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pkirwin
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@ -14,4 +14,4 @@
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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{12}
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\gdef \@abspage@last{13}
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@ -1,8 +1,8 @@
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# Fdb version 3
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["pdflatex"] 1614658426 "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" 1614658632
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@ -86,8 +94,10 @@
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@ -96,11 +106,15 @@
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"doc.aux" 1614655259 1163 a0354534513b31cd9af2bee0ca4d6d1a "pdflatex"
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@ -113,8 +127,8 @@
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(generated)
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"d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.log"
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"doc.pdf"
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"doc.aux"
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"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.aux"
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"d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.log"
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"doc.pdf"
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111
PS2/doc.fls
111
PS2/doc.fls
@ -427,6 +427,56 @@ 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\siunitx\siunitx.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\l3packages\xparse\xparse.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\latex-tools\array.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\latex-tools\array.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\latex-tools\array.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\l3packages\l3keys2e\l3keys2e.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\l3packages\l3keys2e\l3keys2e.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\l3packages\l3keys2e\l3keys2e.sty
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx-abbreviations.cfg
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx-abbreviations.cfg
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INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx-abbreviations.cfg
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BIN
PS2/doc.pdf
BIN
PS2/doc.pdf
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32
PS2/doc.tex
32
PS2/doc.tex
@ -16,6 +16,7 @@
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\usepackage{lipsum}
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\usepackage{enumitem}
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\usepackage{cancel}
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\usepackage{siunitx}
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\titleformat{\subsection}[runin]{\normalfont \large \bfseries}
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{\thesubsection}{1em}{}
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@ -284,7 +285,7 @@ legend({'Numerical','Analytical'},'Location','northwest');
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\item %1ciii
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||||
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Applying the approximation $cos(\Theta) = 1 - \frac{\Theta^2}{2}$ for small $\Theta$,
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Applying the approximation $cos(\theta) = 1 - \frac{\theta^2}{2}$ for small $\theta$,
|
||||
on equation (\ref{eq:numerical_E}), we get the following expression
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||||
\begin{equation*}
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@ -469,7 +470,27 @@ axis([0 1e-9 0 0.04]);
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\item %2d
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For the 1s level, \(E = -13.4978\) eV.
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\item %2e
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\lipsum[1]
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||||
Beginning with equation (11) from the assignment, with \(l_o=0\):
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||||
\begin{align*}
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\left[-\frac{\hbar^2}{2m}\frac{d^2}{dr^2} - \frac{q^2}{4\pi\epsilon_0r}\right]f(r) &= Ef(r)\\
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||||
-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}\left(\frac{2r}{a_0^{3/2}}e^{-r/a_0}\right) - \frac{q^2}{4\pi\epsilon_0r}f(r) &= Ef(r)\\
|
||||
-\frac{\hbar^2}{2m}\frac{2}{a_0^{3/2}}\frac{d}{dr}\left(e^{-r/a_0} - \frac{r}{a_0}e^{-r/a_0}\right) - \frac{q^2}{4\pi\epsilon_0r}f(r) &= Ef(r)\\
|
||||
-\frac{\hbar^2}{2m}\frac{2}{a_0^{3/2}}\left(-\frac{1}{a_0}e^{-r/a_0} - \frac{1}{a_0}e^{-r/a_0} + \frac{r}{a_0^2}e^{-r/a_0}\right) - \frac{q^2}{4\pi\epsilon_0r}f(r) &= Ef(r)\\
|
||||
-\frac{\hbar^2}{2m}\left(-\frac{2}{a_0r} + \frac{1}{a_0^2}\right)f(r) - \frac{q^2}{4\pi\epsilon_0r}f(r) &= Ef(r)\\
|
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-\frac{\hbar^2}{2m}\left(-\frac{2}{a_0r} + \frac{1}{a_0^2}\right) - \frac{q^2}{4\pi\epsilon_0r} &= E.
|
||||
\end{align*}
|
||||
Recalling that \(a_0 = 4\pi\epsilon_0\hbar^2/mq^2\), we can eliminate \(r\):
|
||||
\begin{align*}
|
||||
\cancel{\frac{\hbar^2}{2m}}\frac{\cancel{2m}q^2}{4\pi\epsilon_0\cancel{\hbar^2}}\frac{1}{r} - \frac{\hbar^2}{2m}\frac{1}{a_0^2} - \frac{q^2}{4\pi\epsilon_0}\frac{1}{r} &= E\\
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||||
\cancel{\frac{q^2}{4\pi\epsilon_0}\frac{1}{r}} - \frac{\hbar^2}{2m}\frac{1}{a_0^2} - \cancel{\frac{q^2}{4\pi\epsilon_0}\frac{1}{r}} &= E.
