  The hypothesis by Sparenberg et al. (2013) that the particular linear tracks
appearing in the measurement of a spherically-emitting radioactive source in a
cloud chamber are determined by the (random) positions of atoms or molecules
inside the chamber is further explored in the framework of a recently
established one-dimensional model by Carlone et al. (2015). In this model,
meshes of localized spins 1/2 play the role of the cloud-chamber atoms and the
spherical wave is replaced by a linear superposition of two wave packets moving
from the origin to the left and to the right, evolving deterministically
according to the Schr\"odinger equation. We first revisit these results using a
time-dependent approach, where the wave packets impinge on a symmetric
two-sided detector. We discuss the evolution of the wave function in the
configuration space and stress the interest of a non-symmetric detector in a
quantum-measurement perspective. Next we use a time-independent approach to
study the scattering of a plane wave on a single-sided detector. Preliminary
results are obtained, analytically for the single-spin case and numerically for
up to 8 spins. They show that the spin-excitation probabilities are sometimes
very sensitive to the parameters of the model, which corroborates the idea that
the measurement result could be determined by the atom positions. The possible
origin of decoherence and entropy increase in future models is finally
discussed.
