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main.m
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main.m
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% Numerical analysis of Time-Dependent Schrödinger equation
% Written by: Ahmet Burak Yıldırım, 2023
% i hbar d(psi(r,t))/dt = -(hbar^2)/(2m) nabla^2 psi(r,t) + V(r) psi(r,t)
% 1 dimensional -> r = (x):
% i hbar d(psi(x,t))/dt = -(hbar^2)/(2m) (d^2/dx^2) psi(x,t) + V(x) psi(x,t)
% 2 dimensional -> r = (x,y):
% i hbar d(psi(x,y,t))/dt = -(hbar^2)/(2m) (d^2/dx^2 + d^2/dy^2) psi(x,y,t) + V(x,y) psi(x,y,t)
%% Time-dependent Schrödinger equation
% Parabolic PDE
% OBSERVATIONS:
% EE: dt = 1e-4, tmax = 10, oscillations for finite-well,
% 1 - Explicit Euler
% 2 - Crank-Nicolson
% 2a - Jacobi
% 2b - Gauss-Seidel
% 2c - SOR
% 2d - ADI
clear; close all; clc;
% Initialization of parameters
dimension = 1; % 1 or 2
% Potential choices are applicable according to selected dimension
% List of available potentials:
% FOR 1D: harmonic / finite-well / tunneling / free-particle
% FOR 2D: harmonic / finite-well / double-slit / free-particle
potential1d = "tunneling"; % harmonic / finite-well / tunneling / free-particle
potential2d = "single-slit"; %double-slit / harmonic / finite-well / tunneling / free-particle / single-slit
m = 1; % mass of the particle (in amu)
hbar = 1; % reduced Planck's constant (in eV*fs)
% Parameter table
% 1D - ExplicitEuler : dt: TRUE: 2e-5 / CASES: 8e-5, 2e-4, 8e-4, 2e-3
% 1D - Crank-Nicolson : dt: TRUE: 2e-5 / CASES: 8e-5, 2e-4, 8e-4, 2e-3
% 2D - ExplicitEuler : dt: TRUE: 2e-5 / CASES: 8e-5, 2e-4, 8e-4, 2e-3
% 2D - Crank-Nicolson (ADI): dt: TRUE: 2e-5 / CASES: 8e-5, 2e-4, 8e-4, 2e-3
dt = 2e-4; % time step size (in fs)
dt_true = 2e-5; % time step size for "true" solution (in fs)
tmax = 1; % maximum time (in fs)
t = 0:dt:tmax; % time grid
t_true = 0:dt_true:tmax; % time grid
if dimension == 1
dx = 0.1; % spatial step size in the x direction (in nm)
Lx = 10; % size of the grid in the x direction (in nm)
x = -Lx/2:dx:Lx/2;
X = reshape(x,[size(x,2) 1]); % 1D grid
kx = 0.2; % wave vector: for momentum
r0 = -2; sigma = 0.5;
psi_0 = exp(-0.5*(((X-r0))/sigma).^2).*exp(1i*(kx*(X)));
psi_0 = Normalizewavefunction1D(psi_0);
psi = zeros(length(x),length(t));
% set the initial wave function
psi(:,1) = psi_0;
V = potential1D(potential1d,x);
% TRUE SOLUTION (using smaller time step)
psi_true = zeros(length(x),length(t_true));
% To compare true solution psi_true with psi, creating a new arraw
% and filling same time steps
psi_true(:,1) = psi_0;
% Solve the time-dependent Schrödinger equation using the finite difference method
disp("Solving TDSE using the finite difference method")
tic
for k = 2:length(t)
% d(psi(r,t))/dt = (i hbar)/(2m) nabla^2 psi(r,t) + 1/(i hbar) V(r) psi(r,t)
% v = (i hbar)/(2m) -> diffusion coefficient
% f = psi(r,t)
% df/dt = v (d^2f/dx^2) + V(x)/(i hbar) f
v = (1i*hbar)/(2*m); % diffusion coefficient
psi_t = psi(:,k-1);
pot = V/(1i*hbar);
% Available solvers: ExplicitEuler1D / CrankNicolson1D
psi(:,k) = CrankNicolson1D(psi_t, hbar, m, dx, dt, pot); % replace function
psi(:,k) = Normalizewavefunction1D(psi(:,k));
end
toc
disp("Solving for the true solution")
tic
% Solve for the true solution
for k = 2:length(t_true)
% d(psi(r,t))/dt = (i hbar)/(2m) nabla^2 psi(r,t) + 1/(i hbar) V(r) psi(r,t)
% v = (i hbar)/(2m) -> diffusion coefficient
% f = psi(r,t)
% df/dt = v (d^2f/dx^2) + V(x)/(i hbar) f
v = (1i*hbar)/(2*m); % diffusion coefficient
psi_t = psi_true(:,k-1);
pot = V/(1i*hbar);
psi_true(:,k) = ExplicitEuler1D(psi_t, hbar, m, dx, dt_true, pot);
psi_true(:,k) = Normalizewavefunction1D(psi_true(:,k));
end
toc
psi_true_c = psi_true(:,1:length(t_true)/length(t):end);
end
if dimension == 2
dx = 0.1; % spatial step size in the x direction (in nm)
dy = 0.1; % spatial step size in the y direction (in nm)
Lx = 10; % size of the grid in the x direction (in nm)
Ly = 10; % size of the grid in the y direction (in nm)
x = -Lx/2:dx:Lx/2;
y = -Ly/2:dy:Ly/2;
[X,Y] = meshgrid(x,y); % 2D grid
k = [10, 0]; % wave vector: for momentum
r0 = [-3, 0]; sigma = 0.5;
psi_0 = exp(-((X-r0(1)).^2+(Y-r0(2)).^2)/(2*sigma)).*exp(1i*(k(1)*X+k(2)*Y));
psi_0 = Normalizewavefunction2D(psi_0);
psi = zeros(length(x),length(y),length(t));
% set the initial wave function
psi(:,:,1) = psi_0;
V = potential2D(potential2d,x,y);
% TRUE SOLUTION (using smaller time step)
psi_true = zeros(length(x),length(y),length(t_true));
% To compare true solution psi_true with psi, creating a new arraw
% and filling same time steps
psi_true(:,:,1) = psi_0;
% Solve the time-dependent Schrödinger equation using the finite difference method
disp("Solving TDSE using the finite difference method")
tic
for k = 2:length(t)
% d(psi(r,t))/dt = (i hbar)/(2m) nabla^2 psi(r,t) + 1/(i hbar) V(r) psi(r,t)
% v = (i hbar)/(2m) -> diffusion coefficient
% f = psi(r,t)
% df/dt = v (d^2f/dx^2 + d^2f/dy^2) + V(x,y)/(i hbar) f
v = (1i*hbar)/(2*m); % diffusion coefficient
psi_t = psi(:,:,k-1);
pot = V/(1i*hbar);
% Available solvers: ExplicitEuler2D / CrankNicolson2D (ADI)
psi(:,:,k) = ExplicitEuler2D(psi_t, hbar, m, dx, dy, dt, pot);
psi(:,:,k) = Normalizewavefunction2D(psi(:,:,k));
end
toc
disp("Solving for the true solution")
tic
% Solve for the true solution
for k = 2:length(t_true)
% d(psi(r,t))/dt = (i hbar)/(2m) nabla^2 psi(r,t) + 1/(i hbar) V(r) psi(r,t)
% v = (i hbar)/(2m) -> diffusion coefficient
% f = psi(r,t)
% df/dt = v (d^2f/dx^2 + d^2f/dy^2) + V(x,y)/(i hbar) f
v = (1i*hbar)/(2*m); % diffusion coefficient
psi_t = psi_true(:,:,k-1);
pot = V/(1i*hbar);
psi_true(:,:,k) = ExplicitEuler2D(psi_t, hbar, m, dx, dy, dt_true, pot);
psi_true(:,:,k) = Normalizewavefunction2D(psi_true(:,:,k));
end
toc
psi_true_c = psi_true(:,:,1:length(t_true)/length(t):end);
end
disp("Stability condition: alpha = " + dt*abs(v)/dx^2)
%% Save data: Dimension = 1, Method = Explicit Euler
% Run after each simulation
if dt == 8e-5
X_1DEULER_dt8e5 = X;
V_1DEULER_dt8e5 = V;
t_1DEULER_dt8e5 = t;
psi_1DEULER_dt8e5 = psi;
psi_true_c_1DEULER_dt8e5 = psi_true_c;
save 1DEULER_dt8e5.mat V_1DEULER_dt8e5 X_1DEULER_dt8e5 t_1DEULER_dt8e5 ...
