Conduction bands in classical periodic potentials

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 ΔE, 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.