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21
PS2/doc.fls
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PS2/doc.fls
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Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (12 pages, 3676324 bytes).
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Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (12 pages, 3711631 bytes).
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@ -155,7 +155,7 @@ legend({'Numerical','Analytical'},'Location','northwest');
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\begin{subfigure}{0.4\linewidth}
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\centering
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\includegraphics[width=\textwidth]{q1a_1and50.png}
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\caption{Probability densities for \(n=1\)\\and \(n=50\).}
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\caption{}
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\label{fig:q3_ne}
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\end{subfigure}%
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\begin{subfigure}{0.4\linewidth}
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@ -164,6 +164,8 @@ legend({'Numerical','Analytical'},'Location','northwest');
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\caption{Comparison of first 101 numerical and analytic eigenvalues.}
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\label{fig:q3_current}
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\end{subfigure}
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\caption{(a) Probability densities for \(n=1\) and \(n=50\).
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(b) Comparison of first 101 numerical and analytic eigenvalues.}
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\end{figure}
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\item %1b
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\begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)]
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@ -185,7 +187,7 @@ legend({'Numerical','Analytical'},'Location','northwest');
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We use the following identity:
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\begin{equation}
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\int\sin^2{(\alpha x)}\:dx = \frac{1}{2} x -\frac{1}{4\alpha}\sin{(2\alpha x)}.
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\int\sin^2{(a x)}\:dx = \frac{1}{2} x -\frac{1}{4a}\sin{(2a x)}.
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\label{eq:integral_ident}
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\end{equation}
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@ -247,25 +249,25 @@ legend({'Numerical','Analytical'},'Location','northwest');
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\item %1ci
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% TODO check that B is expressed correctly in all these equations (alpha, not ell, wait for pdf gen)
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From the base form of $\phi _\ell = Bsin \left( \frac {n \pi}{L} \alpha \ell \right)$, we can see that
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From the base form of $\phi _\ell = Bsin \left( \frac {n \pi}{L} a \ell \right)$, we can see that
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$\phi _ {\ell + 1}$ and $\phi _ {\ell - 1}$ correspond to the trigonometric identities
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$sin(\alpha + B) = sin(\alpha)cos(B) + cos(\alpha)sin(B)$ and
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$sin(\alpha + B) = sin(\alpha)cos(B) + cos(\alpha)sin(B)$, respectively, where
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$\alpha = \frac{n \pi \alpha \ell}{L}$ and $B = \frac{n \pi \alpha}{L}$.
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$sin(a + B) = sin(a)cos(B) + cos(a)sin(B)$ and
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$sin(a + B) = sin(a)cos(B) + cos(a)sin(B)$, respectively, where
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$a = \frac{n \pi a \ell}{L}$ and $B = \frac{n \pi a}{L}$.
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Plugging these identities into equation (7) from the assignment and simplifying, we get to this equation:
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\begin{equation*}
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-t_0 B sin \left( \frac{n \pi \alpha \ell}{L} \right) + 2 t_0 \phi _{\ell} - t_0 sin \left( \frac{n \pi \alpha \ell}{L} \right)
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-t_0 B sin \left( \frac{n \pi a \ell}{L} \right) + 2 t_0 \phi _{\ell} - t_0 sin \left( \frac{n \pi a \ell}{L} \right)
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\end{equation*}
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At this point, we notice that $\phi _ \ell = B sin \left( \frac {n \pi}{L} \alpha \ell \right)$,
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At this point, we notice that $\phi _ \ell = B sin \left( \frac {n \pi}{L} a \ell \right)$,
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so we can factor it out.
