().split("\t") + arr = np.asarray(list(map(lambda x: complex(*ast.literal_eval(x)), arr))) + + arr = arr.reshape(size,size) + x = np.linspace(0,1, size) + slice = arr[column, :] + norm = 1. / np.sqrt(sum([(i*i.conjugate()).real for i in slice])) + slice *= norm + slice = np.asarray([i * i.conjugate() for i in slice]) + + ax.plot(x, slice) + plt.savefig(output) + plt.close(fig) + + +if __name__ == "__main__": + plot() diff --git a/python_scripts/normalization.py b/python_scripts/normalization.py new file mode 100644 index 0000000..8a63f28 --- /dev/null +++ b/python_scripts/normalization.py @@ -0,0 +1,25 @@ +import numpy as np +import matplotlib.pyplot as plt +import matplotlib +from matplotlib.animation import FuncAnimation +import ast + +def plot(): + with open("data/probability_deviation.txt") as f: + lines = f.readlines(); + x = [] + arr_narrow = [] + arr_wide = [] + for line in lines: + tmp = line.strip().split(",") + x.append(float(tmp[0])) + arr_narrow.append(float(tmp[1])) + arr_wide.append(float(tmp[2])) + + plt.plot(x,arr_narrow) + plt.plot(x,arr_wide) + plt.savefig("latex/images/probability_deviation.pdf") + +if __name__ == "__main__": + plot() + -- 2.20.1 From 137c5a89dd7592fe955a0d65c61b123cce4ec79b Mon Sep 17 00:00:00 2001 From: Cory Date: Tue, 19 Dec 2023 16:09:43 +0100 Subject: [PATCH 12/32] Add plots --- latex/images/color_map_0.pdf | Bin 0 -> 13236 bytes latex/images/color_map_1.pdf | Bin 0 -> 12471 bytes latex/images/color_map_2.pdf | Bin 0 -> 17820 bytes latex/images/double_slit_detector.pdf | Bin 0 -> 7818 bytes latex/images/probability_deviation.pdf | Bin 0 -> 6926 bytes latex/images/single_slit_detector.pdf | Bin 0 -> 7048 bytes latex/images/triple_slit_detector.pdf | Bin 0 -> 7971 bytes 7 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 latex/images/color_map_0.pdf create mode 100644 latex/images/color_map_1.pdf create mode 100644 latex/images/color_map_2.pdf create mode 100644 latex/images/double_slit_detector.pdf create mode 100644 latex/images/probability_deviation.pdf create mode 100644 latex/images/single_slit_detector.pdf create mode 100644 latex/images/triple_slit_detector.pdf diff --git a/latex/images/color_map_0.pdf b/latex/images/color_map_0.pdf new file mode 100644 index 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Willumsen Date: Fri, 22 Dec 2023 10:00:18 +0100 Subject: [PATCH 13/32] Add parameters for python script, edit method, and add result for problem 7. --- latex/schrodinger_simulation.tex | 2 +- latex/sections/appendices.tex | 55 ++++++++++++++++++++++++++++++-- latex/sections/methods.tex | 44 ++++++++++++++++++------- latex/sections/results.tex | 17 ++++++++++ python_scripts/colormap.py | 11 +++++++ python_scripts/detector.py | 11 +++++++ python_scripts/heat_map.py | 13 ++++++++ python_scripts/normalization.py | 14 ++++++++ python_scripts/plot_v.py | 13 ++++++++ 9 files changed, 165 insertions(+), 15 deletions(-) diff --git a/latex/schrodinger_simulation.tex b/latex/schrodinger_simulation.tex index 3341bb6..92c22b5 100644 --- a/latex/schrodinger_simulation.tex +++ b/latex/schrodinger_simulation.tex @@ -99,7 +99,7 @@ \subfile{sections/methods} % Results -% \subfile{sections/results} +\subfile{sections/results} % Conclusion % \subfile{sections/conclusion} diff --git a/latex/sections/appendices.tex b/latex/sections/appendices.tex index b14a0e9..ef9a514 100644 --- a/latex/sections/appendices.tex +++ b/latex/sections/appendices.tex @@ -34,15 +34,64 @@ Using Equation \eqref{eq:schrodinger_dimensionless}, we get &= -\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}} \\ & \quad - \frac{u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1}}{2 \Delta y^{2}} + \frac{1}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\ & \quad - \frac{u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n}}{2 \Delta x^{2}} \\ - & \quad - \frac{u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n}}{2 \Delta y^{2}} + \frac{1}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \bigg] \\ + & \quad - \frac{u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n}}{2 \Delta y^{2}} + \frac{1}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \bigg] \end{align*} -We rewrite the expression, +We rewrite the expression and gather all terms containing the $n+1$ time step on +the left hand side, and the terms containing $n$ time step on the right hand side. \begin{align*} & u_{\ivec, \jvec}^{n+1} - \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1} \big] \\ & - \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1} \big] + \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\ &= u_{\ivec, \jvec}^{n} + \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n} \big] \\ - & \quad + \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n} \big] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \\ + & \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*} +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*} +and define +\begin{align*} + r \equiv \frac{i \Delta t}{2 \Delta h^{2}} +\end{align*} +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} \\ + &= u_{\ivec, \jvec}^{n} + r \big[ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n} \big] \\ + & \quad + r \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*} +\section{Matrix structure}\label{ap:matrix_structure} +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 = + \begin{bmatrix} + a_{0} & -r & 0 & -r & 0 & 0 & 0 & 0 & 0 \\ + -r & a_{1} & -r & 0 & -r & 0 & 0 & 0 & 0 \\ + 0 & -r & a_{2} & 0 & 0 & -r & 0 & 0 & 0 \\ + -r & 0 & 0 & a_{3} & -r & 0 & -r & 0 & 0 \\ + 0 & -r & 0 & -r & a_{4} & -r & 0 & -r & 0 \\ + 0 & 0 & -r & 0 & -r & a_{5} & 0 & 0 & -r \\ + 0 & 0 & 0 & -r & 0 & 0 & a_{6} & -r & 0 \\ + 0 & 0 & 0 & 0 & -r & 0 & -r & a_{7} & -r \\ + 0 & 0 & 0 & 0 & 0 & -r & 0 & -r & a_{8} \\ + \end{bmatrix} +\end{align*} + +\begin{align*} + B = + \begin{bmatrix} + b_{0} & r & 0 & r & 0 & 0 & 0 & 0 & 0 \\ + r & b_{1} & r & 0 & r & 0 & 0 & 0 & 0 \\ + 0 & r & b_{2} & 0 & 0 & r & 0 & 0 & 0 \\ + r & 0 & 0 & b_{3} & r & 0 & r & 0 & 0 \\ + 0 & r & 0 & r & b_{4} & r & 0 & r & 0 \\ + 0 & 0 & r & 0 & r & b_{5} & 0 & 0 & r \\ + 0 & 0 & 0 & r & 0 & 0 & b_{6} & r & 0 \\ + 0 & 0 & 0 & 0 & r & 0 & r & b_{7} & r \\ + 0 & 0 & 0 & 0 & 0 & r & 0 & r & b_{8} \\ + \end{bmatrix} +\end{align*} \end{document} diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 68455e0..ccac132 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -68,11 +68,15 @@ We get the Crank-Nicolson method (CN) when $\theta = 1/2$ gives Crank-Nicolson \end{align} % Using CN, we derive the discretized Schrödinger equation given by \begin{align*} - & u_{\ivec, \jvec}^{n+1} - \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec-1, \jvec}^{n+1} \big] \\ - & - \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n+1} - 2u_{\ivec, \jvec}^{n+1} + u_{\ivec, \jvec-1}^{n+1} \big] + \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n+1} \\ - &= u_{\ivec, \jvec}^{n} + \frac{i \Delta t}{2 \Delta x^{2}} \big[ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n} \big] \\ - & \quad + \frac{i \Delta t}{2 \Delta y^{2}} \big[ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n} \big] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \numberthis \ . - \label{eq:schrodinger_discretized} + & 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} \\ + &= u_{\ivec, \jvec}^{n} + r \big[ u_{\ivec+1, \jvec}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec-1, \jvec}^{n} \big] \\ + & \quad + r \big[ u_{\ivec, \jvec+1}^{n} - 2u_{\ivec, \jvec}^{n} + u_{\ivec, \jvec-1}^{n} \big] - \frac{i \Delta t}{2} v_{\ivec, \jvec} u_{\ivec, \jvec}^{n} \numberthis \ , + \label{eq:schrodinger_discretized} +\end{align*} % +where $r$ is defined as +\begin{align*} + r \equiv \frac{i \Delta t}{2 \Delta h^{2}} \end{align*} % The full derivation of both Equation \eqref{eq:crank_nicolson_method} and Equation \eqref{eq:schrodinger_discretized} can be found in Appendix \ref{ap:crank_nicolson}. @@ -97,8 +101,20 @@ A, B are sparse csc matrix % & \bullet & \bullet & & & \bullet & & & % \end{pNiceArray} % \end{equation*} - -\begin{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} + A u^{n+1} = B u^{n} \ . +\end{align} +Here, both $u^{n+1}$ and $u^{n}$ are column vectors containing the internal points +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 can be decomposed as submatrices of size $(M-2) \times (M-2)$, with +the following pattern +\begin{align*} + A, B = \begin{bmatrix} \begin{matrix} \bullet & \bullet & \phantom{\bullet} \\ @@ -154,7 +170,13 @@ A, B are sparse csc matrix \phantom{\bullet} & \bullet & \bullet \end{matrix} \end{bmatrix} -\end{equation*} +\end{align*} +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 filled matrices can be found in Appendix \ref{ap:matrix_structure}. Notations: In addition, we use an equal step size in x- and y-direction, $h$ such that @@ -172,7 +194,7 @@ which gives a matrix $U^{n}$ that contains elements $u_{\ivec, \jvec}^{n}$, and a matrix $V$ that contains elements $v_{\ivec, \jvec}$. \begin{table}[H] \centering - \begin{tabular}{l l l} % @{\extracolsep{\fill}} + \begin{tabular}{l r} % @{\extracolsep{\fill}} \hline Position & Value \\ \hline @@ -189,7 +211,7 @@ a matrix $V$ that contains elements $v_{\ivec, \jvec}$. For the general setup of the wall, we used \begin{table}[H] \centering - \begin{tabular}{l l} % @{\extracolsep{\fill}} + \begin{tabular}{l r} % @{\extracolsep{\fill}} \hline Parameter & Value \\ \hline @@ -205,7 +227,7 @@ For the general setup of the wall, we used \begin{table}[H] \centering - \begin{tabular}{l l l} % @{\extracolsep{\fill}} + \begin{tabular}{l r r} % @{\extracolsep{\fill}} \hline Simulation & $1$ & $2$ \\ \hline diff --git a/latex/sections/results.tex b/latex/sections/results.tex index d482e1c..deeb5ea 100644 --- a/latex/sections/results.tex +++ b/latex/sections/results.tex @@ -4,8 +4,25 @@ \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 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 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 for $t \in [0, T]$ where $T=0.008$.