  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.
