115 lines
6.4 KiB
TeX
115 lines
6.4 KiB
TeX
\documentclass[../schrodinger_simulation.tex]{subfiles}
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\begin{document}
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\section{Results}\label{sec:results}
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\subsection{Deviation}\label{ssec:deviation}
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% Problem 3: Discuss approaches to solve Au^{n+1} = b, dealing with sparse matrix...
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We used the \verb|superlu| solver, which is a solver for sparse matrices. It is
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generally used to solve nonsymmetric, sparse matrices. However, as the alternative
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solver \verb|lapack| converts a sparse matrix to a dense matrix, it will increase
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memory usage compared to \verb|superlu|.
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% Problem 7: Consequenses of solver choice, in regards to accuracy of probability conserved
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% Add plot of deviation for both single- and double-slit
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Since we used a solver for sparse matrices, we decrease the number of computations performed
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compared to number of computations using a solver for dense matrices.
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We checked if the total probability was conserved over time, by plotting the deviation
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from $1.0$.
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/probability_deviation.pdf}
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\caption{Deviation of total probability, for time $t \in [0, T]$ where $T=0.008$.}
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\label{fig:deviation}
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\end{figure}
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We simulated the wave equation with the barrier switched off, using setting 1 in
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Table \ref{tab:sim_settings} found in Section \ref{ssec:implementation}. When the
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barrier was switched on, we used setting 2 in \ref{tab:sim_settings}. We observed
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a larger deviation of total probability for a barrier with double slits compared
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to no barrier, the result is showed in Figure \ref{fig:deviation}. The wave interacts
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with the barrier resulting in a change in kinetic energy. The result is more prone
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to computational errors, than if the wave propagates without interacting with a
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barrier. No interaction results in a more stable deviation from the total probability.
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In addition, we have to consider the limitation of a computer, some computational
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error is to be expected.
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\subsection{Time evolution}\label{ssec:time_evolution}
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% Problem 8: Colormap, include plot of both Re and Im for different time steps
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% Account for color scale
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We studied the time evolution of the probability function, using setting 2 in
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Table \ref{tab:sim_settings}, found in Section \ref{ssec:implementation}. To visualize
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the time evolution, we created colormap plots for different time steps. Figure \ref{fig:colormap_0_prob},
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Figure \ref{fig:colormap_1_prob}, and Figure \ref{fig:colormap_2_prob} show the
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results for time steps $t=[0, 0.001, 0.002]$, respectively. In addition, we created
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separate plots for the real and imaginary part of $u_{\ivec, \jvec}$, for the same
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time steps. The results can be found in Appendix \ref{ap:figures}, in Figure \ref{fig:colormap}.
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/color_map_0_prob.pdf}
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\caption{The probability function $p_{\ivec, \jvec}^{n}$, at time $t=0$.}
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\label{fig:colormap_0_prob}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/color_map_1_prob.pdf}
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\caption{The probability function $p_{\ivec, \jvec}^{n}$, at time $t=0.001$.}
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\label{fig:colormap_1_prob}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/color_map_2_prob.pdf}
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\caption{The probability function $p_{\ivec, \jvec}^{n}$, at time $t=0.002$.}
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\label{fig:colormap_2_prob}
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\end{figure}
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At time step $t=0.001$, Figure \ref{fig:colormap_1_prob}, when the wave interacts
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with the double slit barrier, we observe a clear diffraction pattern in the
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probability function. At time step $t=0$ (Figure \ref{fig:colormap_0_prob}) and
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$t=0.002$ (Figure \ref{fig:colormap_2_prob}), the diffraction pattern is not as
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clear. It is, however, more visible when we observe the real and imaginary part
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separately in Figure \ref{fig:colormap}, found in Appendix \ref{ap:figures}. Since
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the probability function is a product of $u_{\ivec, \jvec}$ and its conjugate $u_{\ivec, \jvec}^{*}$,
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initialized by a Gaussian wavepacket, the result is a sum of the real and imaginary part.
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% This can be found using Euler's formula, and the diffraction pattern is determined by interference given by \eqref{eq:interference}
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In Figure \ref{fig:colormap_2_prob}, the probability function result in positive
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areas at both sides of the barries. Some of the probability function is reflected
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by the barrier, while the the rest spread out after passing the barrier. This is
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a consequence of the wave-particle duality.
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\subsection{Particle detection}\label{ssec:particle_detection}
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% Problem 9: Plot detection probability for single-, double- and triple-slit
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We simulation the wave equation using setting 2 in Table \ref{tab:sim_settings},
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and assumed a detector screen located at $x=0.8$. To visualize the pattern of constructive
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and destructive interference, we plotted the probability of particle detection,
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along the screen, at time $t=0.002$. We adjusted the parameters to include single-, double-, and triple-slit
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barriers. The results is found in Figure \ref{fig:particle_detection_single},
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Figure \ref{fig:particle_detection_double}, and Figure \ref{fig:particle_detection_triple},
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respectively.
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/single_slit_detector.pdf}
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\caption{Probability of particle detection along a detector screen at time $t=0.002$,
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when using a single-slit barrier.}
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\label{fig:particle_detection_single}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/double_slit_detector.pdf}
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\caption{Probability of particle detection along a detector screen at time $t=0.002$,
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when using a double-slit barrier.}
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\label{fig:particle_detection_double}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{images/triple_slit_detector.pdf}
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\caption{Probability of particle detection along a detector screen at time $t=0.002$,
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when using a triple-slit barrier.}
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\label{fig:particle_detection_triple}
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\end{figure}
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When the barrier has a single slit, there is no destructive interference and we
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observe a single peak in the probability of particle detection. Adding another slit
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result in more peaks, as there are both constructive and destructive interference.
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When we use a triple-slit barrier, we observe an increase in interference which
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result in narrow peaks. In addition, the probability of detecting a particle at
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the ends of the screen increase with number of slits.
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\end{document}
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