Merge branch 'master' of github.com:davidlenfesty/ECE_456
This commit is contained in:
commit
b71ba2a282
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PS2/doc.aux
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\relax
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\gdef \@abspage@last{7}
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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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\newlabel{fig:q3_current}{{1b}{3}}
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\newlabel{sub@fig:q3_current}{{b}{3}}
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\newlabel{eq:analytical}{{1}{3}}
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\newlabel{eq:normalise}{{2}{3}}
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\newlabel{eq:integral_ident}{{3}{3}}
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\newlabel{eq:numerical_E}{{4}{4}}
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Analytic solution to numeric system, plotted.\relax }}{4}{}\protected@file@percent }
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\newlabel{fig:q1b_electrons}{{2}{4}}
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\gdef \@abspage@last{9}
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@ -1,9 +1,12 @@
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# Fdb version 3
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["pdflatex"] 1614634922 "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf" "doc" 1614634924
|
||||
["pdflatex"] 1614640686 "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf" "doc" 1614640702
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"C:/Users/speed/AppData/Local/MiKTeX/fonts/pk/ljfour/jknappen/ec/dpi600/tcrm1000.pk" 1614620319 11548 693fc704971bedc6e913b37b081cf40e ""
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"C:/Users/speed/AppData/Local/MiKTeX/miktex/data/le/pdftex/pdflatex.fmt" 1612675286 9293900 83195ab665706b6f706823352908aa95 ""
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@ -16,19 +19,32 @@
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"C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmmi12.tfm" 1136765053 1524 4414a8315f39513458b80dfc63bff03a ""
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"C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmmi9.tfm" 1136765053 1524 d89e2d087a9828407a196f428428ef4a ""
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"C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1247593067 32569 5e5ddc8df908dea60932f3c484a54c0d ""
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"C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy7.pfb" 1247593066 32716 08e384dc442464e7285e891af9f45947 ""
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@ -44,7 +60,9 @@
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"C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/base/size10.clo" 1610120746 8449 a72d5d4e612221b46000c3d71724e0ef ""
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"C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/caption/caption.sty" 1603736996 51746 e89c4da670ba533e6ab38e045ce6d1d9 ""
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"C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/caption/caption3.sty" 1603736996 67929 d50ae850e1d9604065e8337a75a41029 ""
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"C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/caption/subcaption.sty" 1603736996 6133 46bea384e36e9b74bd53a238fc2d0654 ""
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@ -77,13 +95,15 @@
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"C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/parskip/parskip.sty" 1592833306 4414 25631f9679adf750bac898538ededd61 ""
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"C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/setspace/setspace.sty" 1324294077 22913 a27d7908fc6f0385466454a966a316eb ""
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"doc.aux" 1614634924 32 4d1825978335c8847f72f7c473f9edc8 "pdflatex"
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"doc.tex" 1614634921 5526 6265d6def36657f58ffe0e5a11340609 ""
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"d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.aux" 1614640688 560 a7f45c1ba582090fea381768d02972d1 ""
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"doc.aux" 1614640688 560 a7f45c1ba582090fea381768d02972d1 "pdflatex"
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(generated)
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"doc.log"
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"d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.log"
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"d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf"
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"doc.pdf"
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"doc.log"
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"doc.aux"
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62
PS2/doc.fls
62
PS2/doc.fls
@ -416,6 +416,17 @@ INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\enumitem\enumitem.s
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\enumitem\enumitem.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\enumitem\enumitem.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\enumitem\enumitem.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\listings\lstlang1.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\listings\lstlang1.sty
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@ -471,24 +482,71 @@ INPUT C:\Users\speed\AppData\Local\MiKTeX\pdftex\config\pdftex.map
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\base\omscmr.fd
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\base\omscmr.fd
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\base\omscmr.fd
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INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_1and50.png
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INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_1and50.png
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INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_1and50.png
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INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_1and50.png
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INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_1and50.png
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INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_1and50.png
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INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_eigenvals.png
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INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_eigenvals.png
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INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_eigenvals.png
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INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_eigenvals.png
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This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 21.1) (preloaded format=pdflatex 2021.2.6) 1 MAR 2021 14:42
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This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 21.1) (preloaded format=pdflatex 2021.2.6) 1 MAR 2021 16:18
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entering extended mode
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**d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex
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File: lstlang1.sty 2020/03/24 1.8d listings language file
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@ -14,8 +14,8 @@
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\usepackage{fancyhdr}
|
||||
\usepackage{titlesec}
|
||||
\usepackage{lipsum}
|
||||
|
||||
\usepackage{enumitem}
|
||||
\usepackage{cancel}
|
||||
|
||||
\titleformat{\subsection}[runin]{\normalfont \large \bfseries}
|
||||
{\thesubsection}{1em}{}
|
||||
@ -46,7 +46,7 @@
|
||||
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=[2]{word1,word2}, emphstyle=[2]{style},
|
||||
xleftmargin=5em
|
||||
xleftmargin=2em
|
||||
}
|
||||
%\lstset{basicstyle=\small,
|
||||
% keywordstyle=\color{mauve},
|
||||
@ -81,28 +81,212 @@
|
||||
|
||||
\begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)]
|
||||
\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
|
||||
\begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)]
|
||||
\item %1bi
|
||||
The analytical solution is
|
||||
|
||||
\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}
|
||||
|
||||
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
|
||||
\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}
|
||||
|
||||
\item %1c
|
||||
\begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)]
|
||||
\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
|
||||
\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
|
||||
\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
|
||||
\lipsum[1]
|
||||
\end{enumerate}
|
||||
|
||||
57
PS2/q1a.asv
Normal file
57
PS2/q1a.asv
Normal file
@ -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]);
|
||||
|
||||
62
PS2/q1a.m
62
PS2/q1a.m
@ -1,4 +1,4 @@
|
||||
|
||||
clear all;
|
||||
%physical constants in MKS units
|
||||
|
||||
hbar = 1.054e-34;
|
||||
@ -7,7 +7,59 @@ m = 9.110e-31;
|
||||
|
||||
%generate lattice
|
||||
|
||||
N = 100; %number of lattice points
|
||||
n = [1:N]; %lattice points
|
||||
a = 1e-10; %lattice constant
|
||||
x = a * [1:1:N];
|
||||
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');
|
||||
|
||||
|
||||
|
||||
BIN
PS2/q1a_1and50.png
Normal file
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PS2/q1a_1and50.png
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|
After Width: | Height: | Size: 33 KiB |
BIN
PS2/q1a_eigenvals.png
Normal file
BIN
PS2/q1a_eigenvals.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 22 KiB |
Loading…
Reference in New Issue
Block a user