diff --git a/latex/images/youngs_double_slit.pdf b/latex/images/youngs_double_slit.pdf new file mode 100644 index 0000000..8d6a7cb Binary files /dev/null and b/latex/images/youngs_double_slit.pdf differ 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} %