psi_1DEULER_dt8e5 psi_true_c_1DEULER_dt8e5
clear
elseif dt == 2e-4
X_1DEULER_dt2e4 = X;
V_1DEULER_dt2e4 = V;
t_1DEULER_dt2e4 = t;
psi_1DEULER_dt2e4 = psi;
psi_true_c_1DEULER_dt2e4 = psi_true_c;
save 1DEULER_dt2e4.mat V_1DEULER_dt2e4 X_1DEULER_dt2e4 t_1DEULER_dt2e4 ...
psi_1DEULER_dt2e4 psi_true_c_1DEULER_dt2e4
clear
elseif dt == 8e-4
X_1DEULER_dt8e4 = X;
V_1DEULER_dt8e4 = V;
t_1DEULER_dt8e4 = t;
psi_1DEULER_dt8e4 = psi;
psi_true_c_1DEULER_dt8e4 = psi_true_c;
save 1DEULER_dt8e4.mat V_1DEULER_dt8e4 X_1DEULER_dt8e4 t_1DEULER_dt8e4 ...
psi_1DEULER_dt8e4 psi_true_c_1DEULER_dt8e4
clear
elseif dt == 2e-3
X_1DEULER_dt2e3 = X;
V_1DEULER_dt2e3 = V;
t_1DEULER_dt2e3 = t;
psi_1DEULER_dt2e3 = psi;
psi_true_c_1DEULER_dt2e3 = psi_true_c;
save 1DEULER_dt2e3.mat V_1DEULER_dt2e3 X_1DEULER_dt2e3 t_1DEULER_dt2e3 ...
psi_1DEULER_dt2e3 psi_true_c_1DEULER_dt2e3
clear
end
%% Save data: Dimension = 1, Method = Crank-Nicolson
% Run after each simulation
if dt == 8e-5
X_1DCN_dt8e5 = X;
V_1DCN_dt8e5 = V;
t_1DCN_dt8e5 = t;
psi_1DCN_dt8e5 = psi;
psi_true_c_1DCN_dt8e5 = psi_true_c;
save 1DCN_dt8e5.mat V_1DCN_dt8e5 X_1DCN_dt8e5 t_1DCN_dt8e5 psi_1DCN_dt8e5 psi_true_c_1DCN_dt8e5
clear
elseif dt == 2e-4
X_1DCN_dt2e4 = X;
V_1DCN_dt2e4 = V;
t_1DCN_dt2e4 = t;
psi_1DCN_dt2e4 = psi;
psi_true_c_1DCN_dt2e4 = psi_true_c;
save 1DCN_dt2e4.mat V_1DCN_dt2e4 X_1DCN_dt2e4 t_1DCN_dt2e4 psi_1DCN_dt2e4 psi_true_c_1DCN_dt2e4
clear
elseif dt == 8e-4
X_1DCN_dt8e4 = X;
V_1DCN_dt8e4 = V;
t_1DCN_dt8e4 = t;
psi_1DCN_dt8e4 = psi;
psi_true_c_1DCN_dt8e4 = psi_true_c;
save 1DCN_dt8e4.mat V_1DCN_dt8e4 X_1DCN_dt8e4 t_1DCN_dt8e4 psi_1DCN_dt8e4 psi_true_c_1DCN_dt8e4
clear
elseif dt == 2e-3
X_1DCN_dt2e3 = X;
V_1DCN_dt2e3 = V;
t_1DCN_dt2e3 = t;
psi_1DCN_dt2e3 = psi;
psi_true_c_1DCN_dt2e3 = psi_true_c;
save 1DCN_dt2e3.mat V_1DCN_dt2e3 X_1DCN_dt2e3 t_1DCN_dt2e3 psi_1DCN_dt2e3 psi_true_c_1DCN_dt2e3
clear
end
%% Figures for the Report: 1D TDSE: Euler method: ERROR
load('1DEULER_dt8e5.mat')
load('1DEULER_dt2e4.mat')
load('1DEULER_dt8e4.mat')
load('1DEULER_dt2e3.mat')
error_1DEULER_dt8e5 = abs(trapz(psi_1DEULER_dt8e5-psi_true_c_1DEULER_dt8e5,1));
error_1DEULER_dt2e4 = abs(trapz(psi_1DEULER_dt2e4-psi_true_c_1DEULER_dt2e4,1));
error_1DEULER_dt8e4 = abs(trapz(psi_1DEULER_dt8e4-psi_true_c_1DEULER_dt8e4,1));
error_1DEULER_dt2e3 = abs(trapz(psi_1DEULER_dt2e3-psi_true_c_1DEULER_dt2e3,1));
set(0, 'DefaultLineLineWidth', 2); close all;
tl = tiledlayout(1,2,'TileSpacing','tight','Padding','tight');
t1 = nexttile(1); hold on; grid on
plot(t_1DEULER_dt8e5,error_1DEULER_dt8e5)
plot(t_1DEULER_dt2e4,error_1DEULER_dt2e4)
plot(t_1DEULER_dt8e4,error_1DEULER_dt8e4)
plot(t_1DEULER_dt2e3,error_1DEULER_dt2e3)
legend("$\Delta t = 8\times10^{-5}$","$\Delta t = 2\times10^{-4}$",...
