The Kepler problem on the lattice

TL;DR

This paper investigates the classical and quantum dynamics of a particle in a 3D lattice with Coulomb potential, revealing non-conservation of angular momentum, chaotic trajectories, and wavepacket behavior, using semiclassical and quantum methods.

Contribution

It introduces a semiclassical and quantum analysis of the Kepler problem on a lattice, highlighting non-conservation of angular momentum and complex trajectories in a non-isotropic lattice.

Findings

Trajectories remain in a plane on a rectangular lattice.

Chaotic and precession trajectories observed due to lattice anisotropy.

Quantum wavepacket analysis shows deformation and an intrinsic angular momentum concept.

Abstract

We study the motion of a particle in a 3-dimensional lattice in the presence of a Coulomb potential, but we demonstrate semiclassicaly that the trajectories will always remain in a plane which can be taken as a rectangular lattice. The Hamiltonian model for this problem is the conservative tight-binding one with lattice constants a, b and hopping elements A, B in the XY axes, respectively. We use the semiclassical and quantum formalisms; for the latter we apply the pseudo-spectral algorithm to integrate the Schroedinger equation. Since the lattice discrete subspace is not isotropic, the angular momentum is not conserved, which has interesting consequences as chaotic trajectories and precession trajectories, similar to the astronomical precession trajectories due to non-central gravitational forces, notably, the non-relativistic Mercury's perihelion precession. Although the elements of the mass tensor are naturally different in a rectangular lattice, these can be chosen to be still different in the continuum, which permits to study the motion with the usual Newtonian kinetic energies. We calculate also the contour plots of an initial Gaussian wavepacket as it moves in the lattice and we propose an "intrinsec angular momentum" associated to its asymmetrical deformation, such that the quantum and semiclassical angular momenta could be simply related.