Collapse models: analysis of the free particle dynamics

We study a model of spontaneous wavefunction collapse for a free quantum particle. We analyse 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 behaviour of the general solution. With an appropriate choice for the parameter λ 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 centre of mass of a macro-object behaves, with high accuracy, like a point moving in space according to Newton's laws.