fomatting
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pkirwin
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ce398d08bf
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\newlabel{eq:analytical}{{1}{3}}
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Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (12 pages, 1843727 bytes).
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PS2/doc.tex
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@ -1,6 +1,7 @@
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\documentclass{article}
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\usepackage{graphicx}
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\usepackage{caption}
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\usepackage{setspace}
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\usepackage{listings}
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\usepackage{color}
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@ -22,6 +23,8 @@
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\titleformat{\subsection}[runin]{\normalfont \large \bfseries}
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{\thesubsection}{1em}{}
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\captionsetup[figure]{width=0.75\textwidth}
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\renewcommand{\thesubsection}{\indent(\alph{subsection})}
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\definecolor{dkgreen}{rgb}{0,0.6,0}
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@ -175,10 +178,10 @@ legend({'Numerical','Analytical'},'Location','northwest');
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\item %1b
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\begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)]
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\item %1bi
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The analytical solution is
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The analytical solution is:
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\begin{equation}
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\phi(x) = A sin \left( \frac{n \pi}{L} x \right)
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\phi(x) = A \sin{\left( \frac{n \pi}{L} x \right)}.
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\label{eq:analytical}
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\end{equation}
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@ -196,41 +199,42 @@ legend({'Numerical','Analytical'},'Location','northwest');
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\label{eq:integral_ident}
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\end{equation}
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Given that $sin$ of a real value is always real, we can disregard the norm operation, and directly relate (\ref{eq:analytical})
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Given that the sine of a real value is always real, we can disregard the norm operation, and directly relate (\ref{eq:analytical})
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to the above identity.
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Evaluating the integral gives us the following relationship
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Evaluating the integral gives us the following relationship:
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\begin{equation*}
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\frac{1}{A^2} = \frac{1}{2} L - \cancel{\frac{L}{4n \pi} sin \left( \frac{2n \pi}{L} L \right)} - \cancel{\frac{1}{2} \cdot 0} + \cancel{\frac{L}{4n \pi} sin(0)}
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\frac{1}{A^2} = \frac{1}{2} L - \cancel{\frac{L}{4n \pi} \sin{\left( \frac{2n \pi}{L} L \right)} }
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- \cancel{\frac{1}{2} \cdot 0} + \cancel{\frac{L}{4n \pi} \sin{(0)}}.
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\end{equation*}
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From this, we find:
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\begin{equation*}
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\boxed{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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Starting with the normalization condition for the numerical case:
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\begin{align}
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a\sum_{l=1}^N\left|\phi_l\right|^2 = a \\
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a\sum_{l=1}^N\left|B\sin{\left(\frac{n\pi}{L}x_l\right)}\right|^2 = a,
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a\sum_{\ell=1}^N\left|\phi_l\right|^2 &= a \\
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a\sum_{\ell=1}^N\left|B\sin{\left(\frac{n\pi}{L}x_\ell\right)}\right|^2 &= a, \nonumber
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\label{eq:numerical_norm}
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\end{align}
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recalling that \(x = al\), and allowing \(a \to 0\) while holding \(L\)
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constant implies that \(N \to \infty\), since \(a=\frac{L}{N}\).
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recalling that \(x = a\ell\), and allowing \(a \to 0\), while holding \(L\)
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constant, implies that \(N \to \infty\), since \(a=\frac{L}{N}\).
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An integral is defined as the limit of a Riemann sum as follows:
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\begin{equation}
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\int_c^df(x)\:dx \equiv \lim_{n \to \infty}\sum_{i=1}^n\Delta x\cdot f(x_i).
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\int_c^df(x)\:dx \equiv \lim_{n \to \infty}\sum_{i=1}^n\Delta x\cdot f(x_i),
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\label{eq:Riemann_sum}
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\end{equation}
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where \(\Delta x = \frac{d-c}{n}\) and \(x_i = c + \Delta x \cdot i\). In our case,
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\(n = N\), \(i = l\), \(c = 0\), \(d = L\), and \(\Delta x = a\), \(x_i = x_l\),
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\(n = N\), \(i = \ell\), \(c = 0\), \(d = L\), and \(\Delta x = a\), \(x_i = x_\ell\),
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\(f(x) = \left|B\sin{\left(\frac{n\pi}{L}x\right)}\right|^2\). Therefore we can write
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\begin{equation*}
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\int_0^L \left|B\sin{\left(\frac{n\pi}{L}x\right)}\right|^2\:dx
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= \lim_{N \to \infty}\sum_{l=1}^N a \cdot \left|B\sin{\left(\frac{n\pi}{L}x_l\right)}\right|^2
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= \lim_{N \to \infty}\sum_{\ell=1}^N a \cdot \left|B\sin{\left(\frac{n\pi}{L}x_l\right)}\right|^2
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= a.
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\end{equation*}
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@ -242,7 +246,7 @@ legend({'Numerical','Analytical'},'Location','northwest');
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= \frac{a}{B^2}.
