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Color screening and parton inelastic scattering modify the heavy-quark antiquark potential in the medium that consists of particles from quantum chromodynamics (QCD), leading to suppression of quarkonium production in relativistic heavy-ion collisions. Due to small charm/anti-charm ($c\bar{c}$) pair production number in proton-nucleus (pA) collisions, the correlation between different $c\bar{c}$ pairs is negligible, which makes the Schrödinger equation viable for tracking the evolution of only one $c\bar{c}$ pair. We employ the time-dependent Schrödinger equation with in-medium $c\bar{c}$ potential to study the evolution of charmonium wave functions in the hydrodynamic like QCD medium produced in pA collisions. We explore different parametrizations of real and imaginary parts of $c\bar{c}$ potential and calculate the nuclear modification factors ($R_{\rm pA}$) of $J/ψ$ and $ψ(2S)$ in $\sqrt{s_{NN}}=5.02$ TeV energy p-Pb collisions at Large Hadron Collider (LHC). Comparing a strong and a weak screening scenario with experimental data in this approach, we arrive at the conclusion that the color screening is weak at temperature close to deconfined phase transition. Moreover, the imaginary part of the potential is crucial to describe the experimental data which is consistent with widely studied semi-classical approaches where the dissociation rates are essential.