5
I think that the main problem might be with the solver you are using. The Hamiltonian (matrix) in this case is Hermitian, it is even symmetric since it is purely real. You could use eigh instead of eig to take advantage of this.
Furthermore, you are not removing only the first and last points but intervals of size 1 at each end. Following, I show you a ...
3
First, you're solving the Schrödinger equation not in a box, but rather you apply periodic boundary conditions. That is, it's more like solving the TDSE on a ring. But this just as a comment and shouldn't matter here.
What's happening here is that you evolve a Gaussian wave packet in time. In a suitable hamonic oscillator potential, it would be stable and ...
2
As David said, absorbing boundary conditions won't be completely reflectionless. That said, we can reduce relfections quite a bit, which helps to avoid influence from the boundaries while the particle still travelling inside.
Since this is a time dependent problem, one simple choice of boundary conditions will look like this.
At the left boundary:
$$\frac{\...
1
Problem solved: I just have to include the solution of the homogeneous equation into the solution of the differential equation.
The solution to the differential equation is
$$V_H(r) = -\frac{K}{r} - \frac{r+1}{r} \exp(-2r).$$
By requiring that $\lim_{r \to 0} V_H(r)$ is real we get $K=-1$ and
$$V_H(r) = \frac{1}{r} - \frac{r+1}{r} \exp(-2r).$$
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