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\end{equation*}
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This means that B must be
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This means that \(B\) must be
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\begin{equation*}
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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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@ -254,25 +258,25 @@ 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 $\phi _\ell = Bsin \left( \frac {n \pi}{L} a \ell \right)$, we can see that
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From the base form of $\phi _\ell = B\sin{\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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$sin(a + B) = sin(a)cos(B) + cos(a)sin(B)$ and
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$sin(a + B) = sin(a)cos(B) + cos(a)sin(B)$, respectively, where
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$\sin{(a + B)} = \sin{(a)}\cos{(B)} + \cos{(a)}\sin{(B)}$ 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 a \ell}{L} \right) + 2 t_0 \phi _{\ell} - t_0 sin \left( \frac{n \pi a \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 $\phi _ \ell = B sin \left( \frac {n \pi}{L} a \ell \right)$,
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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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||||
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||||
\begin{equation}
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||||
\boxed{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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@ -282,7 +286,7 @@ legend({'Numerical','Analytical'},'Location','northwest');
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\centering
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\adjustbox{valign=t}{
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\includegraphics[width=\textwidth]{q1cii.png}
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\includegraphics[width=0.5\textwidth]{q1cii.png}
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}
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\captionof{figure}{Analytic solution to numeric system, plotted.}
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\end{minipage}
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@ -292,24 +296,23 @@ legend({'Numerical','Analytical'},'Location','northwest');
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\item %1ciii
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Applying the approximation $cos(\theta) = 1 - \frac{\theta^2}{2}$ for small $\theta$,
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on equation (\ref{eq:numerical_E}), we get the following expression
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Applying the approximation $\cos{(\theta)} \approx 1 - \frac{\theta^2}{2}$ for small $\theta$
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on equation (\ref{eq:numerical_E}), we get the following expression:
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\begin{equation*}
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E = 2 t_0 \left( \frac{n^2 \pi^2 a^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 analytical expression for $E$:
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We can get our final analytical expression for $E$ by fully substituting the explicit form of $t_0$:
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||||
\begin{equation*}
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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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\item %1civ
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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
|
||||
now a constant-amplitude wave, which corresponds to the expected analytic result - in contrast to the
|
||||
With the decreased lattice spacing and increased number of points we can see the numerical solution
|
||||
more closely matches the analytical solution. As well, the \(n=50\) case is
|
||||
now a constant-amplitude wave, which corresponds to the expected analytic result, in contrast to the
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||||
plot in section (a), which has a low-frequency envelope around it.
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\begin{minipage}[t]{\linewidth}
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@ -319,13 +322,16 @@ legend({'Numerical','Analytical'},'Location','northwest');
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||||
\begin{subfigure}{0.5\textwidth}
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||||
\centering
|
||||
\includegraphics[width=\textwidth]{q1civ_fig1.png}
|
||||
\caption{Probability densities for \(n=1\)\\and \(n=50\).}
|
||||
\caption{}
|
||||
\end{subfigure}%
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||||
\begin{subfigure}{0.5\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{q1civ_fig2.png}
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||||
\caption{Comparison of first 101 numerical and analytic eigenvalues.}
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||||
\caption{}
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||||
\end{subfigure}
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||||
|
||||
\caption{(a) Probability densities for \(n=1\) and \(n=50\). (b) Comparison of first 101 numerical and analytic eigenvalues.}
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||||
|
||||
\end{figure}
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||||
\end{minipage}
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||||
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||||
@ -372,7 +378,7 @@ H(N, 1) = -t0;
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\begin{minipage}[t]{\linewidth}
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||||
\centering
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||||
\adjustbox{valign=t}{
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||||
\includegraphics[width=\textwidth]{q1div.jpg}
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||||
\includegraphics[width=0.5\textwidth]{q1div.jpg}
|
||||
}
|
||||
\captionof{figure}{Sketch of degenerate energy levels.}
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||||
\end{minipage}
|
||||
@ -380,8 +386,8 @@ H(N, 1) = -t0;
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||||
\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
|
||||
by the requisite $q$), we get energy levels of \(0\) eV, \(0.0147\) eV, and \(0.0589\) eV for the
|
||||
indices \(n=0\), \(1\), and \(2\), respectively. These match very closely, to within
|
||||
acceptable margin of the numerical results from part (ii).
|
||||
|
||||
|
||||
@ -412,6 +418,7 @@ H(N, 1) = -t0;
|
||||
clear all;
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||||
%physical constants in MKS units
|
||||
|
||||
|
||||
hbar = 1.054e-34;
|
||||
q = 1.602e-19;
|
||||
m = 9.110e-31;
|
||||
@ -475,13 +482,14 @@ axis([0 1e-9 0 0.04]);
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||||
\begin{subfigure}{0.5\textwidth}
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||||
\centering
|
||||
\includegraphics[width=\textwidth]{q2c_fig1_1s.png}
|
||||
\caption{1s probability density.}
|
||||
\caption{}
|
||||
\end{subfigure}%
|
||||
\begin{subfigure}{0.5\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{q2c_fig2_2s.png}
|
||||
\caption{2s probability density.}
|
||||
\caption{}
|
||||
\end{subfigure}
|
||||
\caption{(a) 1s probability density. (b) 2s probability density.}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
\item %2d
|
||||
@ -515,7 +523,7 @@ axis([0 1e-9 0 0.04]);
|
||||
which corresponds to the analytical result for the energy being slightly greater in magnitude (\(-13.6\) eV versus \(-13.4978\) eV).
|
||||
\begin{minipage}[t]{\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{q2f_fig.png}
|
||||
\includegraphics[width=0.5\textwidth]{q2f_fig.png}
|
||||
\captionof{figure}{Numerical result (black line) and analytical solution scaled by \(a\) (orange circles).}
|
||||
\end{minipage}
|
||||
|
||||
|
||||
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