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With some minor rearranging, this leaves us with the final expression for $E$:
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||||
\begin{equation}
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||||
E = 2t_0 \left( 1 - cos \left( \frac{n \pi \alpha}{L} \right) \right)
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E = 2t_0 \left( 1 - cos \left( \frac{n \pi a}{L} \right) \right)
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\label{eq:numerical_E}
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\end{equation}
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@ -286,7 +288,7 @@ legend({'Numerical','Analytical'},'Location','northwest');
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on equation (\ref{eq:numerical_E}), we get the following expression
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||||
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||||
\begin{equation*}
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||||
E = 2 t_0 \left( \frac{n^2 \pi^2 \alpha^2}{2 L^2} \right)
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||||
E = 2 t_0 \left( \frac{n^2 \pi^2 a^2}{2 L^2} \right)
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||||
\end{equation*}
|
||||
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Fully substituting the known value of $t_0$, and we can get our final expression for $E$:
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@ -388,7 +390,7 @@ H(N, 1) = -t0;
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||||
\end{equation*}
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\item %2b
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||||
Homogenous boundary conditions imply that the corner entries of \(\hat{H}\) will be 0.
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\item %1c
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||||
\item %2c
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Code:
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||||
\begin{lstlisting}
|
||||
clear all;
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||||
@ -464,12 +466,22 @@ axis([0 1e-9 0 0.04]);
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\caption{2s probability density.}
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\end{subfigure}
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||||
\end{figure}
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\item %1d
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||||
\item %2d
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||||
For the 1s level, \(E = -13.4978\) eV.
|
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\item %1e
|
||||
\lipsum[1]
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||||
\item %1f
|
||||
\item %2e
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||||
\lipsum[1]
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||||
\item %2f
|
||||
In the figure below we can see that the numerical and analytical results agree up to scaling by \(a\). The scale difference is expected,
|
||||
as discussed in Problem 1. From (d), we also expect agreement in the curve shapes because the numerical and analytical energies for the 1s
|
||||
level are very similiar. We can see that the peak value of the analytic result is very slightly higher than that of the numerical result,
|
||||
which corresponds to the analytical result for the energy being slightly greater in magnitude (\(-13.6\) eV versus \(-13.4978\) eV).
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{q2f_fig.png}
|
||||
\caption{Numerical result (black line) and analytical solution scaled by \(a\) (orange circles).}
|
||||
\end{figure}
|
||||
|
||||
|
||||
|
||||
\end{enumerate}
|
||||
|
||||
@ -477,28 +489,28 @@ axis([0 1e-9 0 0.04]);
|
||||
|
||||
\begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)]
|
||||
|
||||
\item %2a
|
||||
\item %3a
|
||||
\lipsum[1]
|
||||
\item %2b
|
||||
\item %3b
|
||||
\begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)]
|
||||
\item %1di
|
||||
\item %3bi
|
||||
\lipsum[1]
|
||||
\item %1dii
|
||||
\item %3bii
|
||||
\lipsum[1]
|
||||
\item %1diii
|
||||
\item %3biii
|
||||
\lipsum[1]
|
||||
\item %1div
|
||||
\item %3biv
|
||||
\lipsum[1]
|
||||
\item %1dv
|
||||
\item %3bv
|
||||
\lipsum[1]
|
||||
\item %1dvi
|
||||
\item %3bvi
|
||||
\lipsum[1]
|
||||
\item %1dvii
|
||||
\item %3bvii
|
||||
\lipsum[1]
|
||||
\end{enumerate}
|
||||
\item %1c
|
||||
\item %3c
|
||||
\lipsum[1]
|
||||
\item %1d
|
||||
\item %3d
|
||||
\lipsum[1]
|
||||
|
||||
\end{enumerate}
|
||||
@ -507,15 +519,15 @@ axis([0 1e-9 0 0.04]);
|
||||
|
||||
\begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)]
|
||||
|
||||
\item %2a
|
||||
\item %4a
|
||||
\lipsum[1]
|
||||
\item %2b
|
||||
\item %4b
|
||||
\begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)]
|
||||
\item %1di
|
||||
\item %4bi
|
||||
\lipsum[1]
|
||||
\item %1dii
|
||||
\item %4bii
|
||||
\lipsum[1]
|
||||
\item %1diii
|
||||
\item %4biii
|
||||
\lipsum[1]
|
||||
\end{enumerate}
|
||||
|
||||
|
||||
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Reference in New Issue
Block a user