The Kepler problem on the lattice

Abstract

We study the motion of a particle in a 3-dimensional lattice in the presence of a potential −V1/r, 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 Schrödinger 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 kinetic energies pi2/2mi (i = x,y). We calculate also the contour plots of an initial Gaussian wavepacket as it moves in the lattice and we propose an “intrinsec angular momentum” S associated to its asymmetrical deformation, such that the quantum and semiclassical angular momenta, Lq, Lc , respectively, could be related as Lq = Lc + αS.