Vignette

Is the “grid effect” due to the square boundary or would it happen in free space?

Thanks for the question. No, there is no boundary here: the xy-domain is unbounded. The grid-like structure of the probability density appears because the problem is separable in x and y.

I think that is not the reason for the interference pattern. I think it appears because in the limit for large x and y the potential works as a box-like problem. This is interesting.

Another argument would be is that the Hamiltonian is C4v symmetric (symmetry of the square). The decomposition in terms of irreps may (or may not) be interesting as well?

So it’s expected that a potential with a triangular symmetry would have a wave function that is triangle-y?

Yes indeed, you would be able to decompose the wave function into representations of the symmetry group, each “following” the defining features of the symmetry group. Therefore, triangle-y Hamiltonians would produce triangle-y decomposition.