Numerical solution to Schrödinger equation - eigenvalues

[CINA]

Not sure whether to post this here or in QM: I trying to numerically solve the Schrödinger equation for the Woods-Saxon Potential and find the energy eigenvalues and eigenfucnctions but I am confused about how exactly the eigenvalues come about. I've solved some differential equations in the past using the Runge–Kutta method, but nothing with eigenvalues. From what I've seen Numerov's method is the way to solve Schrödinger's equation but I don't see how solutions that gives the energy eigenvalues or eigenfunctions. Wouldn't numerically solving the DE just give one solution? I seen some mention of "tridiagonal matrices" being generated somehow, but am not sure what the elements of that matrix would be. Any help would be appreciated.

Thanks

 

[Quantum Defect]

Not sure whether to post this here or in QM: I trying to numerically solve the Schrödinger equation for the Woods-Saxon Potential and find the energy eigenvalues and eigenfucnctions but I am confused about how exactly the eigenvalues come about. I've solved some differential equations in the past using the Runge–Kutta method, but nothing with eigenvalues. From what I've seen Numerov's method is the way to solve Schrödinger's equation but I don't see how solutions that gives the energy eigenvalues or eigenfunctions. Wouldn't numerically solving the DE just give one solution? I seen some mention of "tridiagonal matrices" being generated somehow, but am not sure what the elements of that matrix would be. Any help would be appreciated.

Thanks

Numerov-Cooley (or Cooley Numerov) method is an iterative method to numerically solve the Schrodinger equation.

You guess the eigenenergy, forward and backward integrate the wave fnction to the midpoint, and compare first derivatives. In Cooley's implementation, the difference in first derivatives is used to calculate a correction to the energy. You redo the process until the first derivatives are "the same," going forward and backward. For most systems, there are many solutions possible [ e.g. harmonic oscillator, infinite solutions with E = (n+1/2) E_0 (n= 0, 1, 2, ...) ]