diff --git a/PS2/doc.aux b/PS2/doc.aux index 9aad6e5..3a44f65 100644 --- a/PS2/doc.aux +++ b/PS2/doc.aux @@ -4,6 +4,7 @@ \newlabel{sub@fig:q3_ne}{{a}{3}} \newlabel{fig:q3_current}{{1b}{3}} \newlabel{sub@fig:q3_current}{{b}{3}} +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces (a) Probability densities for \(n=1\) and \(n=50\). (b) Comparison of first 101 numerical and analytic eigenvalues.\relax }}{3}{}\protected@file@percent } \newlabel{eq:analytical}{{1}{3}} \newlabel{eq:normalise}{{2}{3}} \newlabel{eq:integral_ident}{{3}{3}} @@ -12,4 +13,5 @@ \newlabel{eq:numerical_E}{{7}{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} diff --git a/PS2/doc.fdb_latexmk b/PS2/doc.fdb_latexmk index aa584d0..52fd9da 100644 --- a/PS2/doc.fdb_latexmk +++ b/PS2/doc.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1614653239 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"doc.aux" diff --git a/PS2/doc.fls b/PS2/doc.fls index eda6ba3..bbd8d7b 100644 --- a/PS2/doc.fls +++ b/PS2/doc.fls @@ -495,19 +495,14 @@ INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_1and50.png INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_1and50.png INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmr9.tfm -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmmi9.tfm -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmsy9.tfm -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\cmextra\cmex9.tfm INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\cmextra\cmex7.tfm -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msam10.tfm -INPUT 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d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_eigenvals.png INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_eigenvals.png INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1a_eigenvals.png @@ -529,6 +524,11 @@ INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1civ_fig1.png INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1civ_fig1.png INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1civ_fig1.png +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmmi9.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmsy9.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\cmextra\cmex9.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msam10.tfm +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\amsfonts\symbols\msbm10.tfm INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1civ_fig2.png INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1civ_fig2.png INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1civ_fig2.png @@ -571,6 +571,13 @@ INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2c_fig2_2s.png INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2c_fig2_2s.png INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2c_fig2_2s.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2f_fig.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2f_fig.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2f_fig.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2f_fig.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2f_fig.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2f_fig.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q2f_fig.png INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.aux INPUT C:\Users\speed\AppData\Local\MiKTeX\fonts\pk\ljfour\jknappen\ec\dpi600\tcrm1000.pk INPUT C:\Users\speed\AppData\Local\MiKTeX\fonts\pk\ljfour\jknappen\ec\dpi600\tcrm1000.pk diff --git a/PS2/doc.log b/PS2/doc.log index d95fea4..31f4e7b 100644 --- a/PS2/doc.log +++ b/PS2/doc.