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\newlabel{eq:Riemann_sum}{{6}{4}}
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\newlabel{eq:numerical_E}{{7}{4}}
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entering extended mode
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@ -311,7 +311,6 @@ legend({'Numerical','Analytical'},'Location','northwest');
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\end{subfigure}
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\end{figure}
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% TODO idk if this is enough
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With the decreased lattice spacing, and increased number of points, we can see the numerical solution
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of eigenvalues matches the analytical solution much closer. As well, we can see that the probability
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density function appears to be "squeezed" in the centre, for the $n = 50$ case. The \(n=50\) case is
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@ -325,7 +324,12 @@ legend({'Numerical','Analytical'},'Location','northwest');
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In order to modify the computations to those for a particle in a "ring"
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we simply had to add $-t_0$ elements as the "corner" elements of the
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hamiltonian operator array.
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hamiltonian operator array:
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\begin{lstlisting}[language=Matlab]
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% Modify hamiltonian for circular boundary conditions
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H(1, N) = -t0;
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H(N, 1) = -t0;
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\end{lstlisting}
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\begin{figure}[H]
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\centering
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@ -344,15 +348,23 @@ legend({'Numerical','Analytical'},'Location','northwest');
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\item %1dii
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The energy levels for eigenvalues number 4 and 5 are both $0.06eV$.
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Because these eigenstates both have the same eigenvalue (0.06).
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These eigenstates are degenerate because they both have the same
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eigenvalue/energy (0.06).
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\item %1diii
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% TODO add figure for sketch
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\begin{figure}[H]
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\centering
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\includegraphics[width=\textwidth, angle=-90]{q1div.jpg}
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\caption{Sketch of appropriate energy levels}
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\end{figure}
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\item %1div
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Plugging the valid levels for $n$ into equation (10) from the assignment (and dividing
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by the requisite $q$), we get the energy levels of $0eV, 0.0147eV, and 0.0589eV$, for the
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eigenvalue numbers $n = 0, 1, and 2$, respectively. These match very closely, to within
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acceptable margin of the numerical results from part (ii).
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\end{enumerate}
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