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Merge branch 'master' of github.com:davidlenfesty/ECE_456

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David Lenfesty 2021-03-01 16:39:45 -07:00
commit b71ba2a282
11 changed files with 436 additions and 35 deletions
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13
PS2/doc.aux
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\relax \relax
\gdef \@abspage@last{7} \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
\newlabel{fig:q3_ne}{{1a}{3}}
\newlabel{sub@fig:q3_ne}{{a}{3}}
\newlabel{fig:q3_current}{{1b}{3}}
\newlabel{sub@fig:q3_current}{{b}{3}}
\newlabel{eq:analytical}{{1}{3}}
\newlabel{eq:normalise}{{2}{3}}
\newlabel{eq:integral_ident}{{3}{3}}
\newlabel{eq:numerical_E}{{4}{4}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Analytic solution to numeric system, plotted.\relax }}{4}{}\protected@file@percent }
\newlabel{fig:q1b_electrons}{{2}{4}}
\gdef \@abspage@last{9}

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\usepackage{fancyhdr} \usepackage{fancyhdr}
\usepackage{titlesec} \usepackage{titlesec}
\usepackage{lipsum} \usepackage{lipsum}
\usepackage{enumitem} \usepackage{enumitem}
\usepackage{cancel}
\titleformat{\subsection}[runin]{\normalfont \large \bfseries} \titleformat{\subsection}[runin]{\normalfont \large \bfseries}
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@ -46,7 +46,7 @@
numbersep=9pt, % this defines how far the numbers are from the text numbersep=9pt, % this defines how far the numbers are from the text
emph=[1]{for,end,break},emphstyle=[1]\color{red}, %some words to emphasise emph=[1]{for,end,break},emphstyle=[1]\color{red}, %some words to emphasise
%emph=[2]{word1,word2}, emphstyle=[2]{style}, %emph=[2]{word1,word2}, emphstyle=[2]{style},
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%\lstset{basicstyle=\small, %\lstset{basicstyle=\small,
% keywordstyle=\color{mauve}, % keywordstyle=\color{mauve},
@ -81,28 +81,212 @@
\begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)] \begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)]
\item %1a \item %1a
\lipsum[1] Code:
\begin{lstlisting}[language=Matlab]
clear all;
%physical constants in MKS units
hbar = 1.054e-34;
q = 1.602e-19;
m = 9.110e-31;
%generate lattice
N = 100; %number of lattice points
n = [1:N]; %lattice points
a = 1e-10; %lattice constant
x = a * n; %x-coordinates
t0 = (hbar^2)/(2*m*a^2)/q; %encapsulating factor
L = a * (N+1); %total length of consideration
%set up Hamiltonian matrix
U = 0*x; %0 potential at all x
main_diag = diag(2*t0*ones(1,N)+U,0); %create main diagonal matrix
lower_diag = diag(-t0*ones(1,N-1),-1); %create lower diagonal matrix
upper_diag = diag(-t0*ones(1,N-1),+1); %create upper diagonal matrix
H = main_diag + lower_diag + upper_diag; %sum to get Hamiltonian matrix
[eigenvectors,E_diag] = eig(H); %"eigenvectors" is a matrix wherein each
%column is an eigenvector
%"E_diag" is a diagonal matrix where the
%corresponding eigenvalues are on the
%diagonal.
E_col = diag(E_diag); %folds E_diag into a column vector of eigenvalues
% return eigenvectors for the 1st and 50th eigenvalues
phi_1 = eigenvectors(:,1);
phi_50 = eigenvectors(:,50);
% find the probability densities of position for 1st and 50th eigenvectors
P_1 = phi_1 .* conj(phi_1);
P_50 = phi_50 .* conj(phi_50);
% Find first N analytic eigenvalues
E_col_analytic = (1/q) * (hbar^2 * pi^2 * n.*n) / (2*m*L^2);
% Plot the probability densities for 1st and 50th eigenvectors
figure(1); clf; h = plot(x,P_1,'kx',x,P_50,'k-');
grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
xlabel('POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]');
legend('n=1','n=50');
% Plot numerical eigenvalues
figure(2); clf; h = plot(n,E_col,'kx'); grid on;
set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
xlabel('EIGENVALUE NUMBER'); ylabel('ENERGY [eV]');
axis([0 100 0 40]);
% Add analytic eigenvalues to above plot
hold on;
plot(n,E_col_analytic,'k-');
legend({'Numerical','Analytical'},'Location','northwest');
\end{lstlisting}
%
\begin{figure}[H]
\centering
\begin{subfigure}{0.4\linewidth}
\centering
\includegraphics[width=\textwidth]{q1a_1and50.png}
\caption{probability densities for \(n=1\)\\and \(n=50\).}
\label{fig:q3_ne}
\end{subfigure}%
\begin{subfigure}{0.4\linewidth}