"$\Delta t = 8\times10^{-4}$","$\Delta t = 2\times10^{-3}$")
t2 = nexttile(2); hold on; grid on
plot(t_1DEULER_dt8e5,error_1DEULER_dt8e5)
plot(t_1DEULER_dt2e4,error_1DEULER_dt2e4)
plot(t_1DEULER_dt8e4,error_1DEULER_dt8e4)
plot(t_1DEULER_dt2e3,error_1DEULER_dt2e3)
xlim([0 0.4])
ylim([0 0.2*1e-2])
title(t1,"(a) Error for explicit Euler method in 1D")
title(t2,"(b) Close-up to initial behavior of error")
title(tl,"POTENTIAL: Free-particle", 'Interpreter', 'latex')
xlabel(tl,'$\textrm{Time step}$','Interpreter','latex');
ylabel(tl,'$\textrm{Error}= |\int \psi - \psi_{true} \; dx|$','Interpreter','latex');
picturewidth = 3*34.4/3; hw_ratio = 0.2; fs = 13;
set(findall(gcf,'-property', 'FontSize'), 'FontSize', fs) % never change fontsize anymore!
set(findall(gcf,'-property', 'Box'), 'Box', 'on') % optional
set(findall(gcf, '-property', 'Interpreter'), 'Interpreter', 'latex')
set(findall(gcf, '-property', 'TickLabelInterpreter'), 'TickLabelInterpreter', 'latex')
set(gcf, 'Units', 'centimeters', 'Position', [2 1 picturewidth hw_ratio*picturewidth])
pos = get(gcf, 'Position');
set(findobj(gcf, 'Type', 'legend'), 'FontSize', fs-3, 'Location', 'northeast')
set(gca,'LooseInset',get(gca,'TightInset'))
set(gcf, 'PaperPositionMode', 'Auto', 'PaperUnits', 'centimeters', 'PaperSize', [pos(3), pos(4)])
%% Figures for the Report: 1D TDSE: Crank-Nicolson method: ERROR
load('1DCN_dt8e5.mat')
load('1DCN_dt2e4.mat')
load('1DCN_dt8e4.mat')
load('1DCN_dt2e3.mat')
error_1DCN_dt8e5 = abs(trapz(psi_1DCN_dt8e5-psi_true_c_1DCN_dt8e5,1));
error_1DCN_dt2e4 = abs(trapz(psi_1DCN_dt2e4-psi_true_c_1DCN_dt2e4,1));
error_1DCN_dt8e4 = abs(trapz(psi_1DCN_dt8e4-psi_true_c_1DCN_dt8e4,1));
error_1DCN_dt2e3 = abs(trapz(psi_1DCN_dt2e3-psi_true_c_1DCN_dt2e3,1));
set(0, 'DefaultLineLineWidth', 2); close all;
tl = tiledlayout(1,2,'TileSpacing','tight','Padding','tight');
t1 = nexttile(1); hold on; grid on
plot(t_1DCN_dt8e5,error_1DCN_dt8e5)
plot(t_1DCN_dt2e4,error_1DCN_dt2e4)
plot(t_1DCN_dt8e4,error_1DCN_dt8e4)
plot(t_1DCN_dt2e3,error_1DCN_dt2e3)
legend("$\Delta t = 8\times10^{-5}$","$\Delta t = 2\times10^{-4}$",...
"$\Delta t = 8\times10^{-4}$","$\Delta t = 2\times10^{-3}$")
t2 = nexttile(2); hold on; grid on
plot(t_1DCN_dt8e5,error_1DCN_dt8e5)
plot(t_1DCN_dt2e4,error_1DCN_dt2e4)
plot(t_1DCN_dt8e4,error_1DCN_dt8e4)
plot(t_1DCN_dt2e3,error_1DCN_dt2e3)
xlim([0 0.4])
ylim([0 0.2*1e-2])
title(t1,"(a) Error for explicit Crank-Nicolson method in 1D")
title(t2,"(b) Close-up to initial behavior of error")
title(tl,"POTENTIAL: Free-particle", 'Interpreter', 'latex')
xlabel(tl,'$\textrm{Time step}$','Interpreter','latex');
ylabel(tl,'$\textrm{Error}= |\int \psi - \psi_{true} \; dx|$','Interpreter','latex');
picturewidth = 3*34.4/3; hw_ratio = 0.2; fs = 13;
set(findall(gcf,'-property', 'FontSize'), 'FontSize', fs) % never change fontsize anymore!
set(findall(gcf,'-property', 'Box'), 'Box', 'on') % optional
set(findall(gcf, '-property', 'Interpreter'), 'Interpreter', 'latex')
set(findall(gcf, '-property', 'TickLabelInterpreter'), 'TickLabelInterpreter', 'latex')
set(gcf, 'Units', 'centimeters', 'Position', [2 1 picturewidth hw_ratio*picturewidth])
pos = get(gcf, 'Position');
set(findobj(gcf, 'Type', 'legend'), 'FontSize', fs-3, 'Location', 'northwest')
set(gca,'LooseInset',get(gca,'TightInset'))
set(gcf, 'PaperPositionMode', 'Auto', 'PaperUnits', 'centimeters', 'PaperSize', [pos(3), pos(4)])
%% Figures for the Report: 1D TDSE: Euler method: SNAPSHOTS
load('1DEULER_dt8e5.mat')
load('1DEULER_dt2e4.mat')
load('1DEULER_dt8e4.mat')
load('1DEULER_dt2e3.mat')
set(0, 'DefaultLineLineWidth', 2); lw = 2;
close all; f = figure;
tl = tiledlayout(3,1,'TileSpacing','tight','Padding','tight');
X = X_1DEULER_dt8e5; V = V_1DEULER_dt8e5;
V(1,1) = 1; V(1,end) = 1; % for visualization purposes
t1 = nexttile(1); hold on; grid on; set(gcf,'color','w');
potplot1 = plot(X, V,'-r','LineWidth',lw);
psiplot1 = plot(X,abs(psi_true_c_1DEULER_dt8e5(:,1)),'-k','LineWidth',lw);
realplot1 = plot(X,real(psi_true_c_1DEULER_dt8e5(:,1)),'-b','LineWidth',lw*0.5);
imagplot1 = plot(X,imag(psi_true_c_1DEULER_dt8e5(:,1)),'-g','LineWidth',lw*0.5);
ylim([-0.5 0.5])
t2 = nexttile(2); hold on; grid on; set(gcf,'color','w');
potplot2 = plot(X, V,'-r','LineWidth',lw);
psiplot2 = plot(X,abs(psi_1DEULER_dt2e4(:,1)),'-k','LineWidth',lw);
realplot2 = plot(X,real(psi_1DEULER_dt2e4(:,1)),'-b','LineWidth',lw*0.5);
imagplot2 = plot(X,imag(psi_1DEULER_dt2e4(:,1)),'-g','LineWidth',lw*0.5);
ylim([-0.5 0.5])
t3 = nexttile(3); hold on; grid on; set(gcf,'color','w');
potplot3 = plot(X, V,'-r','LineWidth',lw);
psiplot3 = plot(X,abs(psi_1DEULER_dt8e4(:,1)),'-k','LineWidth',lw);
realplot3 = plot(X,real(psi_1DEULER_dt8e4(:,1)),'-b','LineWidth',lw*0.5);
imagplot3 = plot(X,imag(psi_1DEULER_dt8e4(:,1)),'-g','LineWidth',lw*0.5);
ylim([-0.5 0.5])
legend("V(x)","|$\psi$|","Re($\psi$)","Im($\psi$)")
picturewidth = 3*34.4/3/4; hw_ratio = 0.2*3*4; fs = 13;
no_snap = 5;
for snap = 1:no_snap
delete(potplot1)
delete(psiplot1)
delete(realplot1)
delete(imagplot1)
delete(potplot2)
delete(psiplot2)
delete(realplot2)
delete(imagplot2)
delete(potplot3)
delete(psiplot3)
delete(realplot3)
delete(imagplot3)
ts_1 = (length(psi_true_c_1DEULER_dt8e5)-1) / (no_snap-1);
ts_2 = (length(psi_1DEULER_dt2e4)-1) / (no_snap-1);
