The neutron at low momentum is quantum mechancal during the scattering, and the bouncing off is by wave mechanics, not by bouncing off. The neutron gets reflected into all directions by the small region with a potential, and the amount of scattering is given by the Born approximation to lowest order in the momentum of the neutron:
$$ A(k-k') = -i\tilde V(k-k') $$
For scattering from incoming direction k to outgoing direction k'. You decorate this with phase-space factors to take into account the size of the neutron wave. If the neutron wave is large, nearly all the scattering is in the forward direction, and this is expressed by writing the S-matrix as;
$$ S = 1 + i A $$
Where the "1" gives a contribution to the scattering which is $\delta(k-k')$ for the case of single-particle scattering. If you smear this with the incoming wavepacket, you get the outgoing wavepacket, which is mostly the incoming wave, plus a spherical outgoing wave. The delta-function guarantees that as you approach a plane-wave limit, there will be no scattering, because the neutron wavefunction area will be much larger than the nuclear area.
This scattering, when the neutron wavefunction is larger wavelength than the nucleus (almost always in real life) leads to a spherical scattering which is roughly isotropic, which you add to the incoming wave to get the full outgoing wave. The imaginary part of the scattering in the forward direction only subtracts some weight from the wavefunction that keeps going forward, and unitarity guarantees that this imaginary part is equal to the scattering probability in all directions added together.
This is covered in scattering theory in most quantum mechanics books, but the treatment in Gribov's Regge theory book "The Theory of Complex Angular Momentum" is most instructive to my mind.