diff --git a/PS2/doc.fdb_latexmk b/PS2/doc.fdb_latexmk index 9d9cc59..c161cfd 100644 --- a/PS2/doc.fdb_latexmk +++ b/PS2/doc.fdb_latexmk @@ -1,5 +1,5 @@ # Fdb version 3 -["pdflatex"] 1614650523 "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" 1614650525 +["pdflatex"] 1614651411 "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" 1614651414 "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 "" @@ -97,10 +97,10 @@ "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" 1614650525 740 96d321d28a358a20184b7fe32068111f "" - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" 1614650522 17710 726d1c26d86b0ac704aeac46e41f3d05 "" - "doc.aux" 1614650525 740 96d321d28a358a20184b7fe32068111f "pdflatex" - "doc.tex" 1614650522 17710 726d1c26d86b0ac704aeac46e41f3d05 "" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.aux" 1614651413 740 96d321d28a358a20184b7fe32068111f "" + "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.tex" 1614651410 17776 60a8990fb3ce32251e01cb0001a9ffb6 "" + "doc.aux" 1614651413 740 96d321d28a358a20184b7fe32068111f "pdflatex" + "doc.tex" 1614651410 17776 60a8990fb3ce32251e01cb0001a9ffb6 "" "q1a_1and50.png" 1614636283 34155 72f34c6abf061325ae1f162db0ffd06b "" "q1a_eigenvals.png" 1614637266 23078 9c99a5505761d0b964e665e35fd9c02a "" "q1cii.png" 1614640691 20581 6a6b3aa285fd3f5be369532a6d2a033d "" @@ -111,7 +111,7 @@ "q1div.jpg" 1614643890 3254255 657a32881fc66c752988aa2be34fe320 "" (generated) "doc.pdf" - "d:/Users/Speedee/Documents/University/Year5/Winter/ECE456/ProblemSets/Git/ECE_456/PS2/doc.log" "doc.log" - "doc.aux" + "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.aux" diff --git a/PS2/doc.log b/PS2/doc.log index b31381b..d4b8435 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 19:02 +This is pdfTeX, Version 3.14159265-2.6-1.40.21 (MiKTeX 21.1) (preloaded format=pdflatex 2021.2.6) 1 MAR 2021 19:16 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 @@ -375,7 +375,7 @@ Here is how much of TeX's memory you used: 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 (11 pages, 3617145 bytes). +Output written on d:/Users\Speedee\Documents\University\Year5\Winter\ECE456\ProblemSets\Git\ECE_456\PS2\doc.pdf (11 pages, 3616528 bytes). PDF statistics: 124 PDF objects out of 1000 (max. 8388607) 0 named destinations out of 1000 (max. 500000) diff --git a/PS2/doc.pdf b/PS2/doc.pdf index 3cf29ee..6f28268 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 5854de3..7b66400 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 d8dd5fe..a512529 100644 --- a/PS2/doc.tex +++ b/PS2/doc.tex @@ -387,9 +387,9 @@ H(N, 1) = -t0; \hat{H}_{l(l\pm1)} = -\frac{\hbar^2}{2ma^2}. \end{equation*} \item %2b - \lipsum[1] + Homogenous boundary conditions imply that the corner entries of \(\hat{H}\) will be 0. \item %1c - \lipsum[1] + \item %1d \lipsum[1] \item %1e diff --git a/PS2/q2c.m b/PS2/q2c.m new file mode 100644 index 0000000..87df436 --- /dev/null +++ b/PS2/q2c.m @@ -0,0 +1,65 @@ +clear all; +%physical constants in MKS units + +hbar = 1.054e-34; +q = 1.602e-19; +m = 9.110e-31; + +%generate lattice + +N = 100; %number of lattice points +n = [1:N]; %lattice points +a = 1e-10; %lattice constant +x = a * n; %x-coordinates +t0 = (hbar^2)/(2*m*a^2)/q; %encapsulating factor +L = a * (N+1); %total length of consideration + +%set up Hamiltonian matrix + +U = 0*x; %0 potential at all x +main_diag = diag(2*t0*ones(1,N)+U,0); %create main diagonal matrix +lower_diag = diag(-t0*ones(1,N-1),-1); %create lower diagonal matrix +upper_diag = diag(-t0*ones(1,N-1),+1); %create upper diagonal matrix + +H = main_diag + lower_diag + upper_diag; %sum to get Hamiltonian matrix + +[eigenvectors,E_diag] = eig(H); %"eigenvectors" is a matrix wherein each column is an eigenvector + %"E_diag" is a diagonal matrix where the + %corresponding eigenvalues are on the + %diagonal. + +E_col = diag(E_diag); %folds E_diag into a column vector of eigenvalues + +% return eigenvectors for the 1st and 50th eigenvalues + +phi_1 = eigenvectors(:,1); +phi_50 = eigenvectors(:,50); + +% find the probability densities of position for 1st and 50th eigenvectors + +P_1 = phi_1 .* conj(phi_1); +P_50 = phi_50 .* conj(phi_50); + +% Find first N analytic eigenvalues +E_col_analytic = (1/q) * (hbar^2 * pi^2 * n.*n) / (2*m*L^2); + +% Plot the probability densities for 1st and 50th eigenvectors + +figure(1); clf; h = plot(x,P_1,'kx',x,P_50,'k-'); +grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]); +xlabel('POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]'); +legend('n=1','n=50'); + +% Plot numerical eigenvalues +figure(2); clf; h = plot(n,E_col,'kx'); grid on; +set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]); +xlabel('EIGENVALUE NUMBER'); ylabel('ENERGY [eV]'); +axis([0 100 0 40]); + +% Add analytic eigenvalues to above plot + +hold on; +plot(n,E_col_analytic,'k-'); +legend({'Numerical','Analytical'},'Location','northwest'); + +