I'm currently writing a book on the interpretation of quantum field theory (which you could read online) where you would find, in the first chapter, a formulation of quantum field theory based on the Hamiltonian formalism, so without any path integral. I think it could be interesting to you, but I don't know if you would call it a "particle interpretation"? What do you mean exactly?

Oh come on, don't give up that quickly ;) I remember the solid harmonics are solutions of Schrödinger's hydrogen atom eigenvalue equation, when one tries to diagonalize the squared orbital momentum and its projection on an axis too... What about doing the same in discrete space?

To be quite specific: I would like to express, on the position basis, the Coulomb scattering process calculated in chapter 10 of Quantum Ethics. For the time being, it is expressed on the momentum basis, and I would like to evaluate the angular distribution using a development on the position basis; in continuous space-time, I would develop the result on the spherical harmonics, since they separate the angular and radial variables, but is there something similar on the lattice, where no spherical symmetry is granted?

Thank you for your answer! Maybe I should have been more specific: I'm not interested in scattering by a lattice structure, but on a lattice, i.e. in space modeled itself as a three dimensional, finite, cubic lattice. For instance, by the Coulomb scattering of an electron beam by an ion target. In continuous space-time, the expansion of the incoming plane wave in spherical harmonics helps calculating the expansion of the outgoing wave, which gives direct clues about the angular scattering. But what about the angular distribution on the lattice? Is there any similar "tool" there?