25 lines
1.5 KiB
TeX
25 lines
1.5 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 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 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 setups. We found that increasing the number of slits
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in the barrier, resulted in an increased number of areas of both high and low probability
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for particle detection. It also increased the variance of particle detection probability.
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\end{document}
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