} + \label{fig:deviation} +\end{figure} \subsection{Time evolution}\label{ssec:time_evolution} % Problem 8: Colormap, include plot of both Re and Im for different time steps diff --git a/python_scripts/colormap.py b/python_scripts/colormap.py index 37ef2da..1d4dd55 100644 --- a/python_scripts/colormap.py +++ b/python_scripts/colormap.py @@ -4,6 +4,17 @@ import ast import seaborn as sns sns.set_theme() +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("data/color_map.txt") as f: diff --git a/python_scripts/detector.py b/python_scripts/detector.py index 6795735..7ccea84 100644 --- a/python_scripts/detector.py +++ b/python_scripts/detector.py @@ -4,6 +4,17 @@ import ast import seaborn as sns sns.set_theme() +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(): files = [ diff --git a/python_scripts/heat_map.py b/python_scripts/heat_map.py index 8700bff..2be0ae6 100644 --- a/python_scripts/heat_map.py +++ b/python_scripts/heat_map.py @@ -4,6 +4,19 @@ 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) + wave_arr = [] fig = plt.figure() ax = plt.gca() diff --git a/python_scripts/normalization.py b/python_scripts/normalization.py index 8a63f28..04afe72 100644 --- a/python_scripts/normalization.py +++ b/python_scripts/normalization.py @@ -4,6 +4,19 @@ 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("data/probability_deviation.txt") as f: lines = f.readlines(); @@ -16,6 +29,7 @@ def plot(): arr_narrow.append(float(tmp[1])) arr_wide.append(float(tmp[2])) + plt.plot(x,arr_narrow) plt.plot(x,arr_wide) plt.savefig("latex/images/probability_deviation.pdf") diff --git a/python_scripts/plot_v.py b/python_scripts/plot_v.py index 3e48134..5d5d8b8 100644 --- a/python_scripts/plot_v.py +++ b/python_scripts/plot_v.py @@ -4,6 +4,19 @@ 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: -- 2.20.1 From 5cc6142b2c3153207115439d4182924afc07ce72 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Fri, 22 Dec 2023 13:19:20 +0100 Subject: [PATCH 14/32] Add draft abstract and conclusion. --- 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z-}v|-Al`rU4MpSsH+cpBB`@#4^p}s1mlyK4ez>`q*w|aR5TLXl4I3XomjRY}c}Ghp z1~6)}0ZNU!lM|rt{BbQW#h@o*Vkra$L(TXg0w8`%6QDF9Qy~Zh#0M4N=QRVHLd`5p z&H06ZP{L5)@td0gE+1w>{2+5du%G~+nW+g>l;Quq2!lT~DH9jBUqc4yl^}?ZfSFlZ IMTX#i0hpTt*#H0l literal 0 HcmV?d00001 diff --git a/latex/references.bib b/latex/references.bib index 78cee6f..e7380fa 100644 --- a/latex/references.bib +++ b/latex/references.bib @@ -8,6 +8,14 @@ year = {2020} } +@misc{mit:2004:physics, + author = {Sen-ben Liao and Peter Dourmashkin and John W. Belcher}, + title = {Course notes in Physics 8.02 at MIT}, + year = {2004}, + url = {https://web.mit.edu/8.02t/www/802TEAL3D/visualizations/coursenotes/modules/guide14.pdf}, + urldate = {2023-12-22} +} + @misc{britannica:1999:light, author = {The Editors of Encyclopaedia Britannica}, title = {Light}, diff --git a/latex/schrodinger_simulation.tex b/latex/schrodinger_simulation.tex index 92c22b5..1ace563 100644 --- a/latex/schrodinger_simulation.tex +++ b/latex/schrodinger_simulation.tex @@ -88,7 +88,7 @@ \noaffiliation % ignore this, but keep it. % Abstract -% \subfile{sections/abstract} +\subfile{sections/abstract} \maketitle diff --git a/latex/sections/abstract.tex b/latex/sections/abstract.tex index 8013794..dc3225b 100644 --- a/latex/sections/abstract.tex +++ b/latex/sections/abstract.tex @@ -2,6 +2,17 @@ \begin{document} \begin{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 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} diff --git a/latex/sections/conclusion.tex b/latex/sections/conclusion.tex index 34c3edd..1ed81da 100644 --- a/latex/sections/conclusion.tex +++ b/latex/sections/conclusion.tex @@ -2,6 +2,24 @@ \begin{document} \section{Conclusion}\label{sec:conclusion} +% 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 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. +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... \end{document} diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index ccac132..7b81da4 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -85,9 +85,35 @@ can be found in Appendix \ref{ap:crank_nicolson}. \subsection{The double-slit experiment}\label{ssec:double_slit} % Thomas Young first performed the double-slit experiment in 1801 to demonstrate the principle of interference of light \cite{britannica:2023:young}, while postulating -light as waves rather than particles. The double-slit experiment gives a diffraction -pattern, where constructive interference of light result in bright spots, and destructive -interference result in dark spots. +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 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 wall.} + \label{fig:youngs_double_slit} +\end{figure} + +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 \ , +\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, 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*} + \delta = m \lambda && m = 0, \pm 1, \pm 2 \dots \ , +\end{align*} +and destructive interference when +\begin{align*} + \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} % -- 2.20.1 From 8761ae1048037c329f8083b823211ba616abff00 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Fri, 22 Dec 2023 13:21:42 +0100 Subject: [PATCH 15/32] Add alternative script for colormap, removing grid and adding colorbar to plots. --- python_scripts/colormap_alt.py | 38 ++++++++++++++++++++++++++++++++++ 1 file changed, 38 insertions(+) create mode 100644 python_scripts/colormap_alt.py diff --git a/python_scripts/colormap_alt.py b/python_scripts/colormap_alt.py new file mode 100644 index 0000000..1496a37 --- /dev/null +++ b/python_scripts/colormap_alt.py @@ -0,0 +1,38 @@ +import numpy as np +import matplotlib.pyplot as plt +import ast +import seaborn as sns + +sns.set_theme() +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("data/color_map.txt") as f: + lines = f.readlines() + size = int(lines[0]) + for i, line in enumerate(lines[1:]): + fig, ax = plt.subplots() + arr = line.strip().split("\t") + arr = np.asarray(list(map(lambda x: ((a := complex(*ast.literal_eval(x)))*a.conjugate()).real, arr))) + + arr = arr.reshape(size,size) + + color_map = ax.imshow(arr.T, interpolation="nearest", cmap=sns.color_palette("mako", as_cmap=True)) + fig.colorbar(color_map, ax=ax) + plt.grid(False) + plt.savefig(f"latex/images/color_map_{i}.pdf") + plt.close(fig) + + +if __name__ == "__main__": + plot() \ No newline at end of file -- 2.20.1 From 369c0ae21db78d60c788ca41349491675422af49 Mon Sep 17 00:00:00 2001 From: Cory Date: Fri, 22 Dec 2023 16:01:11 +0100 Subject: [PATCH 16/32] New data --- data/color_map.txt | 4 +- data/probability_deviation.txt | 320 ----------------------- data/probability_deviation_no_slits.txt | 321 ++++++++++++++++++++++++ data/probability_deviation_slits.txt | 321 ++++++++++++++++++++++++ python_scripts/normalization.py | 30 +-- 5 files changed, 659 insertions(+), 337 deletions(-) delete mode 100644 data/probability_deviation.txt create mode 100644 data/probability_deviation_no_slits.txt create mode 100644 data/probability_deviation_slits.txt diff --git a/data/color_map.txt b/data/color_map.txt index 79e2f46..2779ec4 100644 --- a/data/color_map.txt +++ b/data/color_map.txt @@ -1,4 +1,4 @@ 198 (4.62005e-09,0) (4.91648e-09,0) (5.22866e-09,0) (5.55719e-09,0) (5.90267e-09,0) (6.26572e-09,0) (6.64693e-09,0) (7.04694e-09,0) (7.46634e-09,0) (7.90577e-09,0) (8.36583e-09,0) (8.84713e-09,0) (9.35027e-09,0) 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(-5.01266e-50,6.58755e-50) (-4.80559e-50,6.31542e-50) (-4.60419e-50,6.05075e-50) (-4.40848e-50,5.79354e-50) (-4.21844e-50,5.54381e-50) (-4.03408e-50,5.30152e-50) (-3.85537e-50,5.06666e-50) (-3.68227e-50,4.83917e-50) (-3.51474e-50,4.61902e-50) (-3.35274e-50,4.40612e-50) (-3.19621e-50,4.20041e-50) (-3.04509e-50,4.0018e-50) (-2.89929e-50,3.8102e-50) (-2.75875e-50,3.62551e-50) (-2.62339e-50,3.44761e-50) (-2.49311e-50,3.2764e-50) (-2.36781e-50,3.11174e-50) (-2.24741e-50,2.95351e-50) (-2.1318e-50,2.80157e-50) (-2.02087e-50,2.6558e-50) (-1.91452e-50,2.51603e-50) (-1.81263e-50,2.38213e-50) (-1.71509e-50,2.25394e-50) (-1.62179e-50,2.13133e-50) (-1.5326e-50,2.01412e-50) (-1.44741e-50,1.90217e-50) (-1.36611e-50,1.79532e-50) (-1.28856e-50,1.69341e-50) (-1.21466e-50,1.59629e-50) (-1.14428e-50,1.5038e-50) (-1.07731e-50,1.41578e-50) diff --git a/data/probability_deviation.txt b/data/probability_deviation.txt deleted file mode 100644 index f9609b8..0000000 --- a/data/probability_deviation.txt +++ /dev/null @@ -1,320 +0,0 @@ -0, 1.0000000000e+00, 1.0000000000e+00 -1, 1.0000000000e+00, 1.0000000000e+00 -2, 1.0000000000e+00, 1.0000000000e+00 -3, 1.0000000000e+00, 1.0000000000e+00 -4, 1.0000000000e+00, 1.0000000000e+00 -5, 1.0000000000e+00, 1.0000000000e+00 -6, 1.0000000000e+00, 1.0000000000e+00 -7, 1.0000000000e+00, 1.0000000000e+00 -8, 1.0000000000e+00, 1.0000000000e+00 -9, 1.0000000000e+00, 1.0000000000e+00 -10, 1.0000000000e+00, 1.0000000000e+00 -11, 1.0000000000e+00, 1.0000000000e+00 -12, 1.0000000000e+00, 1.0000000000e+00 -13, 1.0000000000e+00, 1.0000000000e+00 -14, 1.0000000000e+00, 1.0000000000e+00 -15, 1.0000000000e+00, 1.0000000000e+00 -16, 1.0000000000e+00, 1.0000000000e+00 -17, 1.0000000000e+00, 1.0000000000e+00 -18, 1.0000000000e+00, 1.0000000000e+00 -19, 1.0000000000e+00, 1.0000000000e+00 -20, 1.0000000000e+00, 1.0000000000e+00 -21, 1.0000000000e+00, 1.0000000000e+00 -22, 1.0000000000e+00, 1.0000000000e+00 -23, 1.0000000000e+00, 1.0000000000e+00 -24, 1.0000000000e+00, 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FuncAnimation -import ast def plot(): - with open("data/probability_deviation.txt") as f: - lines = f.readlines(); - x = [] - arr_narrow = [] - arr_wide = [] - for line in lines: - tmp = line.strip().split(",") - x.append(float(tmp[0])) - arr_narrow.append(float(tmp[1])) - arr_wide.append(float(tmp[2])) + files = [ + "data/probability_deviation_no_slits.txt", + "data/probability_deviation_slits.txt", + ] + for file in files: + with open(file) as f: + lines = f.readlines(); + x = [] + arr = [] + for line in lines: + tmp = line.strip().split("\t") + x.append(float(tmp[0])) + arr.append(float(tmp[1])) + + plt.plot(x,arr) - plt.plot(x,arr_narrow) - plt.plot(x,arr_wide) plt.savefig("latex/images/probability_deviation.pdf") if __name__ == "__main__": -- 2.20.1 From 8f3c27ac4051e6ee4fc1f7b189e1f03abde3fd0b Mon Sep 17 00:00:00 2001 From: Cory Date: Fri, 22 Dec 2023 16:01:31 +0100 Subject: [PATCH 17/32] Fix return values --- lib/utils.cpp | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/lib/utils.cpp b/lib/utils.cpp index 5dabba2..89752ae 100644 --- a/lib/utils.cpp +++ b/lib/utils.cpp @@ -45,14 +45,14 @@ bool mkpath(std::string path, int mode) cur_dir = path.substr(0, pos); if (mkdir(cur_dir.c_str(), mode) != 0 && stat(cur_dir.c_str(), &buf) != 0) { - return -1; + return false; } } else { break; } } - return 0; + return true; } std::string dirname(const std::string &path) -- 2.20.1 From 7b83a3eed37ebeed1fe0a6a8c91406b965c5b5bc Mon Sep 17 00:00:00 2001 From: Cory Date: Fri, 22 Dec 2023 16:01:57 +0100 Subject: [PATCH 18/32] Modify class --- include/WaveSimulation.hpp | 85 +++++++-- lib/WaveSimulation.cpp | 348 ++++++++++++++++++++++++------------- 2 files changed, 306 insertions(+), 127 deletions(-) diff --git a/include/WaveSimulation.hpp b/include/WaveSimulation.hpp index ae9ba7e..7e6a0c2 100644 --- a/include/WaveSimulation.hpp +++ b/include/WaveSimulation.hpp @@ -21,31 +21,96 @@ class WaveSimulation { protected: uint32_t M; + int32_t N; + arma::cx_mat V; + arma::cx_mat U; arma::sp_cx_mat B; + arma::sp_cx_mat A; double h; double dt; double T; - void build_A(); - void build_B(); + /* @brief Initialize the U matrix using an unormalized Gaussian wave + * packet. + * + * @param x_c The center of the packet in the x direction. + * @param y_c The center of the packet in the y direction. + * @param sigma_x The The initial width in the x direction. + * @param sigma_y The The initial width in the y direction. + * @param p_x The wave packet momentum in the x direction. + * @param p_y The wave packet momentum in the y direction. + * **/ void initialize_U(double x_c, double y_c, double sigma_x, double sigma_y, double p_x, double p_y); - void build_V(double thickness, double pos_x, double aperture_separation, - double aperture, uint32_t slits); + + /* @brief Initialize the V matrix. + * + * @param thickness The thickness of the wall in the x direction. + * @param pos_x The center of the wall in the x direction. + * @param ap_sep The separation between each aperture. + * @param ap The aperture width. + * @param slits The number of slits. + * **/ + void initialize_V(double thickness, double pos_x, + double aperture_separation, double aperture, + uint32_t slits); + + /* @brief Initialize the V matrix with no wall. + * **/ + void initialize_V(); + + /* @brief Initialize the A matrix according to the Crank-Nicolson method + * **/ + void initialize_A(); + + /* @brief Initialize the B matrix according to the Crank-Nicolson method + * **/ + void initialize_B(); + public: - int32_t N; - arma::cx_mat V; - arma::cx_mat U; - arma::sp_cx_mat A; + /* @brief Constructor for the WaveSimulation class. + * + * @param h The step size in the x and y direction. + * @param dt The step size in the temporal dimension. + * @param T The total time to simulate. + * @param x_c The center of the packet in the x direction. + * @param y_c The center of the packet in the y direction. + * @param sigma_x The The initial width in the x direction. + * @param sigma_y The The initial width in the y direction. + * @param p_x The wave packet momentum in the x direction. + * @param p_y The wave packet momentum in the y direction. + * @param thickness The thickness of the wall in the x direction. + * @param pos_x The center of the wall in the x direction. + * @param ap_sep The separation between each aperture. + * @param ap The aperture width. + * @param slits The number of slits. + * **/ WaveSimulation(double h, double dt, double T, double x_c, double y_c, double sigma_x, double sigma_y, double p_x, double p_y, double thickness, double pos_x, double ap_sep, double ap, uint32_t slits); + + /* @brief Constructor for the WaveSimulation class with no wall. + * + * @param h The step size in the x and y direction. + * @param dt The step size in the temporal dimension. + * @param T The total time to simulate. + * @param x_c The center of the packet in the x direction. + * @param y_c The center of the packet in the y direction. + * @param sigma_x The The initial width in the x direction. + * @param sigma_y The The initial width in the y direction. + * @param p_x The wave packet momentum in the x direction. + * @param p_y The wave packet momentum in the y direction. + * **/ WaveSimulation(double h, double dt, double T, double x_c, double y_c, double sigma_x, double sigma_y, double p_x, double p_y); - virtual void solve(std::ofstream &ofile); - void write_U(std::ofstream &ofile); + void step(); + void solve(std::string outfile, bool write_each_step = false); + void solve(std::string outfile, std::vector &steps); + void probability_deviation(std::string outfile, + bool write_each_step = false); + void write_U(std::ofstream &ofile); }; #endif diff --git a/lib/WaveSimulation.cpp b/lib/WaveSimulation.cpp index 7be2fd5..7d425bc 100644 --- a/lib/WaveSimulation.cpp +++ b/lib/WaveSimulation.cpp @@ -17,56 +17,113 @@ #include #include #include +#include -WaveSimulation::WaveSimulation(double h, double dt, double T, double x_c, - double y_c, double sigma_x, double sigma_y, - double p_x, double p_y, double thickness, - double pos_x, double ap_sep, double ap, - uint32_t slits) +// Initializers +void WaveSimulation::initialize_U(double x_c, double y_c, double sigma_x, + double sigma_y, double p_x, double p_y) { - this->dt = dt; - this->h = h; - this->T = T; - this->M = 1. / h; - this->N = M - 2; - this->build_V(thickness, pos_x, ap_sep, ap, slits); - this->initialize_U(x_c, y_c, sigma_x, sigma_y, p_x, p_y); - this->build_A(); - this->build_B(); + this->U.set_size(this->N, this->N); + double x, y, diff_x, diff_y; + std::complex sum = 0.; + for (size_t j = 0; j < this->U.n_cols; j++) { + x = j * h; + diff_x = x - x_c; + for (size_t i = 0; i < this->U.n_rows; i++) { + y = i * h; + diff_y = y - y_c; + this->U(i, j) = + std::exp(-(diff_x * diff_x) / (2. * sigma_x * sigma_x) + - (diff_y * diff_y) / (2. * sigma_y * sigma_y) + + p_x * x * 1._i + p_y * y * 1._i); + sum += this->U(i, j) * std::conj(this->U(i, j)); + } + } + + // The imaginary part of sum should be 0. + if (std::abs(sum.imag()) > 1e-7) { + abort(); + } + + double norm = 1. / std::sqrt(sum.real()); + + // Normalize each element + this->U.for_each([norm](std::complex &el) { el *= norm; }); } -WaveSimulation::WaveSimulation(double h, double dt, double T, double x_c, double y_c, - double sigma_x, double sigma_y, double p_x, double p_y) +void