%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%b % Problem Set 1 % 1.3 Solutions % V. Anant % Mildly modified and more fully anotated by W. Kaminsky 9/19/2006 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % preamble clear; close all; format compact % x row vector dx = 0.02; x = [-20:dx:20]; % x measured in nanometers. % L row vector L = [0.5 1 2]; % L measured in nanometers. % Calculate Psi_L % first make a mesh (see 'help meshgrid') [xmesh, Lmesh] = meshgrid(x,L); % then calculate psi psi = (pi.*Lmesh.^2).^(-1/4).*exp(-xmesh.^2./(2*Lmesh.^2)); % this is the magnitude squared of psi psi_sq = conj(psi).*psi; % now plot magnitude squared figure(1); plot(x, psi_sq) axis([-6 6 0 max(max(psi_sq))]) title('Probability Density of Gaussian Wavefunctions'); xlabel('x (nm)'); ylabel('|\Psi|^2 (nm^{-1})'); legend('L = 0.5nm', 'L = 1nm', 'L = 2nm'); %%%%%%%%%%% % part (c) % Normalization Calculation psi_norm = sum(psi_sq,2)*dx % Check: Sum of each row of psi_sq should be 1. % Syntax Note: As the sum is over the rows, use a 2 in the argument % of the sum function %%%%%%%%%%% % part (d) % Mean Value Calculation: psi_mean = sum(conj(psi).*xmesh.*psi.*dx,2) % measured in nanometers % Check: All the Gaussians have mean 0. % Thus, psi_mean (see below) should be extremely close to 0 % (The discrepancy will be primarily due to the discretization.) % Variance Calculation psi_meansq = sum(conj(psi).*xmesh.^2.*psi.*dx,2) % measured in nanometrers % Check: The Gaussians squared are of the form e^{-x^2 / L^2} % Thus, the variances are \sigma^2 = L^2 / 2, and so psi_meansq (see % below) should take values extremely close to 0.125 nm^2 for L = 0.5 nm, % 0.5 nm^2 for L = 1.0 nm, and 2.0 nm^2 for L = 2.0 nm. % Uncertainty (Standard Deviation) Calculation delta_x = sqrt(psi_meansq - psi_mean.^2) % % As the psi_mean should be [0 nm 0 nm 0 nm] and psi_meansq should be % [0.125 nm^2, 0.5 nm^2, 2.0 nm^2] for L = [0.5 nm 1.0 nm 2.0 nm] % respectively, delta_x should simply be the square root of psi_meansq and % thus [1/sqrt(8) nm 1/sqrt(2) nm sqrt(2) nm] % (to 4 decimal places [0.3536 nm, 0.7071 nm, 1.4142 nm])