|
||||
\end{align*}
|
||||
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]}
|
||||
\end{equation*}
|
||||
This is very similar to the result in (d).
|
||||
|
||||
\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
|
||||
@ -490,8 +511,13 @@ axis([0 1e-9 0 0.04]);
|
||||
\begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)]
|
||||
|
||||
\item %3a
|
||||
\lipsum[1]
|
||||
|
||||
\begin{equation}
|
||||
\theta (x') = \sqrt{\frac{2}{L}} sin \left( \frac{\pi (x' + L/2)}{L} \right) = \sqrt{\frac{2}{L}} cos \left( \frac{\pi x'}{L} \right)
|
||||
\end{equation}
|
||||
|
||||
\item %3b
|
||||
|
||||
\begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)]
|
||||
\item %3bi
|
||||
\lipsum[1]
|
||||
|
||||
74
PS2/q4a.m
Normal file
74
PS2/q4a.m
Normal file
@ -0,0 +1,74 @@
|
||||
clear all;
|
||||
|
||||
% Physical constants in MKS units, including free-electron mass m
|
||||
|
||||
hbar = 1.054e-34;
|
||||
q = 1.602e-19;
|
||||
m = 9.110e-31;
|
||||
|
||||
% SHO frequency
|
||||
|
||||
w0 = (0.2879*q/hbar);
|
||||
|
||||
% Lattice data: lattice constant "a"; SHO distance parameter beta;
|
||||
% a discretized spatial grid x; total number of lattice points N;
|
||||
% Hamiltonian parameter t0, as introduced in class, but now
|
||||
% divided by q so that it has units of eV; total length of the lattice,
|
||||
% L = a*(N+1)
|
||||
|
||||
% YOU MUST ENTER AN APPROPRIATE VALUE OF "a"
|
||||
|
||||
a = ;
|
||||
beta = sqrt(m*w0/hbar);
|
||||
x = [-6/beta:a:6/beta];
|
||||
N = length(x);
|
||||
t0 = (hbar^2)/(2*m*a^2)/q;
|
||||
L = a*(N+1);
|
||||
|
||||
% Set up the Hamiltonian matrix, in pieces, by constructing square
|
||||
% matrices containing information for the main
|
||||
% diagonal, lower diagonal, and upper diagonal, and then adding them
|
||||
% together; you may need to look up the diag command using help
|
||||
% to see how this works; U is the potential energy
|
||||
|
||||
U = (1/q)*(0.5*m*w0^2*x.^2);
|
||||
main_D = diag(2*t0*ones(1,N)+U,0);
|
||||
lower_D = diag(-t0*ones(1,N-1),-1);
|
||||
upper_D = diag(-t0*ones(1,N-1),+1);
|
||||
|
||||
H = main_D + lower_D + upper_D;
|
||||
|
||||
% Find the eigenvectors and eigenvalues of H; V will contain the
|
||||
% eigenvectors in its columns, and D will contain the corresponding
|
||||
% eigenvalues on its main diagonal
|
||||
|
||||
[V,D] = eig(H);
|
||||
|
||||
% Find the numerical eigenvalues
|
||||
|
||||
E_numerical = diag(D,0);
|
||||
|
||||
% Find the eigenvectors corresponding to sorted eigenvalues 1 and 50;
|
||||
% we can get these from the matrix V, provided we reference the
|
||||
% appropriate UNSORTED eigenvalue number
|
||||
|
||||
phi_numerical_1 = V(:,1);
|
||||
phi_numerical_2 = V(:,2);
|
||||
|
||||
% Find the position-probability densities
|
||||
|
||||
P_numerical_1 = phi_numerical_1 .* conj(phi_numerical_1);
|
||||
P_numerical_2 = phi_numerical_2 .* conj(phi_numerical_2);
|
||||
|
||||
% Plot out the relevant results
|
||||
|
||||
figure(1); h = plot(x,P_numerical_1,'k-'); grid on;
|
||||
set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
|
||||
xlabel('POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]');
|
||||
legend('Eigenvector 1');
|
||||
|
||||
figure(2); h = plot(x,P_numerical_2,'k-'); grid on;
|
||||
set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
|
||||
xlabel('POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]');
|
||||
legend('Eigenvector 2');
|
||||
|
||||
Loading…
Reference in New Issue
Block a user