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 21.1) (preloaded format=pdflatex 2021.2.6) 1 MAR 2021 19:47 +This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 21.1) (preloaded format=pdflatex 2021.2.6) 1 MAR 2021 20:20 entering extended mode **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.tex @@ -333,62 +333,67 @@ File: q1a_eigenvals.png Graphic file (type png) Package pdftex.def Info: q1a_eigenvals.png used on input line 163. (pdftex.def) Requested size: 202.3529pt x 151.76526pt. [3 ] - + File: q1cii.png Graphic file (type png) -Package pdftex.def Info: q1cii.png used on input line 276. +Package pdftex.def Info: q1cii.png used on input line 278. (pdftex.def) Requested size: 505.89pt x 379.42523pt. [4] - + File: q1civ_fig1.png Graphic file (type png) -Package pdftex.def Info: q1civ_fig1.png used on input line 304. +Package pdftex.def Info: q1civ_fig1.png used on input line 306. (pdftex.def) Requested size: 252.94499pt x 189.7102pt. - + File: q1civ_fig2.png Graphic file (type png) -Package pdftex.def Info: q1civ_fig2.png used on input line 309. +Package pdftex.def Info: q1civ_fig2.png used on input line 311. (pdftex.def) Requested size: 252.94499pt x 189.7102pt. [5 ] - + File: q1di_fig1.png Graphic file (type png) -Package pdftex.def Info: q1di_fig1.png used on input line 338. +Package pdftex.def Info: q1di_fig1.png used on input line 340. (pdftex.def) Requested size: 252.94499pt x 189.7102pt. - + File: q1di_fig2.png Graphic file (type png) -Package pdftex.def Info: q1di_fig2.png used on input line 343. +Package pdftex.def Info: q1di_fig2.png used on input line 345. (pdftex.def) Requested size: 252.94499pt x 189.7102pt. - + File: q1div.jpg Graphic file (type jpg) -Package pdftex.def Info: q1div.jpg used on input line 358. +Package pdftex.def Info: q1div.jpg used on input line 360. (pdftex.def) Requested size: 505.89pt x 379.41656pt. [6 ] [7 ] [8] File: q2c_fig1_1s.png Graphic file (type png) -Package pdftex.def Info: q2c_fig1_1s.png used on input line 458. +Package pdftex.def Info: q2c_fig1_1s.png used on input line 460. (pdftex.def) Requested size: 252.94499pt x 189.7102pt. File: q2c_fig2_2s.png Graphic file (type png) -Package pdftex.def Info: q2c_fig2_2s.png used on input line 463. +Package pdftex.def Info: q2c_fig2_2s.png used on input line 465. (pdftex.def) Requested size: 252.94499pt x 189.7102pt. - [9 ] [10] [11] [12] (d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.aux) ) + +File: q2f_fig.png Graphic file (type png) + +Package pdftex.def Info: q2f_fig.png used on input line 480. +(pdftex.def) Requested size: 505.89pt x 379.42523pt. + [9 ] [10 ] [11] [12] (d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.aux) ) Here is how much of TeX's memory you used: - 8398 strings out of 479618 - 126163 string characters out of 2865027 + 8404 strings out of 479618 + 126248 string characters out of 2865027 981750 words of memory out of 3000000 - 25454 multiletter control sequences out of 15000+200000 + 25459 multiletter control sequences out of 15000+200000 412866 words of font info for 64 fonts, out of 3000000 for 9000 1141 hyphenation exceptions out of 8191 68i,12n,75p,1278b,2408s stack positions out of 5000i,500n,10000p,200000b,50000s -Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (12 pages, 3676324 bytes). +Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (12 pages, 3711631 bytes). PDF statistics: - 129 PDF objects out of 1000 (max. 8388607) + 130 PDF objects out of 1000 (max. 8388607) 0 named destinations out of 1000 (max. 500000) - 51 words of extra memory for PDF output out of 10000 (max. 10000000) + 56 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/PS2/doc.pdf b/PS2/doc.pdf index 7c2e79e..7b37162 100644 Binary files a/PS2/doc.pdf and b/PS2/doc.pdf differ diff --git a/PS2/doc.synctex.gz b/PS2/doc.synctex.gz index bc94896..f554f76 100644 Binary files a/PS2/doc.synctex.gz and b/PS2/doc.synctex.gz differ diff --git a/PS2/doc.tex b/PS2/doc.tex index 7dae119..fc13037 100644 --- a/PS2/doc.tex +++ b/PS2/doc.tex @@ -155,7 +155,7 @@ legend({'Numerical','Analytical'},'Location','northwest'); \begin{subfigure}{0.4\linewidth} \centering \includegraphics[width=\textwidth]{q1a_1and50.png} - \caption{Probability densities for \(n=1\)\\and \(n=50\).} + \caption{} \label{fig:q3_ne} \end{subfigure}% \begin{subfigure}{0.4\linewidth} @@ -164,6 +164,8 @@ legend({'Numerical','Analytical'},'Location','northwest'); \caption{Comparison of first 101 numerical and analytic eigenvalues.