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pkirwin 2021-03-01 17:02:24 -07:00
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\newlabel{eq:Riemann_sum}{{6}{4}} \newlabel{eq:Riemann_sum}{{6}{4}}
\newlabel{eq:numerical_E}{{7}{4}} \newlabel{eq:numerical_E}{{7}{4}}
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@ -314,27 +314,46 @@ legend({'Numerical','Analytical'},'Location','northwest');
% TODO idk if this is enough % TODO idk if this is enough
With the decreased lattice spacing, and increased number of points, we can see the numerical solution With the decreased lattice spacing, and increased number of points, we can see the numerical solution
of eigenvalues matches the analytical solution much closer. As well, we can see that the probability of eigenvalues matches the analytical solution much closer. As well, we can see that the probability
density function appears to be "squeezed" in the centre, for the $n = 50$ case. density function appears to be "squeezed" in the centre, for the $n = 50$ case. The \(n=50\) case is
now a constant-amplitude wave, which corresponds to the expected analytic result - in contrast to the
plot in section (a), which has a low-frequency envelope around it.
\end{enumerate} \end{enumerate}
\item %1d \item %1d
\begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)] \begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)]
\item %1di \item %1di
In order to modify the computations to those for a particle in a "ring" In order to modify the computations to those for a particle in a "ring"
we simply had to add $-t_0$ elements as the "corner" elements of the we simply had to add $-t_0$ elements as the "corner" elements of the
hamiltonian operator array. hamiltonian operator array.
% TODO do we need a listing here \begin{figure}[H]
\centering
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{q1di_fig1.png}
\caption{Probability densities for \(n=4\)\\and \(n=5\).}
\end{subfigure}%
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{q1di_fig2.png}
\caption{Comparison of first 101 numerical and analytic eigenvalues.}
\end{subfigure}
\end{figure}
\item %1dii \item %1dii
The energy levels for eigenvalues number 4 and 5 are both $0.06eV$.
Because these eigenstates both have the same eigenvalue (0.06).
\item %1diii \item %1diii
% TODO add figure for sketch
\item %1div \item %1div
\end{enumerate} \end{enumerate}