The Kepler problem on the lattice
DOI:
https://doi.org/10.31349/RevMexFisE.22.010210Keywords:
Kepler problem, tight-binding dynamics, semiclassical model, pseudo-spectral method, physics educationAbstract
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.
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This calculation is a natural extension of the study entitled The dynamics of a particle in a tight-binding lattice under a Coulomb potential [14] wherein we have used both quantum and semiclassical approaches, showing that the results of the former can be quite well approximated by the semiclassical approach
The numerical procedure to verify this assertion on the basis of the algorithm given in (14) is too demanding for a computer since the limit of the continuum a → 0 implies the discretization of the lattice in a greater number of cells. For the numerical results presented in this work, we have taken a region of 127 × 127 cells where a typical wavepacket evolution lasts around 30 minutes. The further segmentation a → a/2 would imply a 4-fold time increment, and so on.
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