Conduction bands in classical periodic potentials

TANWA~ARPORNTHIP and CARL M BENDER^*

Department of Physics, Washington University, St. Louis, MO 63130, USA

^*Corresponding author

E-mail: tarpornthip@wustl.edu; cmb@wustl.edu

Abstract. The energy of a quantum particle cannot be determined

exactly unless there is an infinite amount of time 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 behaviour 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 behaviours: (i) The particle may

hop from classically allowed site to nearest-neighbour classically

allowed site in the potential, behaving as if it were a quantum

particle in an energy gap and undergoing repeated tunnelling

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 tunnelling. The

classical conduction bands for this potential are determined

numerically with high precision.

Keywords. PT symmetry; complex trajectories; complex

classical mechanics; conduction bands.

PACS Nos 05.45.-a; 05.45.Pq; 11.30.Er; 02.30.Hq