Dirichlet and Nuemann condition on the same boundary

[Amit Abir]

Hi,

My final goal is to solve numerically Schrödinger's equation in 3D with some potential for the unbounded states, meaning that far away from the potential (at infinity) we may find a free wave and not something that goes to zero.

The basic idea is that I have a particle in (0,0,0) that receives an initial momentum (kx,ky,kz), which is equivalent to knowing the wave function derivatives at the origin. My idea was to create a very small sphere, and say that within this sphere we know the solution: its just a plane wave e^(i k * r) with the given initial momentum k (hbar = 1).

So basically I want to numerically solve the problem of Schrödinger's equation between two spheres (the outer sphere will be taken to be large enough to contain all the data we need), and to find the wave function in this region, with Dirichlet boundary condition and Neumann boundary condition just on the inner sphere.

The problem: I don't know any boundary conditions on the outer sphere, only on the inner one.

The question is: is it solvable? Is knowing the function and its derivatives on the inner sphere enough to find a unique solution for the wave function everywhere?

 

[Orodruin]

You need to put some boundary conditions at infinity if you are after the energy eigenstates (and pick Dirichlet or Neumann at any other boundary).

You also cannot have a particle at the origin with known momentum.

 

[Amit Abir]

You need to put some boundary conditions at infinity if you are after the energy eigenstates (and pick Dirichlet or Neumann at any other boundary).

You also cannot have a particle at the origin with known momentum.

Why must I have conditions on every boundary? Is it true for every PDE or just for this case?