Time-dependent quantum tunnelling via crossover processes

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Abstract
It is shown that the propagator K(xt mod x'0) for a particle in a potential field is derivable from the classical path starting from x' and reaching x at time t. Considering a wavepacket as the particle initial state lying mainly on one side of a barrier the propagator is used for obtaining the wavefunction on the other side. Tunnelling is then discussed in terms of the evolving wavefunction. It is argued that while the particle's expectation energy is taken to be lower than the barrier's height the transformation by which the wavefunction is produced entails energetically crossover flights as required by the dynamics for the classical paths involved in the propagation process. The parabolic repeller exemplifies the formalism and exact results for the probability and current densities are given. A computation involving ballistic tunnelling shows that the current density rises to a saturation value proportional to the particles' injection rate.

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