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arXiv:2406.19423 · Revista Mexicana de Física E

The Kepler problem on the lattice

Diego Sanjinés C., Evaristo Mamani, Javier Velasco

Published Jan 1, 2025• 0 citations

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.

DOI: 10.31349/revmexfise.22.010210

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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.

We study with the semiclassical model the dynamics of a particle moving in a orthorrombic lattice by the hopping mechanism in the presence of a static Coulomb field and a rapidly oscillating homogeneous field. Based on results for the anisotropic Kepler problem, where a chaotic trajectory arises due to different masses of the particle in the X and Y directions, and since this result is also obtained in the rectangular lattice at the continuum limit, the application of the oscillating field allows reparameterizing the effective masses so that they become equal in both directions. Thus, the result is an elliptic trajectory in the continuum limit of the lattice. This chaos-order transition is confirmed by the calculation of Lyapunov exponents for the different trajectories. It is also found that the application of an external magnetic field in the Z direction of the lattice yields chaos control which can be generalized beyond the simple case of the circular trajectory shown in this work. The previous scheme of chaos control was extended for trajectories in 3D in the continuum limit of the orthorrombic lattice which is also equivalent to the far region approximation where the particle moves with respect to the Coulomb field origin. In this case it found that the chaos control technique permits to observe two different dynamical regimes which, although consistent with Bohr-van Leeuwen theorem, suggests that an observable way to distinguish one from the other is to be sought.

The one-particle sector of the simplest one-dimensional quantum lattice-gas automaton has been observed to simulate both the (relativistic) Dirac and (nonrelativistic) Schrödinger equations, in different continuum limits. By analyzing the discrete analogues of plane waves in this sector we find conserved quantities corresponding to energy and momentum. We show that the Klein paradox obtains so that in some regimes the model must be considered to be relativistic and the negative energy modes interpreted as positive energy modes of antiparticles. With a formally similar approach — the Bethe ansatz — we find the evolution eigenfunctions in the two-particle sector of the quantum lattice-gas automaton and conclude by discussing consequences of these calculations and their extension to more particles, additional velocities, and higher dimensions.

Bloch oscillation (BO), i.e., the oscillatory motion of a quantum particle in a periodic potential, is one of the most striking effects of coherent quantum transport in matter. In the semiclassical picture, it is well known that BOs can be explained owing to the periodic band structure of the crystal and the so-called acceleration theorem: since in the momentum space the particle wave packet drifts with a constant speed without being distorted, in real space the probability distribution of the particle undergoes a periodic motion following a trajectory which exactly reproduces the shape of the lattice band. In non-Hermitian lattices with a complex (i.e., not real) energy band, extension of the semiclassical model is not intuitive. Here we show that the acceleration theorem holds for non-Hermitian lattices with a complex energy band only on average, and that the periodic wave-packet motion of the particle in real space is described by a trajectory in the complex plane, i.e., it generally corresponds to reshaping and breathing of the wave packet in addition to a transverse oscillatory motion. The concept of BOs involving complex trajectories is exemplified by considering two examples of non-Hermitian lattices with a complex band dispersion relation, including the Hatano-Nelson tight-binding Hamiltonian describing the hopping motion of a quantum particle on a linear lattice with an imaginary vector potential and a tight-binding lattice with imaginary hopping rates.

To solve the quantum-mechanical problem the procedure of mapping onto linear space

W

of generators of the (sub)group violated by given classical trajectory is formulated. The formalism is illustrated by the plane H-atom model. The problem is solved noting conservation of the Runge-Lentz vector

n

and reducing the 4-dimensional incident phase space

T

to the 3-dimensional linear subspace

W=T^* V\times R^1

, where

T^* V

is the (angular momentum (

l

) - angle ($\vp$)) phase space and

R^1 =n

. It is shown explicitly that (i) the motion in

R^1

is pure classical as the consequence of the reduction, (ii) motion in the $\vp$ direction is classical since the Kepler orbits are closed independently from initial conditions and (iii) motion in the

l

direction is classical since all corresponding quantum corrections are defined on the bifurcation line (

l=\infty

) of the problem. In our terms the H-atom problem is exactly quasiclassical and is completely integrable by this reasons.

We investigate the semiclassical Kepler/Coulomb problem using the classical constants of the motion in the framework of Nelson's stochastic mechanics. This is done by considering the eigenvalue relations for a family of coherent states (known as the atomic elliptic states) whose wave functions are concentrated on the elliptical orbit corresponding to the associated classical problem. We show that these eigenvalue relations lead to identities for the semiclassical energy, angular momentum and Hamilton–Lenz–Runge vectors in the elliptical case. These identities are then extended to include the cases of circular, parabolic and hyperbolic motions. We show that in all cases the semiclassical wave function is determined by our identities and so our identities can be seen as defining a semiclassical Kepler/Coulomb problem. The results are interpreted in terms of two dynamical systems: one a complex valued solution to the classical mechanics for a Coulomb potential and the other the drift field for a semiclassical Nelson diffusion.