ts_3 = (length(psi_1DEULER_dt8e4)-1) / (no_snap-1);
t1 = nexttile(1); hold on; grid on; set(gcf,'color','w');
potplot1 = plot(X, V,'-r','LineWidth',lw);
psiplot1 = plot(X,abs(psi_true_c_1DEULER_dt8e5(:,(snap-1)*ts_1+1)),'-k','LineWidth',lw);
realplot1 = plot(X,real(psi_true_c_1DEULER_dt8e5(:,(snap-1)*ts_1+1)),'-b','LineWidth',lw*0.5);
imagplot1 = plot(X,imag(psi_true_c_1DEULER_dt8e5(:,(snap-1)*ts_1+1)),'-g','LineWidth',lw*0.5);
t2 = nexttile(2); hold on; grid on; set(gcf,'color','w');
potplot2 = plot(X, V,'-r','LineWidth',lw);
psiplot2 = plot(X,abs(psi_1DEULER_dt2e4(:,(snap-1)*ts_2+1)),'-k','LineWidth',lw);
realplot2 = plot(X,real(psi_1DEULER_dt2e4(:,(snap-1)*ts_2+1)),'-b','LineWidth',lw*0.5);
imagplot2 = plot(X,imag(psi_1DEULER_dt2e4(:,(snap-1)*ts_2+1)),'-g','LineWidth',lw*0.5);
t3 = nexttile(3); hold on; grid on; set(gcf,'color','w');
potplot3 = plot(X, V,'-r','LineWidth',lw);
psiplot3 = plot(X,abs(psi_1DEULER_dt8e4(:,(snap-1)*ts_3+1)),'-k','LineWidth',lw);
realplot3 = plot(X,real(psi_1DEULER_dt8e4(:,(snap-1)*ts_3+1)),'-b','LineWidth',lw*0.5);
imagplot3 = plot(X,imag(psi_1DEULER_dt8e4(:,(snap-1)*ts_3+1)),'-g','LineWidth',lw*0.5);
xlabel('x (nm)');
tx = ['Time step = ', num2str(4/(no_snap-1)*snap-1)];
legend("V(x)","$|\psi|$","Re($\psi$)","Im($\psi$)",'NumColumns',2)
title(t1, tx, 'Interpreter', 'latex')
set(findall(gcf,'-property', 'FontSize'), 'FontSize', fs) % never change fontsize anymore!
set(findall(gcf,'-property', 'Box'), 'Box', 'on') % optional
set(findall(gcf, '-property', 'Interpreter'), 'Interpreter', 'latex')
set(findall(gcf, '-property', 'TickLabelInterpreter'), 'TickLabelInterpreter', 'latex')
set(gcf, 'Units', 'centimeters', 'Position', [2 1 picturewidth hw_ratio*picturewidth])
pos = get(gcf, 'Position');
set(findobj(gcf, 'Type', 'legend'), 'FontSize', fs-3, 'Location', 'southoutside')
set(gca,'LooseInset',get(gca,'TightInset'))
set(gcf, 'PaperPositionMode', 'Auto', 'PaperUnits', 'centimeters', 'PaperSize', [pos(3), pos(4)])
ftx = "1DEULER_" + snap;
drawnow
print(f,ftx,'-dpdf','-vector', '-fillpage')
end
%% Figures for the Report: 1D TDSE: Crank-Nicolson method: SNAPSHOTS
load('1DCN_dt8e5.mat')
load('1DCN_dt2e4.mat')
load('1DCN_dt8e4.mat')
load('1DCN_dt2e3.mat')
set(0, 'DefaultLineLineWidth', 2); lw = 2;
close all; f = figure;
tl = tiledlayout(3,1,'TileSpacing','tight','Padding','tight');
X = X_1DCN_dt8e5; V = V_1DCN_dt8e5;
V(1,1) = 1; V(1,end) = 1; % for visualization purposes
t1 = nexttile(1); hold on; grid on; set(gcf,'color','w');
potplot1 = plot(X, V,'-r','LineWidth',lw);
psiplot1 = plot(X,abs(psi_true_c_1DCN_dt8e5(:,1)),'-k','LineWidth',lw);
realplot1 = plot(X,real(psi_true_c_1DCN_dt8e5(:,1)),'-b','LineWidth',lw*0.5);
imagplot1 = plot(X,imag(psi_true_c_1DCN_dt8e5(:,1)),'-g','LineWidth',lw*0.5);
ylim([-0.5 0.5])
t2 = nexttile(2); hold on; grid on; set(gcf,'color','w');
potplot2 = plot(X, V,'-r','LineWidth',lw);
psiplot2 = plot(X,abs(psi_1DCN_dt2e4(:,1)),'-k','LineWidth',lw);
realplot2 = plot(X,real(psi_1DCN_dt2e4(:,1)),'-b','LineWidth',lw*0.5);
imagplot2 = plot(X,imag(psi_1DCN_dt2e4(:,1)),'-g','LineWidth',lw*0.5);
ylim([-0.5 0.5])
t3 = nexttile(3); hold on; grid on; set(gcf,'color','w');
potplot3 = plot(X, V,'-r','LineWidth',lw);
psiplot3 = plot(X,abs(psi_1DCN_dt8e4(:,1)),'-k','LineWidth',lw);
realplot3 = plot(X,real(psi_1DCN_dt8e4(:,1)),'-b','LineWidth',lw*0.5);
imagplot3 = plot(X,imag(psi_1DCN_dt8e4(:,1)),'-g','LineWidth',lw*0.5);
ylim([-0.5 0.5])
legend("V(x)","|$\psi$|","Re($\psi$)","Im($\psi$)")
picturewidth = 3*34.4/3/4; hw_ratio = 0.2*3*4; fs = 13;
no_snap = 5;
for snap = 1:no_snap
delete(potplot1)
delete(psiplot1)
delete(realplot1)
delete(imagplot1)
delete(potplot2)
delete(psiplot2)
delete(realplot2)
delete(imagplot2)
delete(potplot3)
delete(psiplot3)
delete(realplot3)
delete(imagplot3)
ts_1 = (length(psi_true_c_1DCN_dt8e5)-1) / (no_snap-1);
ts_2 = (length(psi_1DCN_dt2e4)-1) / (no_snap-1);
ts_3 = (length(psi_1DCN_dt8e4)-1) / (no_snap-1);
t1 = nexttile(1); hold on; grid on; set(gcf,'color','w');
potplot1 = plot(X, V,'-r','LineWidth',lw);
psiplot1 = plot(X,abs(psi_true_c_1DCN_dt8e5(:,(snap-1)*ts_1+1)),'-k','LineWidth',lw);
realplot1 = plot(X,real(psi_true_c_1DCN_dt8e5(:,(snap-1)*ts_1+1)),'-b','LineWidth',lw*0.5);
imagplot1 = plot(X,imag(psi_true_c_1DCN_dt8e5(:,(snap-1)*ts_1+1)),'-g','LineWidth',lw*0.5);
t2 = nexttile(2); hold on; grid on; set(gcf,'color','w');
potplot2 = plot(X, V,'-r','LineWidth',lw);
psiplot2 = plot(X,abs(psi_1DCN_dt2e4(:,(snap-1)*ts_2+1)),'-k','LineWidth',lw);
realplot2 = plot(X,real(psi_1DCN_dt2e4(:,(snap-1)*ts_2+1)),'-b','LineWidth',lw*0.5);
imagplot2 = plot(X,imag(psi_1DCN_dt2e4(:,(snap-1)*ts_2+1)),'-g','LineWidth',lw*0.5);
t3 = nexttile(3); hold on; grid on; set(gcf,'color','w');
potplot3 = plot(X, V,'-r','LineWidth',lw);
psiplot3 = plot(X,abs(psi_1DCN_dt8e4(:,(snap-1)*ts_3+1)),'-k','LineWidth',lw);
realplot3 = plot(X,real(psi_1DCN_dt8e4(:,(snap-1)*ts_3+1)),'-b','LineWidth',lw*0.5);
imagplot3 = plot(X,imag(psi_1DCN_dt8e4(:,(snap-1)*ts_3+1)),'-g','LineWidth',lw*0.5);
xlabel('x (nm)');
tx = ['Time step = ', num2str(4/(no_snap-1)*snap-1)];
legend("V(x)","$|\psi|$","Re($\psi$)","Im($\psi$)",'NumColumns',2)
title(t1, tx, 'Interpreter', 'latex')
set(findall(gcf,'-property', 'FontSize'), 'FontSize', fs) % never change fontsize anymore!
