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pkirwin 2021-03-01 17:15:53 -07:00
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commit 87af8de693
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\newlabel{eq:Riemann_sum}{{6}{4}} \newlabel{eq:Riemann_sum}{{6}{4}}
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@ -311,7 +311,6 @@ legend({'Numerical','Analytical'},'Location','northwest');
\end{subfigure} \end{subfigure}
\end{figure} \end{figure}
% 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. The \(n=50\) case is density function appears to be "squeezed" in the centre, for the $n = 50$ case. The \(n=50\) case is
@ -325,8 +324,13 @@ legend({'Numerical','Analytical'},'Location','northwest');
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:
\begin{lstlisting}[language=Matlab]
% Modify hamiltonian for circular boundary conditions
H(1, N) = -t0;
H(N, 1) = -t0;
\end{lstlisting}
\begin{figure}[H] \begin{figure}[H]
\centering \centering
\begin{subfigure}{0.5\textwidth} \begin{subfigure}{0.5\textwidth}
@ -344,15 +348,23 @@ legend({'Numerical','Analytical'},'Location','northwest');
\item %1dii \item %1dii
The energy levels for eigenvalues number 4 and 5 are both $0.06eV$. The energy levels for eigenvalues number 4 and 5 are both $0.06eV$.
Because these eigenstates both have the same eigenvalue (0.06). These eigenstates are degenerate because they both have the same
eigenvalue/energy (0.06).
\item %1diii \item %1diii
% TODO add figure for sketch \begin{figure}[H]
\centering
\includegraphics[width=\textwidth, angle=-90]{q1div.jpg}
\caption{Sketch of appropriate energy levels}
\end{figure}
\item %1div \item %1div
Plugging the valid levels for $n$ into equation (10) from the assignment (and dividing
by the requisite $q$), we get the energy levels of $0eV, 0.0147eV, and 0.0589eV$, for the
eigenvalue numbers $n = 0, 1, and 2$, respectively. These match very closely, to within
acceptable margin of the numerical results from part (ii).
\end{enumerate} \end{enumerate}