The response of a tunnelling particle to an impulse

Abstract

A classical particle moving in a one-dimensional periodic potential of height V vibrates if its energy E is less than V and translates if E>V. It responds to an impulse by a change in its mean velocity only if E>V. Due to tunnelling, the quantum states of a particle do not divide sharply into vibrational and translational ones. The author considers the question of whether a quantum particle with E<V responds to an impulse (due to lattice fluctuations) by subsequently increasing its rate of tunnelling in the direction of the impulse and diminishing it in the opposite direction. (The author calls this response driven tunnelling). The question is relevant to all kinds of motion in crystal lattices. It is shown that where conventional transport theory ignores driven tunnelling, it is inconsistent with classical mechanics. A simple theory of driven tunnelling is described in the context of methyl group rotation. It leads to a Hamiltonian which has both a scalar and a vector potential. The latter is changed by an impulse and is responsible for the change in the subsequent tunnelling behaviour.