From 39372354ab2bd8e6aa9209f2cee20957204de14c Mon Sep 17 00:00:00 2001 From: pkirwin Date: Mon, 1 Mar 2021 21:17:28 -0700 Subject: [PATCH] commit --- PS2/doc.aux | 2 +- PS2/doc.fdb_latexmk | 32 ++++++--- PS2/doc.fls | 111 ++++++++++++++++++++++++----- PS2/doc.log | 170 ++++++++++++++++++++++++++++++-------------- PS2/doc.pdf | Bin 3711631 -> 3715189 bytes PS2/doc.synctex.gz | Bin 146684 -> 153591 bytes PS2/doc.tex | 32 ++++++++- PS2/q4a.m | 74 +++++++++++++++++++ 8 files changed, 338 insertions(+), 83 deletions(-) create mode 100644 PS2/q4a.m diff --git a/PS2/doc.aux b/PS2/doc.aux index 3a44f65..d7e9143 100644 --- a/PS2/doc.aux +++ b/PS2/doc.aux @@ -14,4 +14,4 @@ \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Analytic solution to numeric system, plotted.\relax }}{5}{}\protected@file@percent } \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Sketch of appropriate energy levels\relax }}{7}{}\protected@file@percent } \@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Numerical result (black line) and analytical solution scaled by \(a\) (orange circles).\relax }}{10}{}\protected@file@percent } -\gdef \@abspage@last{12} +\gdef \@abspage@last{13} diff --git a/PS2/doc.fdb_latexmk b/PS2/doc.fdb_latexmk index 52fd9da..8fb6934 100644 --- a/PS2/doc.fdb_latexmk +++ b/PS2/doc.fdb_latexmk @@ -1,8 +1,8 @@ # Fdb version 3 -["pdflatex"] 1614655256 "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" 1614655259 +["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 "C:/Users/speed/AppData/Local/MiKTeX/fonts/pk/ljfour/jknappen/ec/dpi600/tcrm1000.pk" 1614620319 11548 693fc704971bedc6e913b37b081cf40e "" "C:/Users/speed/AppData/Local/MiKTeX/miktex/data/le/pdftex/pdflatex.fmt" 1612675286 9293900 83195ab665706b6f706823352908aa95 "" - "C:/Users/speed/AppData/Local/MiKTeX/pdftex/config/pdftex.map" 1614638559 80909 eab91d9745dd2edfd62a31d53cd5fe15 "" + "C:/Users/speed/AppData/Local/MiKTeX/pdftex/config/pdftex.map" 1614658033 80909 eab91d9745dd2edfd62a31d53cd5fe15 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/jknappen/ec/tcrm1000.tfm" 993058908 1436 c7f957a372ef2fbe93c0982f96625e12 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1233951848 1004 54797486969f23fa377b128694d548df "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1233951848 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -25,10 +25,18 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmr6.tfm" 1136765053 1300 b62933e007d01cfd073f79b963c01526 "" 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1116 933a60c408fc0a863a92debe84b2d294 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmsy8.tfm" 1136765053 1120 8b7d695260f3cff42e636090a8002094 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmsy9.tfm" 1136765053 1116 25a7bf822c58caf309a702ef79f4afbb "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmtt10.tfm" 1136765053 768 1321e9409b4137d6fb428ac9dc956269 "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmtt12.tfm" 1136765053 772 9a936b7f5e2ff0557fce0f62822f0bbf "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmtt8.tfm" 1136765053 768 d7b9a2629a0c353102ad947dc9221d49 "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmtt9.tfm" 1136765053 764 c98a2af25c99b73a368cf7336e255190 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmbx10.pfb" 1247593066 34811 78b52f49e893bcba91bd7581cdc144c0 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmbx12.pfb" 1247593066 32080 340ef9bf63678554ee606688e7b5339d "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1247593067 30251 6afa5cb1d0204815a708a080681d4674 "" @@ -86,8 +94,10 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/kvoptions/kvoptions.sty" 1602224496 22521 d2fceb764a442a2001d257ef11db7618 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/l3backend/l3backend-pdftex.def" 1611916124 27097 58278863d97b10ab86e334b8da33df7a "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/l3kernel/expl3.sty" 1610202356 6209 031757bc8d0350c53dd99ad8ae4875eb "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/l3packages/l3keys2e/l3keys2e.sty" 1603800684 4615 0436b95d48df75eb5f264ca6898802f6 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/l3packages/xparse/xparse-generic.tex" 1603800698 80141 edbf9289c99ff37db17116af7a3a423f "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/l3packages/xparse/xparse.sty" 1603800698 5905 c6eb253894f4e808af476e034b49df36 "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/latex-tools/array.sty" 1602844017 13051 9a7bbb9e485cd81cdcc1d56212b088ff "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/lipsum/lipsum.ltd.tex" 1546855561 99745 c6fa29828cc60471827afe275c8bd77f "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/lipsum/lipsum.sty" 1546855560 18705 8cf65af2c4529eed91b5d364b50d3ada "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/listings/listings.cfg" 1585216535 1879 e31effa752c61538383451ae21332364 "" @@ -96,11 +106,15 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/listings/lstmisc.sty" 1585216535 79106 ee25ce086f4a79d8cf73bac6f94c02a5 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/parskip/parskip.sty" 1592833306 4414 25631f9679adf750bac898538ededd61 "" 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9c3bdf9df7dc973686f5105c04140b36 "" "q2f_fig.png" 1614654314 34695 af1d8fa7a33b7f04767332fb391a90f6 "" (generated) - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.log" - "doc.pdf" + "doc.aux" "doc.log" "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf" - "doc.aux" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.log" + "doc.pdf" diff --git a/PS2/doc.fls b/PS2/doc.fls index bbd8d7b..f82d6f2 100644 --- a/PS2/doc.fls +++ b/PS2/doc.fls @@ -427,6 +427,56 @@ INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\cancel\cancel.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\siunitx\siunitx.sty +INPUT 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C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\translator\translator.sty +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\graphics\keyval.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\listings\lstlang1.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\listings\lstlang1.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\listings\lstlang1.sty @@ -450,6 +500,43 @@ INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\epstopdf-pkg\epstop INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\epstopdf-pkg\epstopdf-base.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\epstopdf-pkg\epstopdf-base.sty INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\tex\latex\epstopdf-pkg\epstopdf-base.sty +INPUT 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za7FaK2Y%-y3X4?)!OjRwLXVQ<20Z+IiDJP!KrR!pMF~1=E2MS}_1#Y_dJP_b+?YytpwtN{MY1=yEP=ir}!ikMHJo>uJFJjznY{%x&o z+(&tQJTybUF~8VCnV=Guk$$Cw`FBvJhFWWcnaP5{O>0>%Hh(hVw%e#dF-x82$A;Yq zn9=^|_g}Qb-39;v diff --git a/PS2/doc.tex b/PS2/doc.tex index fc13037..2f51ed3 100644 --- a/PS2/doc.tex +++ b/PS2/doc.tex @@ -16,6 +16,7 @@ \usepackage{lipsum} \usepackage{enumitem} \usepackage{cancel} +\usepackage{siunitx} \titleformat{\subsection}[runin]{\normalfont \large \bfseries} {\thesubsection}{1em}{} @@ -284,7 +285,7 @@ legend({'Numerical','Analytical'},'Location','northwest'); \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 \begin{equation*} @@ -469,7 +470,27 @@ axis([0 1e-9 0 0.04]); \item %2d For the 1s level, \(E = -13.4978\) eV. \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 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] diff --git a/PS2/q4a.m b/PS2/q4a.m new file mode 100644 index 0000000..7917caf --- /dev/null +++ b/PS2/q4a.m @@ -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'); +