diff --git a/PS2/doc.aux b/PS2/doc.aux index ee6ac14..731303a 100644 --- a/PS2/doc.aux +++ b/PS2/doc.aux @@ -7,4 +7,7 @@ \newlabel{eq:analytical}{{1}{3}} \newlabel{eq:normalise}{{2}{3}} \newlabel{eq:integral_ident}{{3}{3}} -\gdef \@abspage@last{8} +\newlabel{eq:numerical_E}{{4}{4}} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Analytic solution to numeric system, plotted.\relax }}{4}{}\protected@file@percent } +\newlabel{fig:q1b_electrons}{{2}{4}} +\gdef \@abspage@last{9} diff --git a/PS2/doc.fdb_latexmk b/PS2/doc.fdb_latexmk index 82849c5..1f0c9a1 100644 --- a/PS2/doc.fdb_latexmk +++ b/PS2/doc.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1614638543 "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" 1614638561 +["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 "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 "" @@ -29,7 +29,6 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmsy6.tfm" 1136765053 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/cmti10.tfm" 1136765053 1480 aa8e34af0eb6a2941b776984cf1dfdc4 "" "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 "" @@ -44,8 +43,8 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr9.pfb" 1247593066 33993 9b89b85fd2d9df0482bd47194d1d3bf3 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1247593067 32569 5e5ddc8df908dea60932f3c484a54c0d "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy7.pfb" 1247593066 32716 08e384dc442464e7285e891af9f45947 "" - "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmti10.pfb" 1247593067 37944 359e864bd06cde3b1cf57bb20757fb06 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/latxfont/line10.pfb" 1247593053 11493 4f5ed183a47d3197cf8cd322325db6de "" + "C:/Users/speed/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/symbols/msbm10.pfb" 1247593073 34694 870c211f62cb72718a00e353f14f254d "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/context/base/mkii/supp-pdf.mkii" 1580390158 71627 94eb9990bed73c364d7f53f960cc8c5b "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/generic/iftex/iftex.sty" 1583523400 6501 4011d89d9621e0b0901138815ba5ff29 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/generic/iftex/ifvtex.sty" 1583523400 1057 525c2192b5febbd8c1f662c9468335bb "" @@ -96,15 +95,15 @@ "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/parskip/parskip.sty" 1592833306 4414 25631f9679adf750bac898538ededd61 "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/setspace/setspace.sty" 1324294077 22913 a27d7908fc6f0385466454a966a316eb "" "C:/Users/speed/AppData/Local/Programs/MiKTeX/tex/latex/titlesec/titlesec.sty" 1571240058 48596 2b6a95da931c07a430b1a61904aaa42d "" - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.aux" 1614638561 334 906353ae3fc5daad100ce8b921730ac7 "" - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" 1614638542 10889 00e896f1f98c88bd40320d9e83a777cb "" - "doc.aux" 1614638561 334 906353ae3fc5daad100ce8b921730ac7 "pdflatex" - "doc.tex" 1614638542 10889 00e896f1f98c88bd40320d9e83a777cb "" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.aux" 1614640688 560 a7f45c1ba582090fea381768d02972d1 "" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" 1614640701 13478 c506f224fad814ac5407ed9564ca556b "" + "doc.aux" 1614640688 560 a7f45c1ba582090fea381768d02972d1 "pdflatex" + "doc.tex" 1614640701 13478 c506f224fad814ac5407ed9564ca556b "" "q1a_1and50.png" 1614636283 34155 72f34c6abf061325ae1f162db0ffd06b "" "q1a_eigenvals.png" 1614637266 23078 9c99a5505761d0b964e665e35fd9c02a "" (generated) + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.log" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf" "doc.pdf" "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" diff --git a/PS2/doc.fls b/PS2/doc.fls index b8cecb6..cbb5f36 100644 --- a/PS2/doc.fls +++ b/PS2/doc.fls @@ -515,7 +515,6 @@ 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_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 -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\tfm\public\cm\cmti10.tfm 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 @@ -547,7 +546,7 @@ INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\c INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy10.pfb INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy7.pfb INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmsy7.pfb -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmti10.pfb -INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\cm\cmti10.pfb INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\latxfont\line10.pfb INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\latxfont\line10.pfb +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\symbols\msbm10.pfb +INPUT C:\Users\speed\AppData\Local\Programs\MiKTeX\fonts\type1\public\amsfonts\symbols\msbm10.pfb diff --git a/PS2/doc.log b/PS2/doc.log index 2dd77cd..def3426 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 15:42 +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 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 @@ -332,19 +332,19 @@ 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 ] [4] [5] [6] [7] [8] (d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.aux) ) + [3 ] [4] [5] [6] [7] [8] [9] (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: - 8334 strings out of 479618 - 125207 string characters out of 2865027 + 8336 strings out of 479618 + 125259 string characters out of 2865027 981750 words of memory out of 3000000 - 25398 multiletter control sequences out of 15000+200000 - 413212 words of font info for 65 fonts, out of 3000000 for 9000 + 25401 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 (8 pages, 248293 bytes). + +Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (9 pages, 244757 bytes). PDF statistics: - 101 PDF objects out of 1000 (max. 8388607) + 104 PDF objects out of 1000 (max. 8388607) 0 named destinations out of 1000 (max. 500000) 11 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/PS2/doc.pdf b/PS2/doc.pdf index cb9bb4e..b4853a9 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 5463743..b316ccf 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 db73568..1e3993b 100644 --- a/PS2/doc.tex +++ b/PS2/doc.tex @@ -205,9 +205,20 @@ legend({'Numerical','Analytical'},'Location','northwest'); \item %1bii - {\em I really don't like this explanation}. - - The output of the numerical sum will change, as the value of $\alpha$ changes, by a factor if $\alpha$. + 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. @@ -226,18 +237,56 @@ legend({'Numerical','Analytical'},'Location','northwest'); \begin{enumerate}[align=left,leftmargin=*,labelsep=0em,label=(\roman*)] \item %1ci - From the base form of $\phi _\ell = Bsin \left( \frac {n \pi}{L} a \ell \right)$, we can see that + % 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 $B = \frac{n \pi \ell}{L}$. + $\alpha = \frac{n \pi \alpha \ell}{L}$ and $B = \frac{n \pi \alpha}{L}$. - Fro + 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}