58 lines
2.0 KiB
Matlab
58 lines
2.0 KiB
Matlab
clear all;
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%physical constants in MKS units
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hbar = 1.054e-34;
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q = 1.602e-19;
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m = 9.110e-31;
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epsilon_0 = 8.854e-12;
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%generate lattice
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N = 100; %number of lattice points
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n = [1:N]; %lattice points
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a = 0.1e-10; %lattice constant
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r = a * n; %x-coordinates
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t0 = (hbar^2)/(2*m*a^2)/q; %encapsulating factor
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L = a * (N+1); %total length of consideration
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%set up Hamiltonian matrix
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U = -q^2./(4*pi*epsilon_0.*r) * (1/q); %potential at r in [eV]
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main_diag = diag(2*t0*ones(1,N)+U,0); %create main diagonal matrix
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lower_diag = diag(-t0*ones(1,N-1),-1); %create lower diagonal matrix
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upper_diag = diag(-t0*ones(1,N-1),+1); %create upper diagonal matrix
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H = main_diag + lower_diag + upper_diag; %sum to get Hamiltonian matrix
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[eigenvectors,E_diag] = eig(H); %"eigenvectors" is a matrix wherein each column is an eigenvector
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%"E_diag" is a diagonal matrix where the
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%corresponding eigenvalues are on the
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%diagonal.
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E_col = diag(E_diag); %folds E_diag into a column vector of eigenvalues
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% return eigenvectors for the 1st and 50th eigenvalues
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phi_1 = eigenvectors(:,1);
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phi_2 = eigenvectors(:,2);
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% find the probability densities of position for 1st and 50th eigenvectors
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P_1 = phi_1 .* conj(phi_1);
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P_2 = phi_2 .* conj(phi_2);
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% Plot the probability densities for 1st and 2nd eigenvectors
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figure(1); clf; h = plot(r,P_1,'k-');
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grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
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xlabel('RADIAL POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]');
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yticks([0.02 0.04 0.06 0.08 0.10 0.12]);
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legend('n=1');
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axis([0 1e-9 0 0.12]);
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figure(2); clf; h = plot(r,P_2,'k-');
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grid on; set(h,'linewidth',[2.0]); set(gca,'Fontsize',[18]);
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xlabel('RADIAL POSITION [m]'); ylabel('PROBABILITY DENSITY [1/m]');
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yticks([0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04]);
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legend('n=2');
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axis([0 1e-9 0 0.04]); |