  We study a model of spontaneous wavefunction collapse for a free quantum
particle. We analyze in detail the time evolution of the single-Gaussian
solution and the double-Gaussian solution, showing how the reduction mechanism
induces the localization of the wavefunction in space; we also study the
asymptotic behavior of the general solution. With an appropriate choice for the
parameter $\lambda$ which sets the strength of the collapse mechanism, we prove
that: i) the effects of the reducing terms on the dynamics of microscopic
systems are negligible, the physical predictions of the model being very close
to those of standard quantum mechanics; ii) at the macroscopic scale, the model
reproduces classical mechanics: the wavefunction of the center of mass of a
macro-object behaves, with high accuracy, like a point moving in space
according to Newton's laws.
