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This commit is contained in:
pkirwin
parent
0ad1f62bed
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@ -1,4 +1,5 @@
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\relax
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\providecommand \oddpage@label [2]{}
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\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
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\newlabel{fig:q3_ne}{{1a}{3}}
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\newlabel{sub@fig:q3_ne}{{a}{3}}
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@ -12,6 +13,6 @@
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\newlabel{eq:Riemann_sum}{{6}{4}}
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\newlabel{eq:numerical_E}{{7}{4}}
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Analytic solution to numeric system, plotted.\relax }}{5}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Sketch of appropriate energy levels\relax }}{7}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Numerical result (black line) and analytical solution scaled by \(a\) (orange circles).\relax }}{10}{}\protected@file@percent }
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\gdef \@abspage@last{13}
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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Sketch of degenerate energy levels.\relax }}{7}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Numerical result (black line) and analytical solution scaled by \(a\) (orange circles).\relax }}{11}{}\protected@file@percent }
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\gdef \@abspage@last{14}
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@ -1,8 +1,8 @@
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# Fdb version 3
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92
PS2/doc.fls
92
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 (13 pages, 3715189 bytes).
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Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (14 pages, 1845313 bytes).
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@ -17,6 +17,7 @@
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\usepackage{enumitem}
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\usepackage{cancel}
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\usepackage{siunitx}
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\usepackage{adjustbox}
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\titleformat{\subsection}[runin]{\normalfont \large \bfseries}
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{\thesubsection}{1em}{}
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@ -77,7 +78,7 @@
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\thispagestyle{empty}
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\singlespacing
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\newpage
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hello
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\section*{Problem 1}
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\begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)]
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@ -203,7 +204,7 @@ legend({'Numerical','Analytical'},'Location','northwest');
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From this, we find:
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\begin{equation*}
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A = \sqrt{\frac{2}{L}}
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\boxed{A = \sqrt{\frac{2}{L}}}
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\end{equation*}
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\item %1bii
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@ -240,7 +241,7 @@ legend({'Numerical','Analytical'},'Location','northwest');
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This means that B must be
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\begin{equation*}
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B = \sqrt{\frac{2 a}{L}} = \sqrt{a} \times A.
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\boxed{B = \sqrt{\frac{2 a}{L}} = \sqrt{a} \times A.}
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\end{equation*}
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\end{enumerate}
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@ -268,20 +269,20 @@ legend({'Numerical','Analytical'},'Location','northwest');
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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 a}{L} \right) \right)
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\boxed{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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\item %1cii
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We can see here that the "predicted" numerical response matches nearly exactly the actual
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calculated numerical solution.
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\begin{figure}[H]
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\centering
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\includegraphics[width=\textwidth]{q1cii.png}
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\caption{Analytic solution to numeric system, plotted.}
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\end{figure}
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We can see here that the "predicted" numerical response matches nearly exactly the actual
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calculated numerical solution.
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\item %1ciii
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@ -292,14 +293,18 @@ legend({'Numerical','Analytical'},'Location','northwest');
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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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Fully substituting the known value of $t_0$, and we can get our final analytical expression for $E$:
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|
||||
\begin{equation*}
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E = \frac{ \hbar^2 n^2 \pi^2 }{ 2 m L^2 }
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\boxed{E = \frac{ \hbar^2 n^2 \pi^2 }{ 2 m L^2 }}
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||||
\end{equation*}
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||||
|
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\item %1civ
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||||
|
||||
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
|
||||
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.
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\begin{subfigure}{0.5\textwidth}
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@ -314,11 +319,7 @@ legend({'Numerical','Analytical'},'Location','northwest');
|
||||
\end{subfigure}
|
||||
\end{figure}
|
||||
|
||||
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
|
||||
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}
|
||||
\item %1d
|
||||
@ -350,17 +351,19 @@ H(N, 1) = -t0;
|
||||
|
||||
\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 \boxed{0.06\:\mathrm{eV}}.
|
||||
These eigenstates are degenerate because they both have the same
|
||||
eigenvalue/energy (0.06).
|
||||
eigenvalue/energy.
