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