WaveSimulation::initialize_V() { - this->dt = dt; - this->h = h; - this->T = T; - this->M = 1. / h; - this->N = M - 2; - this->build_V(0.,0.,0.,0., 0); - this->initialize_U(x_c, y_c, sigma_x, sigma_y, p_x, p_y); - this->build_A(); - this->build_B(); - + this->V.set_size(this->N, this->N); + this->V.fill(0.); } -void WaveSimulation::solve(std::ofstream &ofile) +void WaveSimulation::initialize_V(double thickness, double pos_x, double ap_sep, + double ap, uint32_t slits) { - ofile << this->N << '\n'; - uint32_t iterations = this->T / this->dt; - for (size_t i = 0; i < iterations; i++) { - this->write_U(ofile); - this->step(); + this->V.set_size(this->N, this->N); + this->V.fill(0.); + if (slits == 0) { + return; + } + + arma::cx_vec res; + + // Differentiate between odd and even number of slits. + uint32_t ap_points, ap_sep_points; + + if (slits % 2 == 0) { + ap_points = ap / this->h; + ap_sep_points = ap_sep / this->h + + ((uint32_t)(ap_sep / this->h) % 2 != this->N % 2); + + res = arma::cx_vec(ap_sep_points, arma::fill::value(1e10)); + + for (size_t i = 0; i < slits; i += 2) { + res = arma::join_cols(res, + arma::cx_vec(ap_points, arma::fill::zeros)); + res = arma::join_cols(arma::cx_vec(ap_points, arma::fill::zeros), + res); + res = arma::join_cols( + res, arma::cx_vec(ap_sep_points, arma::fill::value(1e10))); + res = arma::join_cols( + arma::cx_vec(ap_sep_points, arma::fill::value(1e10)), res); + } + } + else { + ap_points = + ap / this->h + ((uint32_t)(ap / this->h) % 2 != this->N % 2); + ap_sep_points = ap_sep / this->h; + + res = arma::cx_vec(ap_points, arma::fill::value(0)); + + for (size_t i = 0; i < slits - 1; i += 2) { + res = arma::join_cols( + res, arma::cx_vec(ap_sep_points, arma::fill::value(1e10))); + res = arma::join_cols( + arma::cx_vec(ap_sep_points, arma::fill::value(1e10)), res); + res = arma::join_cols(res, + arma::cx_vec(ap_points, arma::fill::zeros)); + res = arma::join_cols(arma::cx_vec(ap_points, arma::fill::zeros), + res); + } + } + + if (res.size() > this->N) { + abort(); + } + + uint32_t fill = (this->N - res.size()) / 2; + res = arma::join_cols(arma::cx_vec(fill, arma::fill::value(1e10)), res); + res = arma::join_cols(res, arma::cx_vec(fill + ((this->N - res.size()) % 2), + arma::fill::value(1e10))); + + uint32_t start = pos_x / this->h - thickness / this->h / 2; + for (size_t i = 0; i < thickness / this->h; i++) { + this->V.col(start + i) = res; } } -void WaveSimulation::step() -{ - arma::cx_vec tmp = this->B * this->U.as_col(); - arma::spsolve(this->U, this->A, tmp); -} - -void WaveSimulation::build_A() +void WaveSimulation::initialize_A() { // Create the diagonal arma::cx_vec diagonal(this->N * this->N); @@ -96,7 +153,7 @@ void WaveSimulation::build_A() this->A.diag(this->N).fill(-r); } -void WaveSimulation::build_B() +void WaveSimulation::initialize_B() { std::complex r = (1._i * this->dt) / (2 * h * h); @@ -126,92 +183,149 @@ void WaveSimulation::build_B() this->B.diag(this->N).fill(r); } -void WaveSimulation::initialize_U(double x_c, double y_c, double sigma_x, - double sigma_y, double p_x, double p_y) +// Constructors +WaveSimulation::WaveSimulation(double h, double dt, double T, double x_c, + double y_c, double sigma_x, double sigma_y, + double p_x, double p_y, double thickness, + double pos_x, double ap_sep, double ap, + uint32_t slits) { - this->U.set_size(this->N, this->N); - double x, y, diff_x, diff_y; - std::complex sum = 0.; - for (size_t j = 0; j < this->U.n_cols; j++) { - x = j * h; - diff_x = x - x_c; - for (size_t i = 0; i < this->U.n_rows; i++) { - y = i * h; - diff_y = y - y_c; - this->U(i, j) = - std::exp(-(diff_x * diff_x) / (2. * sigma_x * sigma_x) - - (diff_y * diff_y) / (2. * sigma_y * sigma_y) - + p_x * x * 1._i + p_y * y * 1._i); - sum += this->U(i, j) * std::conj(this->U(i, j)); - } - } - if (std::abs(sum.imag()) > 1e-7) { + this->dt = dt; + this->h = h; + this->T = T; + this->M = 1. / h; + this->N = M - 2; + this->initialize_V(thickness, pos_x, ap_sep, ap, slits); + this->initialize_U(x_c, y_c, sigma_x, sigma_y, p_x, p_y); + this->initialize_A(); + this->initialize_B(); +} + +WaveSimulation::WaveSimulation(double h, double dt, double T, double x_c, + double y_c, double sigma_x, double sigma_y, + double p_x, double p_y) +{ + this->dt = dt; + this->h = h; + this->T = T; + this->M = 1. / h; + this->N = M - 2; + this->initialize_V(); + this->initialize_U(x_c, y_c, sigma_x, sigma_y, p_x, p_y); + this->initialize_A(); + this->initialize_B(); +} + +// Public methods +void WaveSimulation::step() +{ + // Create the right hand side of Ax = b + arma::cx_vec tmp = this->B * this->U.as_col(); + + // Solve Ax = b + arma::spsolve(this->U, this->A, tmp); +} + +void WaveSimulation::solve(std::string outfile, bool write_each_step) +{ + // Create path and proceed if successful. + if (!utils::mkpath(utils::dirname(outfile))) { abort(); } - double norm = 1. / std::sqrt(sum.real()); - this->U.for_each([norm](std::complex &el) { el *= norm; }); + // Open file + std::ofstream ofile; + ofile.open(outfile); + + // Write the size of the matrix on the first line + ofile << this->N << '\n'; + + uint32_t iterations = this->T / this->dt; + + if (write_each_step) { + for (size_t i = 0; i < iterations; i++) { + this->write_U(ofile); + this->step(); + } + } + else { + for (size_t i = 0; i < iterations; i++) { + this->step(); + } + } + this->write_U(ofile); + ofile.close(); +} +void WaveSimulation::solve(std::string outfile, std::vector &steps) +{ + // Create path and proceed if successful. + if (!utils::mkpath(utils::dirname(outfile))) { + abort(); + } + + // Open file + std::ofstream ofile; + ofile.open(outfile); + + // Write the size of the matrix on the first line + ofile << this->N << '\n'; + + uint32_t iterations; + + for (size_t i=0; i < steps.size(); i++) { + if (i == 0) { + iterations = steps[i] / this->h; + } + else { + iterations = (steps[i] - steps[i-1]) / this->h; + } + + for (size_t j=0; j < iterations; j++) { + this->step(); + } + this->write_U(ofile); + } + + ofile.close(); +} + +void WaveSimulation::probability_deviation(std::string outfile, + bool write_each_step) +{ + // Create path and proceed if successful. + if (!utils::mkpath(utils::dirname(outfile))) { + abort(); + } + + // Open file + std::ofstream ofile; + ofile.open(outfile); + + uint32_t iterations = this->T / this->dt; + + double sum; + + if (write_each_step) { + for (size_t i = 0; i < iterations; i++) { + sum = arma::accu(this->U % arma::conj(this->U)).real(); + ofile << i*this->dt << '\t' << 1. - sum << '\n'; + this->step(); + } + } + else { + for (size_t i = 0; i < iterations; i++) { + this->step(); + } + } + sum = arma::accu(this->U % arma::conj(this->U)).real(); + ofile << this->T << '\t' << 1. - sum << '\n'; + ofile.close(); } void WaveSimulation::write_U(std::ofstream &ofile) { + // Write each element to file in column-major order. this->U.for_each( [&ofile](std::complex el) { ofile << el << '\t'; }); ofile << '\n'; } - -void WaveSimulation::build_V(double thickness, double pos_x, - double aperture_separation, double aperture, - uint32_t slits) -{ - this->V.set_size(this->N, this->N); - this->V.fill(0.); - if (slits == 0) { - return; - } - arma::cx_vec res; - - if (slits % 2 == 0) { - res = arma::cx_vec(aperture_separation / this->h, - arma::fill::value(1e10)); - for (size_t i = 0; i < slits; i += 2) { - res = arma::join_cols( - res, arma::cx_vec(aperture / this->h, arma::fill::zeros)); - res = arma::join_cols( - arma::cx_vec(aperture / this->h, arma::fill::zeros), res); - res = - arma::join_cols(res, arma::cx_vec(aperture_separation / this->h, - arma::fill::value(1e10))); - res = arma::join_cols(arma::cx_vec(aperture_separation / this->h, - arma::fill::value(1e10)), - res); - } - } - else { - res = arma::cx_vec(aperture / this->h, arma::fill::value(0)); - for (size_t i = 0; i < slits - 1; i += 2) { - res = - arma::join_cols(res, arma::cx_vec(aperture_separation / this->h, - arma::fill::value(1e10))); - res = arma::join_cols(arma::cx_vec(aperture_separation / this->h, - arma::fill::value(1e10)), - res); - res = arma::join_cols( - res, arma::cx_vec(aperture / this->h, arma::fill::zeros)); - res = arma::join_cols( - arma::cx_vec(aperture / this->h, arma::fill::zeros), res); - } - } - if (res.size() > this->N) { - abort(); - } - uint32_t fill = (this->N - res.size()) / 2; - res = arma::join_cols(arma::cx_vec(fill, arma::fill::value(1e10)), res); - res = arma::join_cols(res, arma::cx_vec(fill + ((this->N - res.size()) % 2), - arma::fill::value(1e10))); - - uint32_t start = pos_x / this->h - thickness / this->h / 2; - for (size_t i = 0; i < thickness / this->h; i++) { - this->V.col(start + i) = res; - } -} -- 2.20.1 From 8637cad16888e161020b793f1c1788f7c2143891 Mon Sep 17 00:00:00 2001 From: Cory Date: Fri, 22 Dec 2023 16:02:44 +0100 Subject: [PATCH 19/32] Use the modified class --- src/main.cpp | 91 +++++++++++++--------------------------------------- 1 file changed, 23 insertions(+), 68 deletions(-) diff --git a/src/main.cpp b/src/main.cpp index 8c2cb6a..bb4eea1 