Wave functions and characteristic times for transmission and reflection

TL;DR

This paper introduces a wave-packet analysis framework for quantum scattering, distinguishing elementary and combined processes, and defines unique transmission and reflection times for tunneling through a potential barrier.

Contribution

It proposes a novel approach to quantum scattering, separating processes into elementary and combined, and defines exact and asymptotic characteristic times for transmission and reflection.

Findings

Distinct elementary and combined scattering processes identified

Unique solutions for transmission and reflection states derived

Exact and asymptotic characteristic times for tunneling established

Abstract

We present a renewed wave-packet analysis based on the following ideas: if a quantum one-particle scattering process and the corresponding state are described by an indivisible wave packet to move as a whole at all stages of scattering, then they are elementary; otherwise, they are combined; each combined process consists from several alternative elementary ones to proceed simultaneously; the corresponding (normed) state can be uniquely presented as the sum of elementary ones whose (constant) norms give unit, in sum; Born's formula intended for calculating the {\it expectation} values of physical observables, as well as the standard timing procedure are valid only for elementary states and processes; only an elementary time-dependent state can be considered as the quantum counterpart to some classical one-particle rajectory. By our approach, tunneling a non-relativistic particle through a static one-dimensional potential barrier is a combined process consisting from two elementary ones, transmission and reflection. In the standard setting of the problem, we find an unique pair of solutions to the Schr\"odinger equation, which describe separately transmission and reflection. On this basis we introduce (exact and asymptotic) characteristic times for transmission and reflection.