3D stationary Schrodinger - numerical solution

[izh-21251]

Hi!

My problem is connected with solving 3-dimentional Schrodinger equation for descrete spectrum.
I need to consider arbitrary 3D-potential and find characteristics of boundary levels, produced by this potential (energy levels, quantum numbers...)

All I found over the internet were algorithms only for a one-dimentional (cartesian or spherical) problem.

If anyone knows anything about methods for numerical solving 3D stationary Schrodinger for arbitrary potential, I will be most obliged!

Thanks in advance!
Ivan

 

[jvicens]

Hi Ivan,

Thank you for reaching out with your question. Solving the 3-dimensional Schrodinger equation for discrete spectrum can be a complex task, but there are several methods that can be used to tackle this problem. One approach is to use the finite element method, which involves discretizing the problem into small elements and using numerical techniques to solve for the wavefunction and energy levels at each element. This method is commonly used in computational quantum mechanics and has been successfully applied to various 3-dimensional potentials.

Another method is the variational method, where the wavefunction is approximated by a trial function and the energy is minimized with respect to this function. This method has been used to solve for the energy levels of many 3-dimensional potentials, including the hydrogen atom.

Additionally, there are also numerical methods such as the shooting method and the relaxation method that can be used to solve for the energy levels of 3-dimensional potentials. These methods involve solving the Schrodinger equation iteratively and adjusting the parameters until the desired energy level is reached.

I hope these suggestions will be helpful in your research. Best of luck in your studies!