} \label{fig:q3_current} \end{subfigure} + \caption{(a) Probability densities for \(n=1\) and \(n=50\). + (b) Comparison of first 101 numerical and analytic eigenvalues.} \end{figure} \item %1b \begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)] @@ -185,7 +187,7 @@ legend({'Numerical','Analytical'},'Location','northwest'); 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)}. + \int\sin^2{(a x)}\:dx = \frac{1}{2} x -\frac{1}{4a}\sin{(2a x)}. \label{eq:integral_ident} \end{equation} @@ -247,25 +249,25 @@ legend({'Numerical','Analytical'},'Location','northwest'); \item %1ci % 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 + From the base form of $\phi _\ell = Bsin \left( \frac {n \pi}{L} a \ell \right)$, 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 $B = \frac{n \pi \alpha}{L}$. + $sin(a + B) = sin(a)cos(B) + cos(a)sin(B)$ and + $sin(a + B) = sin(a)cos(B) + cos(a)sin(B)$, respectively, where + $a = \frac{n \pi a \ell}{L}$ and $B = \frac{n \pi a}{L}$. 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) + -t_0 B sin \left( \frac{n \pi a \ell}{L} \right) + 2 t_0 \phi _{\ell} - t_0 sin \left( \frac{n \pi a \ell}{L} \right) \end{equation*} - At this point, we notice that $\phi _ \ell = B sin \left( \frac {n \pi}{L} \alpha \ell \right)$, + At this point, we notice that $\phi _ \ell = B sin \left( \frac {n \pi}{L} a \ell \right)$, 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) + E = 2t_0 \left( 1 - cos \left( \frac{n \pi a}{L} \right) \right) \label{eq:numerical_E} \end{equation} @@ -286,7 +288,7 @@ legend({'Numerical','Analytical'},'Location','northwest'); 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) + E = 2 t_0 \left( \frac{n^2 \pi^2 a^2}{2 L^2} \right) \end{equation*} Fully substituting the known value of $t_0$, and we can get our final expression for $E$: @@ -388,7 +390,7 @@ H(N, 1) = -t0; \end{equation*} \item %2b Homogenous boundary conditions imply that the corner entries of \(\hat{H}\) will be 0. - \item %1c + \item %2c Code: \begin{lstlisting} clear all; @@ -464,12 +466,22 @@ axis([0 1e-9 0 0.04]); \caption{2s probability density.} \end{subfigure} \end{figure} - \item %1d + \item %2d For the 1s level, \(E = -13.4978\) eV. - \item %1e - \lipsum[1] - \item %1f + \item %2e \lipsum[1] + \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 + level are very similiar. We can see that the peak value of the analytic result is very slightly higher than that of the numerical result, + which corresponds to the analytical result for the energy being slightly greater in magnitude (\(-13.6\) eV versus \(-13.4978\) eV). + \begin{figure}[H] + \centering + \includegraphics[width=\textwidth]{q2f_fig.png} + \caption{Numerical result (black line) and analytical solution scaled by \(a\) (orange circles).} + \end{figure} + + \end{enumerate} @@ -477,28 +489,28 @@ axis([0 1e-9 0 0.04]); \begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)] - \item %2a + \item %3a \lipsum[1] - \item %2b + \item %3b \begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)] - \item %1di + \item %3bi \lipsum[1] - \item %1dii + \item %3bii \lipsum[1] - \item %1diii + \item %3biii \lipsum[1] - \item %1div + \item %3biv \lipsum[1] - \item %1dv + \item %3bv \lipsum[1] - \item %1dvi + \item %3bvi \lipsum[1] - \item %1dvii + \item %3bvii \lipsum[1] \end{enumerate} - \item %1c + \item %3c \lipsum[1] - \item %1d + \item %3d \lipsum[1] \end{enumerate} @@ -507,15 +519,15 @@ axis([0 1e-9 0 0.04]); \begin{enumerate}[align=left,leftmargin=*,labelsep=1em,label=\bfseries(\alph*)] - \item %2a + \item %4a \lipsum[1] - \item %2b + \item %4b \begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)] - \item %1di + \item %4bi \lipsum[1] - \item %1dii + \item %4bii \lipsum[1] - \item %1diii + \item %4biii \lipsum[1] \end{enumerate} diff --git a/PS2/q2f_fig.png b/PS2/q2f_fig.png index 1f36dce..894d4e3 100644 Binary files a/PS2/q2f_fig.png and b/PS2/q2f_fig.png differ