set(findall(gcf,'-property', 'Box'), 'Box', 'on') % optional
set(findall(gcf, '-property', 'Interpreter'), 'Interpreter', 'latex')
set(findall(gcf, '-property', 'TickLabelInterpreter'), 'TickLabelInterpreter', 'latex')
set(gcf, 'Units', 'centimeters', 'Position', [2 1 picturewidth hw_ratio*picturewidth])
pos = get(gcf, 'Position');
set(findobj(gcf, 'Type', 'legend'), 'FontSize', fs-3, 'Location', 'southoutside')
set(gca,'LooseInset',get(gca,'TightInset'))
set(gcf, 'PaperPositionMode', 'Auto', 'PaperUnits', 'centimeters', 'PaperSize', [pos(3), pos(4)])
ftx = "1DCN_" + snap;
drawnow
print(f,ftx,'-dpdf','-vector', '-fillpage')
end
%% Figures for the Report: 1D TDSE: Tunneling: SNAPSHOTS
load('1DEULER_dt8e4.mat')
load('1DCN_dt8e4.mat')
set(0, 'DefaultLineLineWidth', 2); lw = 2;
close all; f = figure;
tl = tiledlayout(3,1,'TileSpacing','tight','Padding','tight');
X = X_1DEULER_dt8e4; V = V_1DEULER_dt8e4;
V(1,1) = 1; V(1,end) = 1; % for visualization purposes
t1 = nexttile(1); hold on; grid on; set(gcf,'color','w');
potplot1 = plot(X, V,'-r','LineWidth',lw);
psiplot1 = plot(X,abs(psi_true_c_1DEULER_dt8e4(:,1)),'-k','LineWidth',lw);
realplot1 = plot(X,real(psi_true_c_1DEULER_dt8e4(:,1)),'-b','LineWidth',lw*0.5);
imagplot1 = plot(X,imag(psi_true_c_1DEULER_dt8e4(:,1)),'-g','LineWidth',lw*0.5);
ylim([-0.5 0.5])
t2 = nexttile(2); hold on; grid on; set(gcf,'color','w');
potplot2 = plot(X, V,'-r','LineWidth',lw);
psiplot2 = plot(X,abs(psi_1DEULER_dt8e4(:,1)),'-k','LineWidth',lw);
realplot2 = plot(X,real(psi_1DEULER_dt8e4(:,1)),'-b','LineWidth',lw*0.5);
imagplot2 = plot(X,imag(psi_1DEULER_dt8e4(:,1)),'-g','LineWidth',lw*0.5);
ylim([-0.5 0.5])
t3 = nexttile(3); hold on; grid on; set(gcf,'color','w');
potplot3 = plot(X, V,'-r','LineWidth',lw);
psiplot3 = plot(X,abs(psi_1DCN_dt8e4(:,1)),'-k','LineWidth',lw);
realplot3 = plot(X,real(psi_1DCN_dt8e4(:,1)),'-b','LineWidth',lw*0.5);
imagplot3 = plot(X,imag(psi_1DCN_dt8e4(:,1)),'-g','LineWidth',lw*0.5);
ylim([-0.5 0.5])
legend("V(x)","|$\psi$|","Re($\psi$)","Im($\psi$)")
picturewidth = 3*34.4/3/4; hw_ratio = 0.2*3*4; fs = 13;
no_snap = 5;
for snap = 1:no_snap
delete(potplot1)
delete(psiplot1)
delete(realplot1)
delete(imagplot1)
delete(potplot2)
delete(psiplot2)
delete(realplot2)
delete(imagplot2)
delete(potplot3)
delete(psiplot3)
delete(realplot3)
delete(imagplot3)
ts_1 = (length(psi_true_c_1DEULER_dt8e4)-1) / (no_snap-1);
ts_2 = (length(psi_1DEULER_dt8e4)-1) / (no_snap-1);
ts_3 = (length(psi_1DCN_dt8e4)-1) / (no_snap-1);
t1 = nexttile(1); hold on; grid on; set(gcf,'color','w');
potplot1 = plot(X, V,'-r','LineWidth',lw);
psiplot1 = plot(X,abs(psi_true_c_1DEULER_dt8e4(:,(snap-1)*ts_1+1)),'-k','LineWidth',lw);
realplot1 = plot(X,real(psi_true_c_1DEULER_dt8e4(:,(snap-1)*ts_1+1)),'-b','LineWidth',lw*0.5);
imagplot1 = plot(X,imag(psi_true_c_1DEULER_dt8e4(:,(snap-1)*ts_1+1)),'-g','LineWidth',lw*0.5);
t2 = nexttile(2); hold on; grid on; set(gcf,'color','w');
potplot2 = plot(X, V,'-r','LineWidth',lw);
psiplot2 = plot(X,abs(psi_1DEULER_dt8e4(:,(snap-1)*ts_2+1)),'-k','LineWidth',lw);
realplot2 = plot(X,real(psi_1DEULER_dt8e4(:,(snap-1)*ts_2+1)),'-b','LineWidth',lw*0.5);
imagplot2 = plot(X,imag(psi_1DEULER_dt8e4(:,(snap-1)*ts_2+1)),'-g','LineWidth',lw*0.5);
t3 = nexttile(3); hold on; grid on; set(gcf,'color','w');
potplot3 = plot(X, V,'-r','LineWidth',lw);
psiplot3 = plot(X,abs(psi_1DCN_dt8e4(:,(snap-1)*ts_3+1)),'-k','LineWidth',lw);
realplot3 = plot(X,real(psi_1DCN_dt8e4(:,(snap-1)*ts_3+1)),'-b','LineWidth',lw*0.5);
imagplot3 = plot(X,imag(psi_1DCN_dt8e4(:,(snap-1)*ts_3+1)),'-g','LineWidth',lw*0.5);
xlabel('x (nm)');
tx = ['Time step = ', num2str(4/(no_snap-1)*snap-1)];
legend("V(x)","$|\psi|$","Re($\psi$)","Im($\psi$)",'NumColumns',2)
title(t1, tx, 'Interpreter', 'latex')
set(findall(gcf,'-property', 'FontSize'), 'FontSize', fs) % never change fontsize anymore!
