diff --git a/PS2/doc.aux b/PS2/doc.aux index 12f0e58..913f089 100644 --- a/PS2/doc.aux +++ b/PS2/doc.aux @@ -11,4 +11,5 @@ \newlabel{eq:Riemann_sum}{{6}{4}} \newlabel{eq:numerical_E}{{7}{4}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Analytic solution to numeric system, plotted.\relax }}{5}{}\protected@file@percent } -\gdef \@abspage@last{10} +\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Sketch of appropriate energy levels\relax }}{7}{}\protected@file@percent } +\gdef \@abspage@last{11} diff --git a/PS2/doc.fdb_latexmk b/PS2/doc.fdb_latexmk index 1588bfb..8276c31 100644 --- a/PS2/doc.fdb_latexmk +++ b/PS2/doc.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1614643264 "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" 1614643267 +["pdflatex"] 1614644079 "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" 1614644142 "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 "" @@ -95,10 +95,10 @@ "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" 1614643267 595 b019f47a873e4d7577a5d1c178942da9 "" - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" 1614643260 16400 c38e0dd0cf4d7c750a7961b5b02ff142 "" - "doc.aux" 1614643267 595 b019f47a873e4d7577a5d1c178942da9 "pdflatex" - "doc.tex" 1614643260 16400 c38e0dd0cf4d7c750a7961b5b02ff142 "" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.aux" 1614644082 740 96d321d28a358a20184b7fe32068111f "" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" 1614644141 17159 07b79832ca92fcda7192a0b7f2369bc5 "" + "doc.aux" 1614644082 740 96d321d28a358a20184b7fe32068111f "pdflatex" + "doc.tex" 1614644141 17159 07b79832ca92fcda7192a0b7f2369bc5 "" "q1a_1and50.png" 1614636283 34155 72f34c6abf061325ae1f162db0ffd06b "" "q1a_eigenvals.png" 1614637266 23078 9c99a5505761d0b964e665e35fd9c02a "" "q1cii.png" 1614640691 20581 6a6b3aa285fd3f5be369532a6d2a033d "" @@ -106,9 +106,10 @@ "q1civ_fig2.png" 1614640691 18950 09e638d39de7c71e3285b2507fc9bb06 "" "q1di_fig1.png" 1614642205 23575 17449aebb9c23a5139435b3506cbca3d "" "q1di_fig2.png" 1614642205 13867 650413e942c9c7a655204496b4f3c21b "" + "q1div.jpg" 1614643890 3254255 657a32881fc66c752988aa2be34fe320 "" (generated) "doc.pdf" - "doc.aux" "doc.log" - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf" "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.log" + "doc.aux" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.pdf" diff --git a/PS2/doc.fls b/PS2/doc.fls index 13082a1..b6f3ddd 100644 --- a/PS2/doc.fls +++ b/PS2/doc.fls @@ -550,6 +550,13 @@ 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\q1di_fig2.png INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1di_fig2.png INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1di_fig2.png +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1div.jpg +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1div.jpg +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1div.jpg +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1div.jpg +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1div.jpg +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1div.jpg +INPUT d:\Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\q1div.jpg 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 1b295e0..a86d69c 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 17:01 +This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 21.1) (preloaded format=pdflatex 2021.2.6) 1 MAR 2021 17:14 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 @@ -353,26 +353,31 @@ Package pdftex.def Info: q1civ_fig2.png used on input line 309. File: q1di_fig1.png Graphic file (type png) -Package pdftex.def Info: q1di_fig1.png used on input line 334. +Package pdftex.def Info: q1di_fig1.png used on input line 338. (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 339. +Package pdftex.def Info: q1di_fig2.png used on input line 343. (pdftex.def) Requested size: 252.94499pt x 189.7102pt. - [6 ] [7] [8] [9] [10] (d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.aux) ) + +File: q1div.jpg Graphic file (type jpg) + +Package pdftex.def Info: q1div.jpg used on input line 358. +(pdftex.def) Requested size: 505.89pt x 379.41656pt. + [6 ] [7 ] [8] [9] [10] [11] (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: - 8367 strings out of 479618 - 125748 string characters out of 2865027 + 8374 strings out of 479618 + 125825 string characters out of 2865027 981750 words of memory out of 3000000 - 25427 multiletter control sequences out of 15000+200000 + 25432 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 (10 pages, 346266 bytes). +Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (11 pages, 3602577 bytes). PDF statistics: - 112 PDF objects out of 1000 (max. 8388607) + 116 PDF objects out of 1000 (max. 8388607) 0 named destinations out of 1000 (max. 500000) - 36 words of extra memory for PDF output out of 10000 (max. 10000000) + 41 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/PS2/doc.pdf b/PS2/doc.pdf index 03785e4..82baff6 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 a2fffc4..2961d3b 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 98ce2a1..bb2283d 100644 --- a/PS2/doc.tex +++ b/PS2/doc.tex @@ -311,7 +311,6 @@ legend({'Numerical','Analytical'},'Location','northwest'); \end{subfigure} \end{figure} - % TODO idk if this is enough With the decreased lattice spacing, and increased number of points, we can see the numerical solution of eigenvalues matches the analytical solution much closer. As well, we can see that the probability density function appears to be "squeezed" in the centre, for the $n = 50$ case. The \(n=50\) case is @@ -325,8 +324,13 @@ legend({'Numerical','Analytical'},'Location','northwest'); In order to modify the computations to those for a particle in a "ring" we simply had to add $-t_0$ elements as the "corner" elements of the - hamiltonian operator array. - + hamiltonian operator array: + \begin{lstlisting}[language=Matlab] +% Modify hamiltonian for circular boundary conditions +H(1, N) = -t0; +H(N, 1) = -t0; + \end{lstlisting} + \begin{figure}[H] \centering \begin{subfigure}{0.5\textwidth} @@ -344,15 +348,23 @@ legend({'Numerical','Analytical'},'Location','northwest'); \item %1dii The energy levels for eigenvalues number 4 and 5 are both $0.06eV$. - Because these eigenstates both have the same eigenvalue (0.06). + These eigenstates are degenerate because they both have the same + eigenvalue/energy (0.06). \item %1diii - % TODO add figure for sketch + \begin{figure}[H] + \centering + \includegraphics[width=\textwidth, angle=-90]{q1div.jpg} + \caption{Sketch of appropriate energy levels} + \end{figure} \item %1div - + Plugging the valid levels for $n$ into equation (10) from the assignment (and dividing + by the requisite $q$), we get the energy levels of $0eV, 0.0147eV, and 0.0589eV$, for the + eigenvalue numbers $n = 0, 1, and 2$, respectively. These match very closely, to within + acceptable margin of the numerical results from part (ii). \end{enumerate}