\centering
\includegraphics[width=\textwidth]{q1a_eigenvals.png}
\caption{Comparison of first 101 numerical and analytic eigenvalues.}
\label{fig:q3_current}
\end{subfigure}
\end{figure}
\item %1b \item %1b
\begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)] \begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)]
\item %1bi \item %1bi
The analytical solution is The analytical solution is
\begin{equation} \begin{equation}
\Phi(x) = A sin (\frac{n \pi}{L} x) \phi(x) = A sin \left( \frac{n \pi}{L} x \right)
\label{eq:analytical}
\end{equation} \end{equation}
In order to normalise this equation it must conform to the following:
\begin{equation}
\int_{0}^{L} \left| \phi(x) \right| ^2 dx = 1.
\label{eq:normalise}
\end{equation}
We use the following identity:
\begin{equation}
\int\sin^2{(\alpha x)}\:dx = \frac{1}{2} x -\frac{1}{4\alpha}\sin{(2\alpha x)}.
\label{eq:integral_ident}
\end{equation}
Given that $sin$ of a real value is always real, we can disregard the norm operation, and directly relate (\ref{eq:analytical})
to the above identity.
Evaluating the integral gives us the following relationship
\begin{equation*}
\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)}
\end{equation*}
From this, we find:
\begin{equation*}
A = \sqrt{\frac{2}{L}}
\end{equation*}
\item %1bii \item %1bii
\lipsum[1]
Starting with the normalization condition for the numerical case:
\begin{align*}
a\sum_{l=1}^N|\phi_l|^2 = a \\
a\sum_{l=1}^N|B\sin{(\frac{n\pi}{L}x_l)}|^2 = a,
\end{align*}
recalling that \(x = al\), and allowing \(a \to 0\) while holding \(L\)
constant implies that \(N \to \infty\), since \(a=\frac{L}{N}\).
An integral is defined as the limit of a Riemann sum as follows:
\begin{equation*}
\int_a^bf(x)\:dx = \lim_{n \to \infty}\sum_{i=1}^n\Delta x\cdot f(x_i).
\end{equation*}
The output of the numerical sum will change, as the value of $\alpha$ changes, by a factor of $\alpha$.
To correct for this, we need to factor the result of the summation by $\alpha$, which corresponds to
% Idk how to explain this at all, I need to chew on it internally and come back later.
\begin{equation*}
foo
\end{equation*}
This means that B must be
\begin{equation*}
B = \sqrt{\frac{2 \alpha}{L}}
\end{equation*}
\end{enumerate} \end{enumerate}
\item %1c \item %1c
\begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)] \begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)]
\item %1ci \item %1ci
\lipsum[1]
% TODO check that B is expressed correctly in all these equations (alpha, not ell, wait for pdf gen)
From the base form of $\phi _\ell = Bsin \left( \frac {n \pi}{L} \alpha \ell \right)$, we can see that
$\phi _ {\ell + 1}$ and $\phi _ {\ell - 1}$ correspond to the trigonometric identities
$sin(\alpha + B) = sin(\alpha)cos(B) + cos(\alpha)sin(B)$ and
$sin(\alpha + B) = sin(\alpha)cos(B) + cos(\alpha)sin(B)$, respectively, where
$\alpha = \frac{n \pi \alpha \ell}{L}$ and $B = \frac{n \pi \alpha}{L}$.
Plugging these identities into equation (7) from the assignment and simplifying, we get to this equation:
\begin{equation*}
-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)
\end{equation*}
At this point, we notice that $\phi _ \ell = B sin \left( \frac {n \pi}{L} \alpha \ell \right)$,
so we can factor it out.
With some minor rearranging, this leaves us with the final expression for $E$:
\begin{equation}
E = 2t_0 \left( 1 - cos \left( \frac{n \pi \alpha}{L} \right) \right)
\label{eq:numerical_E}
\end{equation}
\item %1cii \item %1cii
\lipsum[1]
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{q1cii.png}
\caption{Analytic solution to numeric system, plotted.}
\label{fig:q1b_electrons}
\end{figure}
We can see here that the "predicted" numerical response matches nearly exactly the actual
calculated numerical solution.
\item %1ciii \item %1ciii
\lipsum[1]
Applying the approximation $cos(\Theta) = 1 - \frac{\Theta^2}{2}$ for small $\Theta$,
on equation (\ref{eq:numerical_E}), we get the following expression
\begin{equation*}
E = 2 t_0 \left( \frac{n^2 \pi^2 \alpha^2}{2 L^2} \right)
\end{equation*}
Fully substituting the known value of $t_0$, and we can get our final expression for $E$:
\begin{equation*}
E = \frac{ \hbar^2 n^2 \pi^2 }{ 2 m L^2 }
\end{equation*}
\item %1civ \item %1civ
\lipsum[1] \lipsum[1]
\end{enumerate} \end{enumerate}

