time -independe​nt-Schrodi​nger-like-​1D-pde

I have [-\nabla^2+ (h*G(x)+H(x)) ] y(x) = h^2 y(x), where, for a range of x values, G(x) and H(x) values are known (not their functions are known). How to solve h and y(x) simultaneously.

Answer 1

[Amish]

Hi Pritha,

I see that you are looking for a way to solve time-independent Schrödinger-like 1D Partial Differential Equation. Solving the time-independent Schrödinger-like 1D PDE, where "G(x)" and "H(x)" are given as data points rather than as analytical functions, typically requires numerical methods.

The equation you've provided resembles the time-independent Schrödinger equation with a potential "V(x) = hG(x) + H(x)", and you're looking to solve for the eigenvalues "h^2" and corresponding eigenfunctions "y(x)".

Follwing are the high-level steps you might do using the finite difference methods:

1. Discretize the Spatial Domain
2. Discretize the Equation
3. Construct the Matrix Equation
4. Solve the Eigenvalue Problem
5. Post-Process

You can find a generic MATLAB code for the same below:

% Assuming that the Gx and Hx are vectors containing the values of G(x) and H(x) at discrete points x
N = length(Gx); % Number of points
dx = x(2) - x(1); % Uniform spacing

% Make a tridiagonal matrix for the discretized -nabla^2 operator
main_diag = -2 * ones(N, 1) / dx^2;
off_diag = ones(N-1, 1) / dx^2;
laplacian_matrix = diag(main_diag) + diag(off_diag, 1) + diag(off_diag, -1);

% Add the potential terms to the diagonal
A = laplacian_matrix + diag(h*Gx + Hx);

% Solve the eigenvalue problem
[eigenvectors, eigenvalues] = eig(A);

% Extract eigenvalues and convert them to h^2
h_squared = diag(eigenvalues);

The above code assumes the conditions of a uniform grid and boundary conditions that the wavefunction "y(x)" goes to zero at the boundaries of the domain.

Here are some of the documentation links for your reference:

diag: https://www.mathworks.com/help/matlab/ref/diag.html

eig: https://www.mathworks.com/help/matlab/ref/eig.html

eig (symbolic matrices): https://www.mathworks.com/help/symbolic/eig.html

Hope this helps!

[pritha]

Thanks for you reply Amish. I'm getting "Unrecognized function or variable 'h'." while running. Could you suggest how to solve this?

[pritha]

Here eigen values are 'h' not 'h^2' and it is also multiplied with the potential G(x). So it cant be extracted simply from eig(A). Could you please help me out?