A one-dimensional barrier and time-dependent tunnelling

Abstract

There has recently been a growth in interest in electron tunnelling through barriers, and in particular in the question of whether it is possible to talk of a tunnelling time and, if so, what is it? A brief account of a study of the signal velocity of wavepackets travelling into a rectangular barrier was given recently, and the present paper is an extended version of this theory. In particular it is shown that the signal velocity of a free electron is the same as its group velocity, which in turn is the same as that which a classical particle particle of the same energy would have. It is also shown that when the wavepacket enters a classically forbidden region the signal velocity continues to have a meaning, and indeed a pulse-like wavefunction travels very much like a light pulse in a dispersive and attenuating medium. The expression for the signal velocity is symmetrical with respect to the barrier height, so that pulses well below the barrier travel faster, though with greater attenuation, than pulses near the top of the barrier. This result agrees with conclusions reached by some other workers. It is also shown that an electron responds to an accelerating field in the barrier as would be expected for a classical particle. It is argued that the not uncommon statement that the non nor eigenstates of a free particle describe a beam of particles is incorrect for electrons, for such a solution can at most describe two electrons. The theory indicates that by using orthogonal wavepackets it is possible to construct many-electron wavefunctions which do describe beams of electrons, and then free electrons travel with the group (and signal) velocity.