100644 --- a/src/main.cpp +++ b/src/main.cpp @@ -11,36 +11,20 @@ * */ #include "WaveSimulation.hpp" #include "utils.hpp" + #include void probability_deviation() { - WaveSimulation sim_narrow(.005, 2.5e-5, .008, .25, .5, .05, .05, 200., 0.); + WaveSimulation sim_no_slits(.005, 2.5e-5, .008, .25, .5, .05, .05, 200., + 0.); - WaveSimulation sim_wide(.005, 2.5e-5, .008, .25, .5, .05, .10, 200., 0., - 0.02, .5, .05, .05, 2); + WaveSimulation sim_slits(.005, 2.5e-5, .008, .25, .5, .05, .10, 200., 0., + 0.02, .5, .05, .05, 2); - std::ofstream ofile; - utils::mkpath("data"); - ofile.open("data/probability_deviation.txt"); - - double sum_narrow, sum_wide; - - for (size_t i = 0; i < 320; i++) { - sum_narrow = 0; - sum_wide = 0; - for (size_t j = 0; j < sim_narrow.U.n_elem; j++) { - sum_narrow += (sim_narrow.U(j) * std::conj(sim_narrow.U(j))).real(); - sum_wide += (sim_wide.U(j) * std::conj(sim_wide.U(j))).real(); - } - - sim_narrow.step(); - sim_wide.step(); - ofile << i << ',' << utils::scientific_format(sum_narrow) << ',' - << utils::scientific_format(sum_wide) << '\n'; - } - - ofile.close(); + sim_no_slits.probability_deviation( + "data/probability_deviation_no_slits.txt", true); + sim_slits.probability_deviation("data/probability_deviation_slits.txt", true); } void color_map() @@ -48,19 +32,8 @@ void color_map() WaveSimulation sim(.005, 2.5e-5, .002, .25, .5, .05, .20, 200., 0., 0.02, .5, .05, .05, 2); - std::ofstream ofile; - ofile.open("data/color_map.txt"); - ofile << sim.N << '\n'; - sim.write_U(ofile); - for (size_t i = 0; i < 40; i++) { - sim.step(); - } - sim.write_U(ofile); - for (size_t i = 0; i < 40; i++) { - sim.step(); - } - sim.write_U(ofile); - ofile.close(); + std::vector times{0., .001, .002}; + sim.solve("data/color_map.txt", times); } void detector_screen() @@ -68,53 +41,35 @@ void detector_screen() WaveSimulation *sim = new WaveSimulation( .005, 2.5e-5, .002, .25, .5, .05, .20, 200., 0., 0.02, .5, .05, .05, 1); - std::ofstream ofile; - utils::mkpath("data/screen"); - ofile.open("data/screen/single_slit.txt"); - ofile << sim->N << '\n'; - for (size_t i = 0; i < 80; i++) { - sim->step(); - } - sim->write_U(ofile); - ofile.close(); + sim->solve("data/screen/single_slit.txt"); delete sim; sim = new WaveSimulation(.005, 2.5e-5, .002, .25, .5, .05, .20, 200., 0., 0.02, .5, .05, .05, 2); - ofile.open("data/screen/double_slit.txt"); - ofile << sim->N << '\n'; - for (size_t i = 0; i < 80; i++) { - sim->step(); - } - sim->write_U(ofile); - ofile.close(); + sim->solve("data/screen/double_slit.txt"); delete sim; sim = new WaveSimulation(.005, 2.5e-5, .002, .25, .5, .05, .20, 200., 0., 0.02, .5, .05, .05, 3); - ofile.open("data/screen/triple_slit.txt"); - ofile << sim->N << '\n'; - for (size_t i = 0; i < 80; i++) { - sim->step(); - } - sim->write_U(ofile); - ofile.close(); + sim->solve("data/screen/triple_slit.txt"); } int main() { - //probability_deviation(); + DEBUG("Before probability deviation"); + probability_deviation(); + DEBUG("Before color map"); //color_map(); - detector_screen(); - - //std::ofstream ofile; - //ofile.open("test.txt"); - //WaveSimulation sim(.005, 2.5e-5, .008, .25, .5, .05, .10, 200., 0., - //0.02, .5, .05, .05, 2); - //sim.solve(ofile); + DEBUG("Before detector screen"); + //detector_screen(); + // std::ofstream ofile; + // ofile.open("test.txt"); + // WaveSimulation sim(.005, 2.5e-5, .008, .25, .5, .05, .10, 200., 0., + // 0.02, .5, .05, .05, 2); + // sim.solve(ofile); return 0; } -- 2.20.1 From 51c084ae544b21870569290d9aa00d8a3c38aa11 Mon Sep 17 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07face3dd7d9c57d92e488bf793616c298b6e9c4 Mon Sep 17 00:00:00 2001 From: Cory Date: Tue, 26 Dec 2023 15:31:15 +0100 Subject: [PATCH 21/32] Updated data --- data/color_map.txt | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/data/color_map.txt b/data/color_map.txt index 2779ec4..79e2f46 100644 --- a/data/color_map.txt +++ b/data/color_map.txt @@ -1,4 +1,4 @@ 198 (4.62005e-09,0) (4.91648e-09,0) (5.22866e-09,0) (5.55719e-09,0) (5.90267e-09,0) (6.26572e-09,0) (6.64693e-09,0) (7.04694e-09,0) (7.46634e-09,0) (7.90577e-09,0) (8.36583e-09,0) (8.84713e-09,0) (9.35027e-09,0) (9.87585e-09,0) (1.04245e-08,0) (1.09967e-08,0) (1.1593e-08,0) (1.22141e-08,0) (1.28604e-08,0) (1.35325e-08,0) (1.42308e-08,0) (1.49557e-08,0) (1.57078e-08,0) (1.64873e-08,0) (1.72948e-08,0) (1.81304e-08,0) (1.89946e-08,0) (1.98875e-08,0) (2.08094e-08,0) (2.17604e-08,0) (2.27407e-08,0) (2.37502e-08,0) (2.47891e-08,0) (2.58573e-08,0) (2.69546e-08,0) (2.8081e-08,0) (2.92361e-08,0) (3.04198e-08,0) 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40af4d3b1ae77c4540ee9563faee2e0ad6aa4f3d Mon Sep 17 00:00:00 2001 From: Cory Date: Tue, 26 Dec 2023 15:31:57 +0100 Subject: [PATCH 22/32] Add changes --- python_scripts/colormap.py | 59 ++++++++++++++++++++++++++++------ python_scripts/colormap_alt.py | 38 ---------------------- 2 files changed, 49 insertions(+), 48 deletions(-) delete mode 100644 python_scripts/colormap_alt.py diff --git a/python_scripts/colormap.py b/python_scripts/colormap.py index 1d4dd55..f596e15 100644 --- a/python_scripts/colormap.py +++ b/python_scripts/colormap.py @@ -1,6 +1,7 @@ -import numpy as np -import matplotlib.pyplot as plt import ast + +import matplotlib.pyplot as plt +import numpy as np import seaborn as sns sns.set_theme() @@ -16,22 +17,60 @@ params = { } plt.rcParams.update(params) + def plot(): with open("data/color_map.txt") as f: - lines = f.readlines(); + lines = f.readlines() size = int(lines[0]) for i, line in enumerate(lines[1:]): - fig, ax = plt.subplots() + # Create figures for each plot + fig1, ax1 = plt.subplots() + fig2, ax2 = plt.subplots() + fig3, ax3 = plt.subplots() + arr = line.strip().split("\t") - arr = np.asarray(list(map(lambda x: ((a := complex(*ast.literal_eval(x)))*a.conjugate()).real, arr))) + arr = np.asarray(list(map(lambda x: complex(*ast.literal_eval(x)), arr))) + # Reshape and transpose array + arr = arr.reshape(size, size).T - arr = arr.reshape(size,size) + # Plot color maps + color_map1 = ax1.imshow( + np.multiply(arr, arr.conj()).real, + interpolation="nearest", + cmap=sns.color_palette("mako", as_cmap=True), + ) + color_map2 = ax2.imshow( + arr.real, + interpolation="nearest", + cmap=sns.color_palette("mako", as_cmap=True), + ) + color_map3 = ax3.imshow( + arr.imag, + interpolation="nearest", + cmap=sns.color_palette("mako", as_cmap=True), + ) - ax.imshow(arr.T, interpolation="nearest") - plt.savefig(f"latex/images/color_map_{i}.pdf") - plt.close(fig) + # Create color bar + fig1.colorbar(color_map1, ax=ax1) + fig2.colorbar(color_map2, ax=ax2) + fig3.colorbar(color_map3, ax=ax3) + + # Remove grids + ax1.grid(False) + ax2.grid(False) + ax3.grid(False) + + # Save the figures + 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) + plt.close(fig2) + plt.close(fig3) if __name__ == "__main__": plot() - + diff --git a/python_scripts/colormap_alt.py b/python_scripts/colormap_alt.py deleted file mode 100644 index 1496a37..0000000 --- a/python_scripts/colormap_alt.py +++ /dev/null @@ -1,38 +0,0 @@ -import numpy as np -import matplotlib.pyplot as plt -import ast -import seaborn as sns - -sns.set_theme() -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("data/color_map.txt") as f: - lines = f.readlines() - size = int(lines[0]) - for i, line in enumerate(lines[1:]): - fig, ax = plt.subplots() - arr = line.strip().split("\t") - arr = np.asarray(list(map(lambda x: ((a := complex(*ast.literal_eval(x)))*a.conjugate()).real, arr))) - - arr = arr.reshape(size,size) - - color_map = ax.imshow(arr.T, interpolation="nearest", cmap=sns.color_palette("mako", as_cmap=True)) - fig.colorbar(color_map, ax=ax) - plt.grid(False) - plt.savefig(f"latex/images/color_map_{i}.pdf") - plt.close(fig) - - -if __name__ == "__main__": - plot() \ No newline at end of file -- 2.20.1 From 3ec69195f0fb30182a766c396783d149041318aa Mon Sep 17 00:00:00 2001 From: Cory Date: Tue, 26 Dec 2023 15:32:58 +0100 Subject: [PATCH 23/32] Fix mistake --- lib/WaveSimulation.cpp | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/lib/WaveSimulation.cpp b/lib/WaveSimulation.cpp index 7d425bc..1ccde39 100644 --- a/lib/WaveSimulation.cpp +++ b/lib/WaveSimulation.cpp @@ -274,10 +274,10 @@ void WaveSimulation::solve(std::string outfile, std::vector &steps) for (size_t i=0; i < steps.size(); i++) { if (i == 0) { - iterations = steps[i] / this->h; + iterations = steps[i] / this->dt; } else { - iterations = (steps[i] - steps[i-1]) / this->h; + iterations = (steps[i] - steps[i-1]) / this->dt; } for (size_t j=0; j < iterations; j++) { -- 2.20.1 From 9cdb3c659e6b9cd673f7cfb51e4adfc783924844 Mon Sep 17 00:00:00 2001 From: Cory Date: Tue, 26 Dec 2023 15:33:58 +0100 Subject: [PATCH 24/32] Add seaborn theme --- python_scripts/normalization.py | 2 ++ 1 file changed, 2 insertions(+) diff --git a/python_scripts/normalization.py b/python_scripts/normalization.py index 2a89462..fb43137 100644 --- a/python_scripts/normalization.py +++ b/python_scripts/normalization.py @@ -2,6 +2,8 @@ import numpy as np import matplotlib.pyplot as plt import seaborn as sns + +sns.set_theme() params = { "font.family": "Serif", "font.serif": "Roman", -- 2.20.1 From 583f449c9cdfaf4b8b7aec93f6d127eab18d1889 Mon Sep 17 00:00:00 2001 From: Cory Date: Tue, 26 Dec 2023 15:34:25 +0100 Subject: [PATCH 25/32] Uncomment stuff --- src/main.cpp | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/src/main.cpp b/src/main.cpp index bb4eea1..137fb58 100644 --- a/src/main.cpp +++ b/src/main.cpp @@ -61,9 +61,9 @@ int main() DEBUG("Before probability deviation"); probability_deviation(); DEBUG("Before color map"); - //color_map(); + color_map(); DEBUG("Before detector screen"); - //detector_screen(); + detector_screen(); // std::ofstream ofile; // ofile.open("test.txt"); -- 2.20.1 From 3895b9ce86e57e107ad73771c6ca23f341f898ec Mon Sep 17 00:00:00 2001 From: Cory Date: Tue, 26 Dec 2023 15:34:56 +0100 Subject: [PATCH 26/32] Add plots --- latex/images/color_map_0.pdf | Bin 13236 -> 0 bytes latex/images/color_map_0_imag.pdf | Bin 0 -> 111597 bytes latex/images/color_map_0_prob.pdf | Bin 0 -> 78248 bytes latex/images/color_map_0_real.pdf | Bin 0 -> 112207 bytes latex/images/color_map_1.pdf | Bin 13236 -> 0 bytes latex/images/color_map_1_imag.pdf | Bin 0 -> 111268 bytes latex/images/color_map_1_prob.pdf | Bin 0 -> 77392 bytes latex/images/color_map_1_real.pdf | Bin 0 -> 111506 bytes latex/images/color_map_2.pdf | Bin 13236 -> 0 bytes latex/images/color_map_2_imag.pdf | Bin 0 -> 130292 bytes latex/images/color_map_2_prob.pdf | Bin 0 -> 82837 bytes latex/images/color_map_2_real.pdf | Bin 0 -> 130300 bytes latex/images/double_slit_detector.pdf | Bin 7818 -> 69347 bytes latex/images/probability_deviation.pdf | Bin 11647 -> 136911 bytes latex/images/single_slit_detector.pdf | Bin 7048 -> 69018 bytes latex/images/triple_slit_detector.pdf | Bin 7971 -> 69495 bytes 16 files changed, 0 insertions(+), 0 deletions(-) delete mode 100644 latex/images/color_map_0.pdf create mode 100644 latex/images/color_map_0_imag.pdf create mode 100644 latex/images/color_map_0_prob.pdf create mode 100644 latex/images/color_map_0_real.pdf delete mode 100644 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z@RT_{xT{-enp=FjJz!~b1s3wf4S%>q8YWx()vi`%@{zy&faKP;ni zQ**eUir%i6>wLhTf$E&)K~)B09f0=V-atQoQLu82f8JR^n*j&xZ>7YE@hjYw9bm!( z84#VE%@uQDP|8LKG_0JallW4jpX2##KZjDKv>7mB(I<{ZgEcM^prh)6&BZ?? zO5)^q!CHp7v7r^Cv}fqzO#S)8#0xJn6^{$~lbLx5~ z$p_N!Z$B@0^ zA(oG}qnYc%m{b-1b(~5W zUYV`^%&mxw%t+z4Eo13ma;A;IP4u(O52Lm5NQ|2$PYVIt{^pqrn9Yn+r*m;@=xt(r55xQ>Tn23jrY{_;?9%*b#erHcaO zq;-MYi^|Zy$j>ZYTc|q@zxA;^_s41hb-dEZs!YM*t+#E%U*C|10cVf5T~D@|cY}9fB=Yc4_l+rod)855d8Ru!u zE24ZG27$q$Fc=Jqh9VKkZ0yvUwh$^l2$3x*rGzRrA41xO)T<{0|*EV0s8}kAh9s?pBM^?`lB%j z0&;%YA3QVyhWX!m#Q&ebIS~u}qcaEs3WcBlfBv)nZ&DC60s;L)G!%k_VgD3}K7Xk{ z1)`DA{}5yR&*&jY1nkd(a3l)(M> \delta$ \cite[p. 6]{mit:2004:physics}. In this case, we observe constructive interference when \begin{align*} @@ -111,24 +113,32 @@ 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} % -% 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 +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 +\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 +\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*} +which allowed us to express Equation \eqref{eq:schrodinger_discretized} as a matrix equation \begin{align} A u^{n+1} = B u^{n} \ . @@ -137,8 +147,8 @@ 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 can be decomposed as submatrices of size $(M-2) \times (M-2)$, with -the following pattern +The matrices are sparse and can be decomposed as submatrices of size $(M-2) \times (M-2)$, +with the following pattern \begin{align*} A, B = \begin{bmatrix} @@ -195,7 +205,7 @@ 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 used \begin{align*} @@ -204,37 +214,16 @@ To fill the matrices $A$ and $B$, we used \end{align*} An example of filled matrices can be found in Appendix \ref{ap:matrix_structure}. -Notations: -In addition, we use an equal step size in x- and y-direction, $h$ such that +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}. +In addition, we initialized the wave function using a Gaussian wavepacket, given by \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 + 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*} -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 +$x_{c}$ and $y_{c}$ are the coordinates of the center of the wavepacket, $\sigma_{x}$ +and $\sigma_{y}$ are the width of the wavepacket, when it is initialized. The wave +packet momenta are given by $p_{x}$ and $p_{y}$. +% Insert Heisenberg uncertainty here? Or refer to it? \begin{table}[H] \centering \begin{tabular}{l r} % @{\extracolsep{\fill}} @@ -247,15 +236,14 @@ For the general setup of the wall, we used Slit aperture & $0.05$ \\ \hline \end{tabular} - \caption{Wall setup.} - \label{tab:wall_setup} + \caption{Barrier parameters and values.} + \label{tab:barrier_setup} \end{table} - \begin{table}[H] \centering \begin{tabular}{l r r} % @{\extracolsep{\fill}} \hline - Simulation & $1$ & $2$ \\ + Parameter & Setting 1 & Setting 2 \\ \hline $h$ & $0.005$ & $0.005$ \\ $\Delta t$ & $2.5 \times 10^{-5}$ & $2.5 \times 10^{-5}$ \\ @@ -269,12 +257,39 @@ For the general setup of the wall, we used $v_{0}$ & $0$ & $1 \times10^{10}$ \\ \hline \end{tabular} - \caption{Wall setup.} - \label{tab:sim_setup} + \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 + switched on.} + \label{tab:sim_settings} \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} &= |\sum_{\ivec , \jvec} p_{\ivec , \jvec}^{n} - 1| \\ + &= |\sum_{\ivec , \jvec} u_{\ivec , \jvec}^{n*} u_{\ivec , \jvec}^{n} - 1| \ . +\end{align*} + \subsection{Tools}\label{ssec:tools} % The double-slit experiment is implemented in C++. We use the Python library \verb|matplotlib| \cite{hunter:2007:matplotlib} to produce all the plots, and \verb|seaborn| \cite{waskom:2021:seaborn} to set the theme in the figures. \end{document} + + + +% \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} \ No newline at end of file diff --git a/latex/sections/results.tex b/latex/sections/results.tex index deeb5ea..f1846eb 100644 --- a/latex/sections/results.tex +++ b/latex/sections/results.tex @@ -4,32 +4,89 @@ \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 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. +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|. % 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 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}. +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. +We checked if the total probability was conserved over time, by plotting the deviation +from $1.0$. \begin{figure} \centering \includegraphics[width=\linewidth]{images/probability_deviation.pdf} - \caption{Deviation for $t \in [0, T]$ where $T=0.008$.} + \caption{Deviation of total probability, for time $t \in [0, T]$ where $T=0.008$.} \label{fig:deviation} \end{figure} +In Figure \ref{fig:deviation}, we observe a larger deviation of total probability +for a barrier with double slits compared to no barrier. Interaction with the +barrier result in a change ... of values, as the light passes through, +The result is more prone to computational errors. Using no barrier, the light is not +affected by obstacles resulting in a more stable deviation from the total probability. +In addition, we have to consider the limitation of a computer, 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 ran the simulation using the values in column 2 of Table \ref{tab:sim_setup}, found +in Section \ref{ssec:implementation}. To study the time evolution of the probability +function, 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 +result 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 result can be found in Appendix \ref{ap:figures}, in Figure \ref{fig:colormap}. +\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} +In Figure \ref{fig:colormap_1_prob}, the ... interacts with the double +slitted barrier, which result in a wavelike pattern. However, the waves are more +visible when we observe the real and imaginary part separately in Figure \ref{fig:colormap}. + \subsection{Particle detection}\label{ssec:particle_detection} % Problem 9: Plot detection probability for single-, double- and triple-slit - +We used the simulation from the previous section, assumed a detector screen +located at $x=0.8$, and plotted the detection probability along the screen at time +$t=0.002$. We adjusted the parameters to include single-, double-, and triple-slitted +barrier. The results is found in Figure \ref{}, Figure \ref{}, and Figure \ref{fig:}. +\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} \end{document} diff --git a/python_scripts/normalization.py b/python_scripts/normalization.py index fb43137..cf50368 100644 --- a/python_scripts/normalization.py +++ b/python_scripts/normalization.py @@ -21,9 +21,17 @@ def plot(): "data/probability_deviation_no_slits.txt", "data/probability_deviation_slits.txt", ] - for file in files: + 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): with open(file) as f: - lines = f.readlines(); + lines = f.readlines() x = [] arr = [] for line in lines: @@ -31,10 +39,13 @@ def plot(): x.append(float(tmp[0])) arr.append(float(tmp[1])) - plt.plot(x,arr) + 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.savefig("latex/images/probability_deviation.pdf") if __name__ == "__main__": - plot() - + plot() \ No newline at end of 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zuv`gv4go|OVKhuJW3m>Y&p_!n;b#Y{wp&mB)it<`^sjp_uYE<7mvt?M^cb#N1 zB|lW}WannARAyIK6Ytk+yW!?eIGuSi3`?0mCgGK0yDaltROTKaefuNcR~wfF*T#uF zVUGss*q9ZgbcJN;7w|8w3d?xZ%%F2ZZ=%7G@=rC`#&d_O%8j1fIUkDhKEz4uWzzlN zmEjxc2R94sWImZcI8@M7pM3R`5i|7;eC&Lz-&xb_NLC>&N9HCWV)~@?