From a83a090aeec63e0074ec946cd1cd4ea1e4bcc435 Mon Sep 17 00:00:00 2001 From: pkirwin Date: Mon, 1 Mar 2021 19:41:00 -0700 Subject: [PATCH] commit --- PS2/q2c.asv | 76 ++++++++++++++++++++++++++++++++++++++++++++ PS2/q2c.m | 31 ++++++++++++------ PS2/q2c_fig1_1s.png | Bin 0 -> 23989 bytes PS2/q2c_fig2_2s.png | Bin 0 -> 29139 bytes 4 files changed, 97 insertions(+), 10 deletions(-) create mode 100644 PS2/q2c.asv create mode 100644 PS2/q2c_fig1_1s.png create mode 100644 PS2/q2c_fig2_2s.png diff --git a/PS2/q2c.asv b/PS2/q2c.asv new file mode 100644 index 0000000..830619f --- /dev/null +++ b/PS2/q2c.asv @@ -0,0 +1,76 @@ +clear all; +%physical constants in MKS units + +hbar = 1.054e-34; +q = 1.602e-19; +m = 9.110e-31; +epsilon_0 = 8.854e-12; + +%generate lattice + +N = 100; %number of lattice points +n = [1:N]; %lattice points +a = 0.1e-10; %lattice constant +r = 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 = -q^2./(4*pi*epsilon_0.*r) * (1/q); %potential at r in [eV] +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_2 = eigenvectors(:,2); + +% find the probability densities of position for 1st and 50th eigenvectors + +P_1 = phi_1 .* conj(phi_1); +P_2 = phi_2 .* conj(phi_2); + +% 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 2nd eigenvectors + +figure(1); clf; h = plot(r,P_1,'k-'); +grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]); +xlabel('POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]'); +yticks([0.02 0.04 0.06 0.08 0.10 0.12]); +legend('n=1'); +axis([0 1e-9 0 0.12]); + +figure(2); clf; h = plot(r,P_2,'k-'); +grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]); +xlabel('POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]'); +yticks([0.02 0.04 0.06 0.08 0.10 0.12]); +legend('n=2'); +axis([0 1e-9 0 0.04]); + +%{ +% 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'); +%} + diff --git a/PS2/q2c.m b/PS2/q2c.m index 87df436..9b69784 100644 --- a/PS2/q2c.m +++ b/PS2/q2c.m @@ -4,19 +4,20 @@ clear all; hbar = 1.054e-34; q = 1.602e-19; m = 9.110e-31; +epsilon_0 = 8.854e-12; %generate lattice N = 100; %number of lattice points n = [1:N]; %lattice points -a = 1e-10; %lattice constant -x = a * n; %x-coordinates +a = 0.1e-10; %lattice constant +r = 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 +U = -q^2./(4*pi*epsilon_0.*r) * (1/q); %potential at r in [eV] 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 @@ -33,23 +34,33 @@ 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); +phi_2 = eigenvectors(:,2); % 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); +P_2 = phi_2 .* conj(phi_2); % 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 +% Plot the probability densities for 1st and 2nd eigenvectors -figure(1); clf; h = plot(x,P_1,'kx',x,P_50,'k-'); +figure(1); clf; h = plot(r,P_1,'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'); +xlabel('RADIAL POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]'); +yticks([0.02 0.04 0.06 0.08 0.10 0.12]); +legend('n=1'); +axis([0 1e-9 0 0.12]); +figure(2); clf; h = plot(r,P_2,'k-'); +grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]); +xlabel('RADIAL POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]'); +yticks([0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04]); +legend('n=2'); +axis([0 1e-9 0 0.04]); + +%{ % Plot numerical eigenvalues figure(2); clf; h = plot(n,E_col,'kx'); grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]); @@ -61,5 +72,5 @@ axis([0 100 0 40]); hold on; plot(n,E_col_analytic,'k-'); legend({'Numerical','Analytical'},'Location','northwest'); - +%} diff --git a/PS2/q2c_fig1_1s.png b/PS2/q2c_fig1_1s.png new file mode 100644 index 0000000000000000000000000000000000000000..c71b68645d590f95da74088ee5e53b84350e1985 GIT binary patch literal 23989 zcmbrmcRbdA-#>gvOGBB7B%?$|$;e7^&>^G9-q~ek?@_7jQAP-vEff+eBZaJFWkzLW zml0*&&!g|}Jn!qee%E!~f8598d+K~hj?eLVzu&LdbG<&n=T*<_+{V0(Kp^Z?BoWmK zgbhat1d7b98}Oa`VL?Cf2h|mlt}}r^{f_)Uir6F6EcoUY7ey8MEyI)>HXJ^@ ze2dFPUdQEvgQbi6MJEe_%4rSd)3SUAG^{NSs9bch)^>2RappUqXm2JUrSq8yKhHt_ zc{wMGi!PTO>5H{<8D`zN68%x$)+o*XyGqIo;mPg_RyU zHZtD1qW-JD_D~QAcaz@t;Sa)Yc6JIvz^Mb|uLu6`uR4#s%--JIshwwVY=!cL!$AL| z3*2v?QW6NcN%es-sj1CMvbd=qZ`KFCJ0U4K;X5`v>r~dfi9k@wU}vA|EpD!>Q>;*0 zUS8(p#v8<)Z^v6+`>4>N=jQ3@Nfb7zB$Nr-{bnE#DxDrGlkk_>fjpA9s*OA4cZQe? 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