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1) Start with identical hydrogen atoms. Cool them off to a
few thousandths of a degree above absolute zero. Using a
seriously powerful magnet, align all the electron spins and
the nuclear spins.
In this regime, the atoms are not like baseballs. They are
not hard little spheres that bounce off each other. Rather,
they are big fluffy clouds that diffract through each other.
If you go by the thermal de Broglie length, the atoms are
the size of bacteria.
In this regime, you cannot distinguish between the two
cases of right-angle scattering. The add-before-you-square
law applies. You see interference between the two cases.
This effectively doubles the 90-degree scattering amplitude.
2) Interestingly enough, at higher temperatures you /can/
distinguish the two cases. The thermal de Broglie length is
small compared to the Bohr radius, so the atoms act like little
baseballs. The two possible trajectories are offset from one
another by about one atomic diameter, and if you look closely
enough you can distinguish the two cases on this basis. Even
though the particles are nominally identical, the trajectories
are distinguishable, and that's all the quantum mechanical laws
care about. In this situation the square-before-you-add law
gives the right answer. You can quantify all this in terms
of wavefunction phase angles et cetera.