{*2>Q>8^!9>mqM`#LD;-U9Pk`-8uP?+S7pNmQ~KI=iA=;Y40rK)cPB`YKdFP z5b1zXF89u^{NtKlTaJ;Y0?)WBH_K$H#%>pC&CsyGi0C9U$lJ{iWufNZP#UXmm?nDit=%p zKl3Zf{8{SxP8}(Z38=O41dbVQG^gxj9QWMgA&x$m-Ug zMwks)pV!C%s&C?(@w(dX#@_YrZcv8(vfN#+-qd10aY-o~-t@?3CrDjlRGjM^LnfXJ zzy4Ai*sM}1bHx_iqRh+R?t2gzWusbF+ND+e`5HDy%E0F&>;zDd=`Mgy{K zKZF!y2qQ&VU`1gh5|wrr+;$fXIw&lR)kRS{As~}~#U2DCFQ%`L#3Ip1G(}kwWD|%* z> \delta$ \cite[p. 6]{mit:2004:physics}. In @@ -216,13 +224,6 @@ 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}. -In addition, we initialized 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, when it is initialized. The wave -packet momenta are given by $p_{x}$ and $p_{y}$. % Insert Heisenberg uncertainty here? Or refer to it? \begin{table}[H] \centering diff --git a/latex/sections/results.tex b/latex/sections/results.tex index f1846eb..85d9cbb 100644 --- a/latex/sections/results.tex +++ b/latex/sections/results.tex @@ -20,24 +20,28 @@ from $1.0$. \caption{Deviation of total probability, for time $t \in [0, T]$ where $T=0.008$.} \label{fig:deviation} \end{figure} -In Figure \ref{fig:deviation}, we observe a larger deviation of total probability -for a barrier with double slits compared to no barrier. Interaction with the -barrier result in a change ... of values, as the light passes through, -The result is more prone to computational errors. Using no barrier, the light is not -affected by obstacles resulting in a more stable deviation from the total probability. +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 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 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 ran the simulation using the values in column 2 of Table \ref{tab:sim_setup}, found -in Section \ref{ssec:implementation}. To study the time evolution of the probability -function, we created colormap plots for different time steps. Figure \ref{fig:colormap_0_prob}, +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 -result 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 result 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}. \begin{figure} \centering \includegraphics[width=\linewidth]{images/color_map_0_prob.pdf} @@ -56,17 +60,30 @@ time steps. The result can be found in Appendix \ref{ap:figures}, in Figure \ref \caption{The probability function $p_{\ivec, \jvec}^{n}$, at time $t=0.002$.} \label{fig:colormap_2_prob} \end{figure} -In Figure \ref{fig:colormap_1_prob}, the ... interacts with the double -slitted barrier, which result in a wavelike pattern. However, the waves are more -visible when we observe the real and imaginary part separately in Figure \ref{fig:colormap}. +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}, found in Appendix \ref{ap:figures}. 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. \subsection{Particle detection}\label{ssec:particle_detection} % Problem 9: Plot detection probability for single-, double- and triple-slit -We used the simulation from the previous section, assumed a detector screen -located at $x=0.8$, and plotted the detection probability along the screen at time -$t=0.002$. We adjusted the parameters to include single-, double-, and triple-slitted -barrier. The results is found in Figure \ref{}, Figure \ref{}, and Figure \ref{fig:}. +We simulation 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}, +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} @@ -88,5 +105,10 @@ barrier. The results is found in Figure \ref{}, Figure \ref{}, and Figure \ref{f 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 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. \end{document} diff --git a/python_scripts/colormap.py b/python_scripts/colormap.py index f596e15..ec3720a 100644 --- a/python_scripts/colormap.py +++ b/python_scripts/colormap.py @@ -19,6 +19,7 @@ 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]) @@ -38,16 +39,19 @@ 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 @@ -60,10 +64,26 @@ 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") - fig2.savefig(f"latex/images/color_map_{i}_real.pdf") - fig3.savefig(f"latex/images/color_map_{i}_imag.pdf") + 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") # Close figures plt.close(fig1) diff --git a/python_scripts/colormap_all.py b/python_scripts/colormap_all.py new file mode 100644 index 0000000..6090b97 --- /dev/null +++ b/python_scripts/colormap_all.py @@ -0,0 +1,93 @@ +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, + "axes.titlesize": "large", + "axes.labelsize": "large", + "xtick.labelsize": "large", + "ytick.labelsize": "large", + "legend.fontsize": "medium", +} +plt.rcParams.update(params) + + +def plot(): + fig, axes = plt.subplots(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:]): + # Create figures for each plot + # fig1, ax1 = plt.subplots() + # fig2, ax2 = plt.subplots() + # fig3, ax3 = plt.subplots() + + 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 + color_map1 = axes[i,0].imshow( + np.multiply(arr, arr.conj()).real, + interpolation="nearest", + cmap=sns.color_palette("mako", as_cmap=True) + ) + color_map2 = axes[i,1].imshow( + arr.real, + interpolation="nearest", + cmap=sns.color_palette("mako", as_cmap=True) + ) + color_map3 = axes[i,2].imshow( + arr.imag, + interpolation="nearest", + cmap=sns.color_palette("mako", as_cmap=True) + ) + + # Create color bar + fig.colorbar(color_map1, ax=axes) + # fig2.colorbar(color_map2, ax=ax2) + # fig3.colorbar(color_map3, ax=ax3) + + # # Remove grids + axes.grid(False) + # 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 + fig.savefig(f"latex/images/color_map_all.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") + + # # Close figures + # plt.close(fig1) + # plt.close(fig2) + # plt.close(fig3) + + +if __name__ == "__main__": + plot() + diff --git a/python_scripts/detector.py b/python_scripts/detector.py index 7ccea84..260933d 100644 --- a/python_scripts/detector.py +++ b/python_scripts/detector.py @@ -27,6 +27,7 @@ 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(); @@ -43,8 +44,10 @@ def plot(): slice *= norm slice = np.asarray([i * i.conjugate() for i in slice]) - ax.plot(x, slice) - plt.savefig(output) + 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") plt.close(fig) -- 2.20.1 From c51bcb4369387dde5726f7aef18f4eb4fa6f0ee8 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Mon, 1 Jan 2024 13:05:09 +0100 Subject: [PATCH 29/32] Finished the introduction. --- latex/sections/introduction.tex | 76 +++++++++++++++++++++------------ latex/sections/methods.tex | 8 +++- 2 files changed, 55 insertions(+), 29 deletions(-) diff --git a/latex/sections/introduction.tex b/latex/sections/introduction.tex index b6f40b5..33f2675 100644 --- a/latex/sections/introduction.tex +++ b/latex/sections/introduction.tex @@ -3,45 +3,69 @@ \begin{document} \section{Introduction}\label{sec:introduction} % Light: wave particle -% Wave equation -% Hyugens theory -% Thomas Young -The nature of light has long been a subject of interest and discussion. % +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}. + +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 +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 +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. + +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. + +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} + % 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)} @@ -59,11 +83,7 @@ we will apply the Crank-Nicolson method in 2+1 dimensions. % 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 \ No newline at end of file diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 52859bf..8f0e512 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -6,7 +6,13 @@ % 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 +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}. 