set(findall(gcf,'-property', 'Box'), 'Box', 'on') % optional
set(findall(gcf, '-property', 'Interpreter'), 'Interpreter', 'latex')
set(findall(gcf, '-property', 'TickLabelInterpreter'), 'TickLabelInterpreter', 'latex')
set(gcf, 'Units', 'centimeters', 'Position', [2 1 picturewidth hw_ratio*picturewidth])
pos = get(gcf, 'Position');
set(findobj(gcf, 'Type', 'legend'), 'FontSize', fs-3, 'Location', 'southoutside')
set(gca,'LooseInset',get(gca,'TightInset'))
set(gcf, 'PaperPositionMode', 'Auto', 'PaperUnits', 'centimeters', 'PaperSize', [pos(3), pos(4)])
ftx = "1Dcomparison_" + snap;
drawnow
print(f,ftx,'-dpdf','-vector', '-fillpage')
end
%% GIF: 1D
load('1DEULER_dt8e4.mat')
load('1DCN_dt8e4.mat')
set(0, 'DefaultLineLineWidth', 2); lw = 2;
close all; f = figure; filename = "1d_tunneling.gif"; set(gcf,'color','w');
tl = tiledlayout(3,1,'TileSpacing','tight','Padding','tight');
X = X_1DEULER_dt8e4; V = V_1DEULER_dt8e4;
V(1,1) = 1; V(1,end) = 1; % for visualization purposes
t1 = nexttile(1); hold on; grid on; set(gcf,'color','w');
potplot1 = plot(X, V,'-r','LineWidth',lw);
psiplot1 = plot(X,abs(psi_true_c_1DEULER_dt8e4(:,1)),'-k','LineWidth',lw);
realplot1 = plot(X,real(psi_true_c_1DEULER_dt8e4(:,1)),'-b','LineWidth',lw*0.5);
imagplot1 = plot(X,imag(psi_true_c_1DEULER_dt8e4(:,1)),'-g','LineWidth',lw*0.5);
ylim([-0.5 0.5])
t2 = nexttile(2); hold on; grid on; set(gcf,'color','w');
potplot2 = plot(X, V,'-r','LineWidth',lw);
psiplot2 = plot(X,abs(psi_1DEULER_dt8e4(:,1)),'-k','LineWidth',lw);
realplot2 = plot(X,real(psi_1DEULER_dt8e4(:,1)),'-b','LineWidth',lw*0.5);
imagplot2 = plot(X,imag(psi_1DEULER_dt8e4(:,1)),'-g','LineWidth',lw*0.5);
ylim([-0.5 0.5])
t3 = nexttile(3); hold on; grid on; set(gcf,'color','w');
potplot3 = plot(X, V,'-r','LineWidth',lw);
psiplot3 = plot(X,abs(psi_1DCN_dt8e4(:,1)),'-k','LineWidth',lw);
realplot3 = plot(X,real(psi_1DCN_dt8e4(:,1)),'-b','LineWidth',lw*0.5);
imagplot3 = plot(X,imag(psi_1DCN_dt8e4(:,1)),'-g','LineWidth',lw*0.5);
ylim([-0.5 0.5])
legend("V(x)","|$\psi$|","Re($\psi$)","Im($\psi$)")
picturewidth = 3*34.4/3/4; hw_ratio = 0.2*3*4; fs = 13;
no_snap = 101;
for snap = 1:no_snap
delete(potplot1)
delete(psiplot1)
delete(realplot1)
delete(imagplot1)
delete(potplot2)
delete(psiplot2)
delete(realplot2)
delete(imagplot2)
delete(potplot3)
delete(psiplot3)
delete(realplot3)
delete(imagplot3)
ts_1 = (length(psi_true_c_1DEULER_dt8e4)-1) / (no_snap-1);
ts_2 = (length(psi_1DEULER_dt8e4)-1) / (no_snap-1);
ts_3 = (length(psi_1DCN_dt8e4)-1) / (no_snap-1);
t1 = nexttile(1); hold on; grid on; set(gcf,'color','w');
potplot1 = plot(X, V,'-r','LineWidth',lw);
psiplot1 = plot(X,abs(psi_true_c_1DEULER_dt8e4(:,(snap-1)*ts_1+1)),'-k','LineWidth',lw);
realplot1 = plot(X,real(psi_true_c_1DEULER_dt8e4(:,(snap-1)*ts_1+1)),'-b','LineWidth',lw*0.5);
imagplot1 = plot(X,imag(psi_true_c_1DEULER_dt8e4(:,(snap-1)*ts_1+1)),'-g','LineWidth',lw*0.5);
title("True solution, (Euler, $\Delta t = 2 \times 10^{-5}$)")
t2 = nexttile(2); hold on; grid on; set(gcf,'color','w');
potplot2 = plot(X, V,'-r','LineWidth',lw);
psiplot2 = plot(X,abs(psi_1DEULER_dt8e4(:,(snap-1)*ts_2+1)),'-k','LineWidth',lw);
realplot2 = plot(X,real(psi_1DEULER_dt8e4(:,(snap-1)*ts_2+1)),'-b','LineWidth',lw*0.5);
imagplot2 = plot(X,imag(psi_1DEULER_dt8e4(:,(snap-1)*ts_2+1)),'-g','LineWidth',lw*0.5);
title("Euler method, $\Delta t = 8\times 10^{-4}$")
t3 = nexttile(3); hold on; grid on; set(gcf,'color','w');
potplot3 = plot(X, V,'-r','LineWidth',lw);
psiplot3 = plot(X,abs(psi_1DCN_dt8e4(:,(snap-1)*ts_3+1)),'-k','LineWidth',lw);
realplot3 = plot(X,real(psi_1DCN_dt8e4(:,(snap-1)*ts_3+1)),'-b','LineWidth',lw*0.5);
imagplot3 = plot(X,imag(psi_1DCN_dt8e4(:,(snap-1)*ts_3+1)),'-g','LineWidth',lw*0.5);
title("Crank-Nicolson method, $\Delta t = 8\times 10^{-4}$")
xlabel('x (nm)');
tx = ['Time step = ', num2str(8e-4*(snap-1)*ts_2)];
legend("V(x)","$|\psi|$","Re($\psi$)","Im($\psi$)",'NumColumns',2)
title(tl, tx, 'Interpreter', 'latex')
set(findall(gcf,'-property', 'FontSize'), 'FontSize', fs) % never change fontsize anymore!
