Conduction bands in classical periodic potentials

Abstract

The energy of a quantum particle cannot be determined exactly unless there is an infinite amount of time in which to perform the measurement. This paper considers the possibility that $\Delta E$, the uncertainty in the energy, may be complex. To understand the effect of a particle having a complex energy, the behavior of a classical particle in a one-dimensional periodic potential $V(x)=-\cos (x)$ is studied. On the basis of detailed numerical simulations it is shown that if the energy of such a particle is allowed to be complex, the classical motion of the particle can exhibit two qualitatively different behaviors: (i) The particle may hop from classically-allowed site to nearest-neighbor classically-allowed site in the potential, behaving as if it were a quantum particle in an energy gap and undergoing repeated tunneling processes, or (ii) the particle may behave as a quantum particle in a conduction band and drift at a constant average velocity through the potential as if it were undergoing resonant tunneling. The classical conduction bands for this potential are determined numerically with high precision.

11 pages, 10 figures

References in corpus (9)

• The ODE/IM Correspondence
• PT-symmetric Deformations of the Korteweg-de Vries Equation
• Classical Trajectories for Complex Hamiltonians
• Complexified Dynamical Systems
• Spontaneous Breaking of Classical PT Symmetry
• Cryptogauge symmetry and cryptoghosts for crypto-Hermitian Hamiltonians
• PT-symmetric deformations of Calogero models
• Exact Isospectral Pairs of PT-Symmetric Hamiltonians
• PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation

Cited by in corpus (10)

• Bohmian quantum trajectories from coherent states
• PT-symmetry breaking in complex nonlinear wave equations and their deformations
• Quantum tunneling as a classical anomaly
• Probability Density in the Complex Plane
• Infinitely many inequivalent field theories from one Lagrangian
• Periodic orbits for classical particles having complex energy
• Classical Particle in a Complex Elliptic Potential
• Complex Trajectories in a Classical Periodic Potential
• PT-symmetrically deformed shock waves
• Dynamics, interference effects, and multistability in a Lorenz-like system of a classical wave-particle entity in a periodic potential