Complex Trajectories in a Classical Periodic Potential

TL;DR

This paper analyzes the complex classical trajectories of a particle in a periodic potential, revealing special energy-dependent classes of motion and fractal-like velocity behavior for generic energies.

Contribution

It identifies two unique classes of trajectories determined solely by energy and explores their properties in a classical periodic potential.

Findings

Two classes of trajectories: periodic and shift-variant.

Special trajectories occur only at measure-zero energy sets.

Average velocity becomes fractal-like for most energies.

Abstract

This paper examines the complex trajectories of a classical particle in the potential V(x)=-cos(x). Almost all the trajectories describe a particle that hops from one well to another in an erratic fashion. However, it is shown analytically that there are two special classes of trajectories x(t) determined only by the energy of the particle and not by the initial position of the particle. The first class consists of periodic trajectories; that is, trajectories that return to their initial position x(0) after some real time T. The second class consists of trajectories for which there exists a real time T such that $x(t+T)=x(t) \pm2 \pi$. These two classes of classical trajectories are analogous to valence and conduction bands in quantum mechanics, where the quantum particle either remains localized or else tunnels resonantly (conducts) through a crystal lattice. These two special types of trajectories are associated with sets of energies of measure 0. For other energies, it is shown that for long times the average velocity of the particle becomes a fractal-like function of energy.