  Within a plane-wave approach, a number of scattering events in a collision is
insensitive to a general phase of a transition amplitude, although this phase
is extremely important for a number of problems, especially in hadronic
physics. In reality the particles are better described as wave packets, and
here we show that the observables grow dependent upon this phase if one lays
aside the simplified plane-wave model. We discuss two methods for probing how
the Coulomb- and hadronic phases change with a transferred momentum $t$, either
by colliding two beams at a non-vanishing impact-parameter or by employing such
novel states as the vortex particles carrying orbital angular momentum or the
Airy beams. For electron-electron collision, the phase contribution to a cross
section can reach the values higher than $10^{-4} -- 10^{-3}$ for well-focused
beams with energies of hundreds of keV.