set(findall(gcf,'-property', 'Box'), 'Box', 'on') % optional
set(findall(gcf, '-property', 'Interpreter'), 'Interpreter', 'latex')
set(findall(gcf, '-property', 'TickLabelInterpreter'), 'TickLabelInterpreter', 'latex')
set(gcf, 'Units', 'centimeters', 'Position', [2 1 picturewidth hw_ratio*picturewidth])
pos = get(gcf, 'Position');
set(findobj(gcf, 'Type', 'legend'), 'FontSize', fs-3, 'Location', 'southoutside')
set(gca,'LooseInset',get(gca,'TightInset'))
set(gcf, 'PaperPositionMode', 'Auto', 'PaperUnits', 'centimeters', 'PaperSize', [pos(3), pos(4)])
drawnow
if snap == 1
frame = getframe(f);
im = frame2im(frame);
[imind,cm] = rgb2ind(im,256);
imwrite(imind,cm,filename,'gif','DelayTime',0.01, 'Loopcount',inf);
else
frame = getframe(f);
im = frame2im(frame);
[imind,cm] = rgb2ind(im,256);
imwrite(imind,cm,filename,'gif','DelayTime',0.01, 'WriteMode','append');
end
end
%% Save data: Dimension = 2, Method = Explicit Euler
% Run after each simulation
if dt == 8e-5
X_2DEULER_dt8e5 = X;
Y_2DEULER_dt8e5 = Y;
V_2DEULER_dt8e5 = V;
t_2DEULER_dt8e5 = t;
psi_2DEULER_dt8e5 = psi;
psi_true_c_2DEULER_dt8e5 = psi_true_c;
save 2DEULER_dt8e5.mat V_2DEULER_dt8e5 X_2DEULER_dt8e5 Y_2DEULER_dt8e5 t_2DEULER_dt8e5 ...
psi_2DEULER_dt8e5 psi_true_c_2DEULER_dt8e5
clear
elseif dt == 2e-4
X_2DEULER_dt2e4 = X;
Y_2DEULER_dt2e4 = Y;
V_2DEULER_dt2e4 = V;
t_2DEULER_dt2e4 = t;
psi_2DEULER_dt2e4 = psi;
psi_true_c_2DEULER_dt2e4 = psi_true_c;
save 2DEULER_dt2e4.mat V_2DEULER_dt2e4 X_2DEULER_dt2e4 Y_2DEULER_dt2e4 t_2DEULER_dt2e4 ...
psi_2DEULER_dt2e4 psi_true_c_2DEULER_dt2e4
clear
elseif dt == 8e-4
X_2DEULER_dt8e4 = X;
Y_2DEULER_dt8e4 = Y;
V_2DEULER_dt8e4 = V;
t_2DEULER_dt8e4 = t;
psi_2DEULER_dt8e4 = psi;
psi_true_c_2DEULER_dt8e4 = psi_true_c;
save 2DEULER_dt8e4.mat V_2DEULER_dt8e4 X_2DEULER_dt8e4 Y_2DEULER_dt8e4 t_2DEULER_dt8e4 ...
psi_2DEULER_dt8e4 psi_true_c_2DEULER_dt8e4
clear
elseif dt == 2e-3
X_2DEULER_dt2e3 = X;
Y_2DEULER_dt2e3 = Y;
V_2DEULER_dt2e3 = V;
t_2DEULER_dt2e3 = t;
psi_2DEULER_dt2e3 = psi;
psi_true_c_2DEULER_dt2e3 = psi_true_c;
save 2DEULER_dt2e3.mat V_2DEULER_dt2e3 X_2DEULER_dt2e3 Y_2DEULER_dt2e3 t_2DEULER_dt2e3 ...
psi_2DEULER_dt2e3 psi_true_c_2DEULER_dt2e3
clear
end
%% Save data: Dimension = 2, Method = Crank-Nicolson
% Run after each simulation
if dt == 8e-5
X_2DCN_dt8e5 = X;
Y_2DCN_dt8e5 = Y;
V_2DCN_dt8e5 = V;
t_2DCN_dt8e5 = t;
psi_2DCN_dt8e5 = psi;
psi_true_c_2DCN_dt8e5 = psi_true_c;
save 2DCN_dt8e5.mat V_2DCN_dt8e5 X_2DCN_dt8e5 Y_2DCN_dt8e5 t_2DCN_dt8e5 psi_2DCN_dt8e5 psi_true_c_2DCN_dt8e5
clear
elseif dt == 2e-4
X_2DCN_dt2e4 = X;
Y_2DCN_dt2e4 = Y;
V_2DCN_dt2e4 = V;
t_2DCN_dt2e4 = t;
psi_2DCN_dt2e4 = psi;
psi_true_c_2DCN_dt2e4 = psi_true_c;
save 2DCN_dt2e4.mat V_2DCN_dt2e4 X_2DCN_dt2e4 Y_2DCN_dt2e4 t_2DCN_dt2e4 psi_2DCN_dt2e4 psi_true_c_2DCN_dt2e4
clear
elseif dt == 8e-4
X_2DCN_dt8e4 = X;
Y_2DCN_dt8e4 = Y;
V_2DCN_dt8e4 = V;
t_2DCN_dt8e4 = t;
psi_2DCN_dt8e4 = psi;
psi_true_c_2DCN_dt8e4 = psi_true_c;
save 2DCN_dt8e4.mat V_2DCN_dt8e4 X_2DCN_dt8e4 Y_2DCN_dt8e4 t_2DCN_dt8e4 psi_2DCN_dt8e4 psi_true_c_2DCN_dt8e4
clear
elseif dt == 2e-3
X_2DCN_dt2e3 = X;
Y_2DCN_dt2e3 = Y;
V_2DCN_dt2e3 = V;
t_2DCN_dt2e3 = t;
psi_2DCN_dt2e3 = psi;
psi_true_c_2DCN_dt2e3 = psi_true_c;
save 2DCN_dt2e3.mat V_2DCN_dt2e3 X_2DCN_dt2e3 Y_2DCN_dt2e3 t_2DCN_dt2e3 psi_2DCN_dt2e3 psi_true_c_2DCN_dt2e3
clear
end
%% Figures for the Report: 2D TDSE: Euler method: ERROR
load('2DEULER_dt8e5.mat')
load('2DEULER_dt2e4.mat')
load('2DEULER_dt8e4.mat')
load('2DEULER_dt2e3.mat')
error_2DEULER_dt8e5 = abs(trapz(trapz(psi_2DEULER_dt8e5-psi_true_c_2DEULER_dt8e5,1)));
error_2DEULER_dt2e4 = abs(trapz(trapz(psi_2DEULER_dt2e4-psi_true_c_2DEULER_dt2e4,1)));
error_2DEULER_dt8e4 = abs(trapz(trapz(psi_2DEULER_dt8e4-psi_true_c_2DEULER_dt8e4,1)));
error_2DEULER_dt2e3 = abs(trapz(trapz(psi_2DEULER_dt2e3-psi_true_c_2DEULER_dt2e3,1)));
error_2DEULER_dt8e5 = reshape(error_2DEULER_dt8e5,[length(error_2DEULER_dt8e5) 1]);
error_2DEULER_dt2e4 = reshape(error_2DEULER_dt2e4,[length(error_2DEULER_dt2e4) 1]);
error_2DEULER_dt8e4 = reshape(error_2DEULER_dt8e4,[length(error_2DEULER_dt8e4) 1]);
error_2DEULER_dt2e3 = reshape(error_2DEULER_dt2e3,[length(error_2DEULER_dt2e3) 1]);
set(0, 'DefaultLineLineWidth', 2); close all;
tl = tiledlayout(1,2,'TileSpacing','tight','Padding','tight');
t1 = nexttile(1); hold on; grid on
plot(t_2DEULER_dt8e5,error_2DEULER_dt8e5)
plot(t_2DEULER_dt2e4,error_2DEULER_dt2e4)
plot(t_2DEULER_dt8e4,error_2DEULER_dt8e4)
plot(t_2DEULER_dt2e3,error_2DEULER_dt2e3)
legend("$\Delta t = 8\times10^{-5}$","$\Delta t = 2\times10^{-4}$",...
