Nonlinear Quantum Motions in 2D Nano-channels Part II: Quantization and Wave Motion

Abstract

Quantization and wave motion in nano-channels were conventionally treated from the probabilistic viewpoint. In Part II, we show how quantization and wave motion can be reproduced by a deterministic corpuscular description of electrons moving in the channel. The quantum Hamilton mechanics introduced in Part I is exploited to derive electron trajectories in the presence of the channelized quantum potential. The number of channels within the quantum potential is shown to be just the integer quantization levels of the conductance, and the conductance quantization is found to be a manifestation of the step-change nature of number of channels with respect to the electron incident energy. Three types of trajectory within the channel, tunneling, reflection and transmission, are solved analytically from the Hamilton equations of motion; meanwhile, explicit inequality criteria are derived to predict which type of motion will occur actually under a given incident condition. Upon solving the complex-valued Hamilton equations of motion, multiple paths between two fixed points come out naturally, and the collection of these multiple paths produces, what we know, the electronic matter wave propagating within the channel.