57
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@ -0,0 +1,57 @@
clear all;
%physical constants in MKS units
hbar = 1.054e-34;
q = 1.602e-19;
m = 9.110e-31;
%generate lattice
N = 100; %number of lattice points
n = [1:N]; %lattice points
a = 1e-10; %lattice constant
x = a * n; %x-coordinates
t0 = (hbar^2)/(2*m*a^2)/q; %encapsulating factor
L = a * (N+1); %total length of consideration
%set up Hamiltonian matrix
U = 0*x; %0 potential at all x
main_diag = diag(2*t0*ones(1,N)+U,0); %create main diagonal matrix
lower_diag = diag(-t0*ones(1,N-1),-1); %create lower diagonal matrix
upper_diag = diag(-t0*ones(1,N-1),+1); %create upper diagonal matrix
H = main_diag + lower_diag + upper_diag; %sum to get Hamiltonian matrix
[eigenvectors,E_diag] = eig(H); %"eigenvectors" is a matrix wherein each column is an eigenvector
%"E_diag" is a diagonal matrix where the
%corresponding eigenvalues are on the
%diagonal.
E_col = diag(E_diag); %folds E_diag into a column vector of eigenvalues
% return eigenvectors for the 1st and 50th eigenvalues
phi_1 = eigenvectors(:,1);
phi_50 = eigenvectors(:,50);
% find the probability densities of position for 1st and 50th eigenvectors
P_1 = phi_1 .* conj(phi_1);
P_50 = phi_50 .* conj(phi_50);
% Find first N analytic eigenvalues
E_col_analytic = (1/q) * (hbar^2 * pi^2 * n.*n) / (
% Plot the probability densities for 1st and 50th eigenvectors
figure(1); clf; h = plot(x,P_1,'kx',x,P_50,'k-');
grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
xlabel('POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]');
legend('n=1','n=50');
% Plot numerical eigenvalues
figure(2); clf; h = plot(n,E_col,'kx'); grid on;
set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
xlabel('EIGENVALUE NUMBER'); ylabel('ENERGY [eV]');
axis([0 100 0 40]);

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@ -1,4 +1,4 @@
clear all;
%physical constants in MKS units %physical constants in MKS units
hbar = 1.054e-34; hbar = 1.054e-34;
@ -7,7 +7,59 @@ m = 9.110e-31;
%generate lattice %generate lattice
N = 100; %number of lattice points N = 100; %number of lattice points
n = [1:N]; %lattice points n = [1:N]; %lattice points
a = 1e-10; %lattice constant a = 1e-10; %lattice constant
x = a * [1:1:N]; x = a * n; %x-coordinates
t0 = (hbar^2)/(2*m*a^2)/q; %encapsulating factor
L = a * (N+1); %total length of consideration
%set up Hamiltonian matrix
U = 0*x; %0 potential at all x
main_diag = diag(2*t0*ones(1,N)+U,0); %create main diagonal matrix
lower_diag = diag(-t0*ones(1,N-1),-1); %create lower diagonal matrix
upper_diag = diag(-t0*ones(1,N-1),+1); %create upper diagonal matrix
H = main_diag + lower_diag + upper_diag; %sum to get Hamiltonian matrix
[eigenvectors,E_diag] = eig(H); %"eigenvectors" is a matrix wherein each column is an eigenvector
%"E_diag" is a diagonal matrix where the
%corresponding eigenvalues are on the
%diagonal.
E_col = diag(E_diag); %folds E_diag into a column vector of eigenvalues
% return eigenvectors for the 1st and 50th eigenvalues
phi_1 = eigenvectors(:,1);
phi_50 = eigenvectors(:,50);
% find the probability densities of position for 1st and 50th eigenvectors
P_1 = phi_1 .* conj(phi_1);
P_50 = phi_50 .* conj(phi_50);
% Find first N analytic eigenvalues
E_col_analytic = (1/q) * (hbar^2 * pi^2 * n.*n) / (2*m*L^2);
% Plot the probability densities for 1st and 50th eigenvectors
figure(1); clf; h = plot(x,P_1,'kx',x,P_50,'k-');
grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
xlabel('POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]');
legend('n=1','n=50');
% Plot numerical eigenvalues
figure(2); clf; h = plot(n,E_col,'kx'); grid on;
set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
xlabel('EIGENVALUE NUMBER'); ylabel('ENERGY [eV]');
axis([0 100 0 40]);
% Add analytic eigenvalues to above plot
hold on;
plot(n,E_col_analytic,'k-');
legend({'Numerical','Analytical'},'Location','northwest');

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