Numerical solution to the Schrodinger eqn. using Finite Difference Method

[patrykh18]

As part of my project I was asked to use the finite difference method to solve Schrödinger equation. I see how you can turn it into a matrix equation, but I don't know how to solve it if the energy eigenvalues are unknown. Are there any recommended methods I can use to determine those eigenvalues. Maybe for simplicity assume we are dealing with a finite square well and assume that for some reason you can't determine the energies in advance.

 

[stevendaryl]

As part of my project I was asked to use the finite difference method to solve Schrödinger equation. I see how you can turn it into a matrix equation, but I don't know how to solve it if the energy eigenvalues are unknown. Are there any recommended methods I can use to determine those eigenvalues. Maybe for simplicity assume we are dealing with a finite square well and assume that for some reason you can't determine the energies in advance.

Do you mean to numerically solve the Schrödinger equation? That does not directly lead to eigenvalues.

What you can do, although this is a lot of work, is to pick an energy, numerically solve the Schrödinger equation for the energy, and then look to see if the wave function blows up as $x \rightarrow \infty$. If so, try a slightly different energy. You've found an energy eigenvalue if the numerical solution of Schrödinger's equation leads to a wave function that is well-behaved at both $x=0$ and $x \rightarrow \infty$.

Actually, to solve Schrödinger's equation numerically, you need more than just the energy. You also need boundary conditions at $x=0$.

(The above is about the one-dimensional Schrödinger equation. In three-dimensions, you can often reduce the problem to a one-dimensional case if you have a spherically symmetric potential by using separation of variables.)