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b/latex/schrodinger_simulation.tex @@ -102,7 +102,7 @@ \subfile{sections/results} % Conclusion -% \subfile{sections/conclusion} +\subfile{sections/conclusion} % Notes % \subfile{draft} diff --git a/latex/sections/appendices.tex b/latex/sections/appendices.tex index 14aefbc..54adff5 100644 --- a/latex/sections/appendices.tex +++ b/latex/sections/appendices.tex @@ -3,7 +3,8 @@ \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*} @@ -20,9 +21,11 @@ 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, assume this is known... +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. -Schrödinger contain $i$ at the lhs, factor it as +The Schrödinger equation contains $i$ on the left hand side, we rewrite 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] \ , @@ -44,8 +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*} -In addition, since we will use an equal step size $h$ in both $x$ and $y$ direction, -we can use +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*} @@ -53,7 +55,7 @@ and define \begin{align*} r \equiv \frac{i \Delta t}{2 \Delta h^{2}} \end{align*} -Now, the discretized Schrödinger equation can be written as +Now, the discretized Schrödinger equation can be rewritten 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} \\ @@ -97,6 +99,8 @@ $(M-2)^{2} \times (M-2)^{2} = 9 \times 9$ given by \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} @@ -119,9 +123,7 @@ $(M-2)^{2} \times (M-2)^{2} = 9 \times 9$ given by \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} @@ -142,7 +144,17 @@ $(M-2)^{2} \times (M-2)^{2} = 9 \times 9$ given by \caption{Im$(u_{\ivec, \jvec})$ at time $t=0.002$.} \label{fig:colormap_2_imag} \end{subfigure} - \caption{The time evolution of the probability function $p_{\ivec, \jvec}^{n}$.} - \label{fig:colormap} + \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} diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 8f0e512..1994cdf 100644 --- a/latex/sections/methods.tex +++ b/latex/sections/methods.tex @@ -273,8 +273,8 @@ and for the simulations, we used the parameter settings in Table \ref{tab:sim_se 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} &= |\sum_{\ivec , \jvec} p_{\ivec , \jvec}^{n} - 1| \\ - &= |\sum_{\ivec , \jvec} u_{\ivec , \jvec}^{n*} u_{\ivec , \jvec}^{n} - 1| \ . + 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} % diff --git a/latex/sections/results.tex b/latex/sections/results.tex index 85d9cbb..02543fb 100644 --- a/latex/sections/results.tex +++ b/latex/sections/results.tex @@ -74,6 +74,20 @@ areas at both sides of the barries. Some of the probability function is reflecte 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 diff --git a/python_scripts/colormap_all.py b/python_scripts/colormap_all.py index 6090b97..ae19e9b 100644 --- a/python_scripts/colormap_all.py +++ b/python_scripts/colormap_all.py @@ -9,17 +9,23 @@ 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", + "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(nrows=3, ncols=3) + 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]) @@ -35,39 +41,45 @@ def plot(): arr = arr.reshape(size, size).T # Plot color maps - color_map1 = axes[i,0].imshow( + color_map1 = axes[0, i].imshow( np.multiply(arr, arr.conj()).real, interpolation="nearest", cmap=sns.color_palette("mako", as_cmap=True) ) - color_map2 = axes[i,1].imshow( + color_map2 = axes[1, i].imshow( arr.real, interpolation="nearest", cmap=sns.color_palette("mako", as_cmap=True) ) - color_map3 = axes[i,2].imshow( + color_map3 = axes[2, i].imshow( arr.imag, interpolation="nearest", cmap=sns.color_palette("mako", as_cmap=True) ) # Create color bar - fig.colorbar(color_map1, ax=axes) - # fig2.colorbar(color_map2, ax=ax2) - # fig3.colorbar(color_map3, ax=ax3) + fig.subplots_adjust(wspace=0, hspace=0) + # fig.colorbar(color_map1, ax=axes[0, 2]) + # fig.colorbar(color_map2, ax=axes[1, 2]) + # fig.colorbar(color_map3, ax=axes[2, 2]) - # # Remove grids - axes.grid(False) - # ax2.grid(False) - # ax3.grid(False) + # Remove grids + # for i in range(3): + # axes[i, 0].set_xticks([]) + # axes[i, 0].set_yticks([]) + # axes[i, 0]. + # axes[i, 1].set_xticks([]) + # axes[i, 1].set_yticks([]) + # axes[i, 2].set_xticks([]) + # axes[i, 2].set_yticks([]) - # # 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) + + # axes[0].set_xticks([]) + # axes[0].set_yticks([]) + # axes[1].set_xticks([]) + # axes[].set_yticks([]) + # axes[].set_xticks([]) + # axes[].set_yticks([]) # # Set labels # ax1.set_xlabel("x-axis") @@ -78,7 +90,7 @@ def plot(): # ax3.set_ylabel("y-axis") # # Save the figures - fig.savefig(f"latex/images/color_map_all.pdf", bbox_inches="tight") + plt.savefig(f"latex/images/color_map_all.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") -- 2.20.1 From 18aa574624f188ec274e465401bb7f9a19a15404 Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Mon, 1 Jan 2024 15:38:03 +0100 Subject: [PATCH 31/32] Clean up script --- python_scripts/colormap_all.py | 53 ++++------------------------------ 1 file changed, 5 insertions(+), 48 deletions(-) diff --git a/python_scripts/colormap_all.py b/python_scripts/colormap_all.py index ae19e9b..dc9786b 100644 --- a/python_scripts/colormap_all.py +++ b/python_scripts/colormap_all.py @@ -30,74 +30,31 @@ def plot(): lines = f.readlines() size = int(lines[0]) for i, line in enumerate(lines[1:]): - # Create figures for each plot - # fig1, ax1 = plt.subplots() - # fig2, ax2 = plt.subplots() - # fig3, ax3 = plt.subplots() - 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 - color_map1 = axes[0, i].imshow( + axes[0, i].imshow( np.multiply(arr, arr.conj()).real, interpolation="nearest", cmap=sns.color_palette("mako", as_cmap=True) ) - color_map2 = axes[1, i].imshow( + axes[1, i].imshow( arr.real, interpolation="nearest", cmap=sns.color_palette("mako", as_cmap=True) ) - color_map3 = axes[2, i].imshow( + axes[2, i].imshow( arr.imag, interpolation="nearest", cmap=sns.color_palette("mako", as_cmap=True) ) - # Create color bar + # Create tight subplots and save figure fig.subplots_adjust(wspace=0, hspace=0) - # fig.colorbar(color_map1, ax=axes[0, 2]) - # fig.colorbar(color_map2, ax=axes[1, 2]) - # fig.colorbar(color_map3, ax=axes[2, 2]) - - # Remove grids - # for i in range(3): - # axes[i, 0].set_xticks([]) - # axes[i, 0].set_yticks([]) - # axes[i, 0]. - # axes[i, 1].set_xticks([]) - # axes[i, 1].set_yticks([]) - # axes[i, 2].set_xticks([]) - # axes[i, 2].set_yticks([]) - - - # axes[0].set_xticks([]) - # axes[0].set_yticks([]) - # axes[1].set_xticks([]) - # axes[].set_yticks([]) - # axes[].set_xticks([]) - # axes[].set_yticks([]) - - # # 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 - plt.savefig(f"latex/images/color_map_all.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") - - # # Close figures - # plt.close(fig1) - # plt.close(fig2) - # plt.close(fig3) + fig.savefig(f"latex/images/color_map_all.pdf", bbox_inches="tight") if __name__ == "__main__": -- 2.20.1 From 57abf18fa616ac96597fd42f5a974da38cdf103d Mon Sep 17 00:00:00 2001 From: Janita Willumsen Date: Mon, 1 Jan 2024 17:36:17 +0100 Subject: [PATCH 32/32] Finish report --- latex/references.bib | 23 +++++++++++++++++ latex/schrodinger_simulation.tex | 2 +- latex/sections/abstract.tex | 2 +- latex/sections/appendices.tex | 4 +-- latex/sections/conclusion.tex | 23 ++++------------- latex/sections/introduction.tex | 22 ++++++++-------- latex/sections/methods.tex | 44 +++++++++++++++----------------- latex/sections/results.tex | 28 ++++++++++---------- 8 files changed, 77 insertions(+), 71 deletions(-) diff --git a/latex/references.bib b/latex/references.bib index e7380fa..572bc39 100644 --- a/latex/references.bib +++ b/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}, diff --git a/latex/schrodinger_simulation.tex b/latex/schrodinger_simulation.tex index 0058e3a..258d20b 100644 --- a/latex/schrodinger_simulation.tex +++ b/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. diff --git a/latex/sections/abstract.tex b/latex/sections/abstract.tex index 34bc0f7..e0d384c 100644 --- a/latex/sections/abstract.tex +++ b/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. diff --git a/latex/sections/appendices.tex b/latex/sections/appendices.tex index 54adff5..b4adfe2 100644 --- a/latex/sections/appendices.tex +++ b/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 = diff --git a/latex/sections/conclusion.tex b/latex/sections/conclusion.tex index 3e74e12..4211214 100644 --- a/latex/sections/conclusion.tex +++ b/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 -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} diff --git a/latex/sections/introduction.tex b/latex/sections/introduction.tex index 33f2675..bc58e16 100644 --- a/latex/sections/introduction.tex +++ b/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 -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} diff --git a/latex/sections/methods.tex b/latex/sections/methods.tex index 1994cdf..6ec8279 100644 --- a/latex/sections/methods.tex +++ b/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. -% 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} diff --git a/latex/sections/results.tex b/latex/sections/results.tex index 02543fb..13b1388 100644 --- a/latex/sections/results.tex +++ b/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 -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} -- 2.20.1

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