|
||||
|
||||
\item %1diii
|
||||
|
||||
\begin{figure}[H]
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||||
\begin{minipage}[t]{\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth, angle=-90]{q1div.jpg}
|
||||
\caption{Sketch of appropriate energy levels}
|
||||
\end{figure}
|
||||
\adjustbox{valign=t}{
|
||||
\includegraphics[width=\textwidth]{q1div.jpg}
|
||||
}
|
||||
\captionof{figure}{Sketch of degenerate energy levels.}
|
||||
\end{minipage}
|
||||
|
||||
\item %1div
|
||||
|
||||
@ -374,7 +377,7 @@ H(N, 1) = -t0;
|
||||
|
||||
\end{enumerate}
|
||||
|
||||
|
||||
\newpage
|
||||
\section*{Problem 2}
|
||||
|
||||
\begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)]
|
||||
@ -383,14 +386,14 @@ H(N, 1) = -t0;
|
||||
Since \(t_0 = \frac{\hbar^2}{2ma^2}\), and \(U_l = -\frac{q^2}{4\pi\varepsilon_0r_l}
|
||||
+ \frac{l_o(l_o + 1)\hbar^2}{2mr_l^2}\), the middle diagonal elements will have values
|
||||
\begin{equation*}
|
||||
\hat{H}_{ll} = \frac{\hbar^2}{ma^2}-\frac{q^2}{4\pi\epsilon_0r_l} + \frac{l_o(l_o + 1)\hbar^2}{2mr_l^2},
|
||||
\boxed{\hat{H}_{ll} = \frac{\hbar^2}{ma^2}-\frac{q^2}{4\pi\epsilon_0r_l} + \frac{l_o(l_o + 1)\hbar^2}{2mr_l^2},}
|
||||
\end{equation*}
|
||||
and the upper and lower diagonals elements will have values
|
||||
\begin{equation*}
|
||||
\hat{H}_{l(l\pm1)} = -\frac{\hbar^2}{2ma^2}.
|
||||
\boxed{\hat{H}_{l(l\pm1)} = -\frac{\hbar^2}{2ma^2}.}
|
||||
\end{equation*}
|
||||
\item %2b
|
||||
Homogenous boundary conditions imply that the corner entries of \(\hat{H}\) will be 0.
|
||||
Homogenous boundary conditions imply that the corner entries of \(\hat{H}\) will be \boxed{0}.
|
||||
\item %2c
|
||||
Code:
|
||||
\begin{lstlisting}
|
||||
@ -468,7 +471,7 @@ axis([0 1e-9 0 0.04]);
|
||||
\end{subfigure}
|
||||
\end{figure}
|
||||
\item %2d
|
||||
For the 1s level, \(E = -13.4978\) eV.
|
||||
For the 1s level, \boxed{E = -13.4978\:\mathrm{eV}}.
|
||||
\item %2e
|
||||
Beginning with equation (11) from the assignment, with \(l_o=0\):
|
||||
\begin{align*}
|
||||
@ -487,7 +490,7 @@ axis([0 1e-9 0 0.04]);
|
||||
We can then solve for \(E\):
|
||||
\begin{equation*}
|
||||
E\:\mathrm{[eV]} = -\frac{1}{q}\cdot\frac{\hbar^2}{2ma_0^2} = -\frac{1}{q}\cdot\frac{(\SI{1.054e-34}{\joule\cdot\second})^2}{2(\SI{9.110e-31}{\kilogram})(\SI{0.0529}{\nm})^2}
|
||||
= - 13.6\:\mathrm{[eV]}
|
||||
= \boxed{- 13.6\:\mathrm{eV}.}
|
||||
\end{equation*}
|
||||
This is very similar to the result in (d).
|
||||
|
||||
@ -505,7 +508,8 @@ axis([0 1e-9 0 0.04]);
|
||||
|
||||
|
||||
\end{enumerate}
|
||||
|
||||
|
||||
\newpage
|
||||
\section*{Problem 3}
|
||||
|
||||
\begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)]
|
||||
@ -540,7 +544,7 @@ axis([0 1e-9 0 0.04]);
|
||||
\lipsum[1]
|
||||
|
||||
\end{enumerate}
|
||||
|
||||
\newpage
|
||||
\section*{Problem 4}
|
||||
|
||||
\begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)]
|
||||
|
||||
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