why the wavefunction is wrong by solving the Schrodinger equation numerically.?

2 次查看（过去 30 天）

显示更早的评论

Daniel Niu 2023-8-9

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2006707-why-the-wavefunction-is-wrong-by-solving-the-schrodinger-equation-numerically

评论： Dyuman Joshi 2023-8-10

在 MATLAB Online 中打开

clear

% Constants in atomic units

V0 = 0.37;

a0 = 8;

hbar = 1; % Reduced Planck's constant

m = 1; % Electron mass

e = 1; % Electron charge

% Define the size of the region and the number of points for discretization

L = 4 * a0; % Region size

N = 300; % Number of points for discretization

dx = L / N; % Discretization step

% Define the potential function

x = linspace(-L/2, L/2, N);

V=x.^2;

% Construct the kinetic energy matrix T using finite difference approximation

d2dx2 = -2*diag(ones(1, N)) + diag(ones(1, N-1), 1) + diag(ones(1, N-1), -1);

d2dx2 = d2dx2 / dx^2;

T = -hbar^2 / (2*m) * d2dx2;

% Construct the potential energy matrix V

V_matrix = diag(V);

% Form the Hamiltonian matrix H = T + V

H = T + V_matrix;

% Solve the eigenvalue problem for the Hamiltonian matrix

[eigenfunctions, eigenvalues] = eig(H);

% Sort the eigenvalues and corresponding eigenfunctions in ascending order

[eigenvalues, idx] = sort(diag(eigenvalues));

eigenfunctions = eigenfunctions(:, idx);

% Plot the potential function and the first few eigenfunctions

num_plot_eigenfunctions = 5;

figure;

% Define custom colors for the eigenfunctions

colors = {'y', 'r', 'g', 'k', 'm'};

plot(x, V, 'b', 'LineWidth', 2);

hold on;

for i = 1:num_plot_eigenfunctions

eigenfunction = eigenfunctions(:, i);

eigenfunction = eigenfunction / max(abs(eigenfunction)); % Normalize the eigenfunction

plot(x, eigenfunction + eigenvalues(i), 'Color', colors{i}, 'LineWidth', 1.5);

%plot(x, eigenfunction + eigenvalues(i), 'r', 'LineWidth', 1.5);

end

hold off;

xlabel('x (Atomic Units)');

ylabel('Energy (Atomic Units)');

title('Potential Function and Eigenfunctions');

% Place the legend in the southeast corner

legend('V(x)', 'Eigenfunction 1', 'Eigenfunction 2', 'Eigenfunction 3', 'Eigenfunction 4', 'Eigenfunction 5', 'Location', 'southeast');

legend('boxoff');

xlim([-4 4]);

ylim([0 10]);

%Please pay attention to eigenfunction3 and eigenfunction4, they are inverted compare with quantum mechanics textbook

%for example see:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html

%Your help would be highly appreciated!

1 个评论
显示 -1更早的评论隐藏 -1更早的评论

Dyuman Joshi 2023-8-10

The discretization step should be x(2)-x(1).

However, the problem you are observing here is due to the fact the eigenvectors are not unique i.e. they may vary in sign. You can multiply an eigenvector by -1 (or any constant for that matter), and the generated vector will still be an eigenvector corresponding to the eigenvalue.

And unfortunately, you can not predict which eigenvector will have which sign. Thus, any particular Eigenfunction that you want to plot, might be flipped over y axis. You will observe a similar effect for other Eigenfunctions when you'll plot more of them.

请先登录，再进行评论。

请先登录，再回答此问题。