  The physical interpretation of scattering line polarization offers a novel
diagnostic window for exploring the thermal and magnetic structure of the quiet
regions of the solar atmosphere. Here we evaluate the impact of isotropic
collisions with neutral hydrogen atoms on the scattering polarization signals
of the 13 lines of multiplet 42 of Ti I and on those of the K line and of the
IR triplet of Ca II, with emphasis on the collisional transfer rates between
nearby J-levels. To this end, we calculate the linear polarization produced by
scattering processes considering realistic multilevel models and solving the
statistical equilibrium equations for the multipolar components of the atomic
density matrix. We confirm that the lower levels of the 13 lines of multiplet
42 of Ti I are completely depolarized by elastic collisions. We find that
upper-level collisional depolarization turns out to have an unnoticeable impact
on the emergent linear polarization amplitudes, except for the ${\lambda 4536$
line for which it is possible to notice a rather small depolarization caused by
the collisional transfer rates. Concerning the Ca II lines, we show that the
collisional rates play no role on the polarization of the upper level of the K
line, while they have a rather small depolarizing effect on the atomic
polarization of the metastable lower levels of the Ca II IR triplet.