"$\Delta t = 8\times10^{-4}$","$\Delta t = 2\times10^{-3}$")
t2 = nexttile(2); hold on; grid on
plot(t_2DEULER_dt8e5,error_2DEULER_dt8e5)
plot(t_2DEULER_dt2e4,error_2DEULER_dt2e4)
plot(t_2DEULER_dt8e4,error_2DEULER_dt8e4)
plot(t_2DEULER_dt2e3,error_2DEULER_dt2e3)
xlim([0 0.1])
%ylim([0 0.2*1e-2])
title(t1,"(a) Error for explicit Euler method in 2D")
title(t2,"(b) Close-up to initial behavior of error")
title(tl,"POTENTIAL: Free-particle", 'Interpreter', 'latex')
xlabel(tl,'$\textrm{Time step}$','Interpreter','latex');
ylabel(tl,'$\textrm{Error}= |\int \int \psi - \psi_{true} \; dx dy|$','Interpreter','latex');
picturewidth = 3*34.4/3; hw_ratio = 0.2; fs = 13;
set(findall(gcf,'-property', 'FontSize'), 'FontSize', fs) % never change fontsize anymore!
set(findall(gcf,'-property', 'Box'), 'Box', 'on') % optional
set(findall(gcf, '-property', 'Interpreter'), 'Interpreter', 'latex')
set(findall(gcf, '-property', 'TickLabelInterpreter'), 'TickLabelInterpreter', 'latex')
set(gcf, 'Units', 'centimeters', 'Position', [2 1 picturewidth hw_ratio*picturewidth])
pos = get(gcf, 'Position');
set(findobj(gcf, 'Type', 'legend'), 'FontSize', fs-3, 'Location', 'northeast')
set(gca,'LooseInset',get(gca,'TightInset'))
set(gcf, 'PaperPositionMode', 'Auto', 'PaperUnits', 'centimeters', 'PaperSize', [pos(3), pos(4)])
%% Figures for the Report: 2D TDSE: Crank-Nicolson method: ERROR
load('2DCN_dt8e5.mat')
load('2DCN_dt2e4.mat')
load('2DCN_dt8e4.mat')
load('2DCN_dt2e3.mat')
error_2DCN_dt8e5 = abs(trapz(trapz(psi_2DCN_dt8e5-psi_true_c_2DCN_dt8e5,1)));
error_2DCN_dt2e4 = abs(trapz(trapz(psi_2DCN_dt2e4-psi_true_c_2DCN_dt2e4,1)));
error_2DCN_dt8e4 = abs(trapz(trapz(psi_2DCN_dt8e4-psi_true_c_2DCN_dt8e4,1)));
error_2DCN_dt2e3 = abs(trapz(trapz(psi_2DCN_dt2e3-psi_true_c_2DCN_dt2e3,1)));
error_2DCN_dt8e5 = reshape(error_2DCN_dt8e5,[length(error_2DCN_dt8e5) 1]);
error_2DCN_dt2e4 = reshape(error_2DCN_dt2e4,[length(error_2DCN_dt2e4) 1]);
error_2DCN_dt8e4 = reshape(error_2DCN_dt8e4,[length(error_2DCN_dt8e4) 1]);
error_2DCN_dt2e3 = reshape(error_2DCN_dt2e3,[length(error_2DCN_dt2e3) 1]);
set(0, 'DefaultLineLineWidth', 2); close all;
tl = tiledlayout(1,2,'TileSpacing','tight','Padding','tight');
t1 = nexttile(1); hold on; grid on
plot(t_2DCN_dt8e5,error_2DCN_dt8e5)
plot(t_2DCN_dt2e4,error_2DCN_dt2e4)
plot(t_2DCN_dt8e4,error_2DCN_dt8e4)
plot(t_2DCN_dt2e3,error_2DCN_dt2e3)
legend("$\Delta t = 8\times10^{-5}$","$\Delta t = 2\times10^{-4}$",...
"$\Delta t = 8\times10^{-4}$","$\Delta t = 2\times10^{-3}$")
t2 = nexttile(2); hold on; grid on
plot(t_2DCN_dt8e5,error_2DCN_dt8e5)
plot(t_2DCN_dt2e4,error_2DCN_dt2e4)
plot(t_2DCN_dt8e4,error_2DCN_dt8e4)
plot(t_2DCN_dt2e3,error_2DCN_dt2e3)
xlim([0 0.1])
%ylim([0 0.2*1e-2])
title(t1,"(a) Error for Crank-Nicolson method in 2D")
title(t2,"(b) Close-up to initial behavior of error")
title(tl,"POTENTIAL: Free-particle", 'Interpreter', 'latex')
xlabel(tl,'$\textrm{Time step}$','Interpreter','latex');
ylabel(tl,'$\textrm{Error}= |\int \int \psi - \psi_{true} \; dx dy|$','Interpreter','latex');
picturewidth = 3*34.4/3; hw_ratio = 0.2; fs = 13;
set(findall(gcf,'-property', 'FontSize'), 'FontSize', fs) % never change fontsize anymore!
set(findall(gcf,'-property', 'Box'), 'Box', 'on') % optional
set(findall(gcf, '-property', 'Interpreter'), 'Interpreter', 'latex')
set(findall(gcf, '-property', 'TickLabelInterpreter'), 'TickLabelInterpreter', 'latex')
set(gcf, 'Units', 'centimeters', 'Position', [2 1 picturewidth hw_ratio*picturewidth])
pos = get(gcf, 'Position');
set(findobj(gcf, 'Type', 'legend'), 'FontSize', fs-3, 'Location', 'northwest')
set(gca,'LooseInset',get(gca,'TightInset'))
set(gcf, 'PaperPositionMode', 'Auto', 'PaperUnits', 'centimeters', 'PaperSize', [pos(3), pos(4)])
%% Figures for the Report: 2D TDSE: Euler method: SNAPSHOTS
load('2DEULER_dt8e5.mat')
load('2DEULER_dt2e4.mat')
load('2DEULER_dt8e4.mat')
load('2DEULER_dt2e3.mat')
x = X_2DEULER_dt2e3(1,:); y = Y_2DEULER_dt2e3(:,1);
set(0, 'DefaultLineLineWidth', 2); lw = 2; fs = 13;
close all; f = figure;
tl = tiledlayout(3,1,'TileSpacing','tight','Padding','tight');
X = X_2DEULER_dt8e5; Y = Y_2DEULER_dt8e5; V = V_2DEULER_dt8e5;
V(:,1) = 1; V(:,end) = 1; V(1,:) = 1; V(end,:) = 1; % for visualization purposes
t1 = nexttile(1); hold on; set(gcf,'color','w');
potplot1 = imagesc(x,y,V);
psiplot1 = imagesc(x,y,abs(psi_true_c_2DEULER_dt2e3(:,:,1)));
t2 = nexttile(2); hold on; set(gcf,'color','w');
potplot2 = imagesc(x,y,V);
psiplot2 = imagesc(x,y,abs(psi_2DEULER_dt2e4(:,:,1)));
t3 = nexttile(3); hold on; set(gcf,'color','w');
potplot3 = imagesc(x,y,V);
psiplot3 = imagesc(x,y,abs(psi_2DEULER_dt8e4(:,:,1)));
c = colorbar; clim([0 25]); c.Location = "southoutside";
picturewidth = 3*34.4/3/4; hw_ratio = 0.2*3*4;