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pkirwin 2021-03-01 21:17:28 -07:00
parent 8b7754bd2b
commit 39372354ab
8 changed files with 338 additions and 83 deletions
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PS2/doc.aux
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@ -16,6 +16,7 @@
\usepackage{lipsum} \usepackage{lipsum}
\usepackage{enumitem} \usepackage{enumitem}
\usepackage{cancel} \usepackage{cancel}
\usepackage{siunitx}
\titleformat{\subsection}[runin]{\normalfont \large \bfseries} \titleformat{\subsection}[runin]{\normalfont \large \bfseries}
{\thesubsection}{1em}{} {\thesubsection}{1em}{}
@ -284,7 +285,7 @@ legend({'Numerical','Analytical'},'Location','northwest');
\item %1ciii \item %1ciii
Applying the approximation $cos(\Theta) = 1 - \frac{\Theta^2}{2}$ for small $\Theta$, Applying the approximation $cos(\theta) = 1 - \frac{\theta^2}{2}$ for small $\theta$,
on equation (\ref{eq:numerical_E}), we get the following expression on equation (\ref{eq:numerical_E}), we get the following expression
\begin{equation*} \begin{equation*}
@ -469,7 +470,27 @@ axis([0 1e-9 0 0.04]);
\item %2d \item %2d
For the 1s level, \(E = -13.4978\) eV. For the 1s level, \(E = -13.4978\) eV.
\item %2e \item %2e
\lipsum[1] Beginning with equation (11) from the assignment, with \(l_o=0\):
\begin{align*}
\left[-\frac{\hbar^2}{2m}\frac{d^2}{dr^2} - \frac{q^2}{4\pi\epsilon_0r}\right]f(r) &= Ef(r)\\
-\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)\\
-\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\\
\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 \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, 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 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*)] \begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)]
\item %3a \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 \item %3b
\begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)] \begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)]
\item %3bi \item %3bi
\lipsum[1] \lipsum[1]

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@ -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');