Tunnelling effects of a Gaussian wave packet impinging on a barrier

A general procedure based on momentum-like quantity provides the reflection and transmission amplitudes for a given barrier sandwiched by semiconductor reservoirs is presented. Furthermore, the evolution of the wave function stemming from an initial Gaussian wave packet located on the left hand side of the barrier with ignorable barrier overlap is obtained. The evolving wave function enables obtaining the associated probability and current densities space and time-wise. As application, the case of smooth double barrier is considered. The numerical results exhibit similar picture as obtained via propagator in the limited case of square barrier, e.g. repeated current density reversal at the barrier entrance, while being unidirectional at the exit. Presently, the treatment takes account of any barrier, inclusive of applied voltage. The basic quantity required is the value of the momentum-like quantity at the barrier entrance, which is obtained solving a Riccati equation governing the quantity, in question, whose value is known at the barrier exit in terms of the carrier energy and applied bias.