Schr\"odinger-Newton "collapse" of the wave function

It has been suggested that the nonlinear Schr\"odinger-Newton equation might approximate the coupling of quantum mechanics with gravitation, particularly in the context of the M{\o}ller-Rosenfeld semiclassical theory. Numerical results for the spherically symmetric, time-dependent, single-particle case are presented, clarifying and extending previous work on the subject. It is found that, for a particle mass greater than $\sim 1.14(\hbar^2/(G\sigma))^{1/3}$, a wave packet of width $\sigma$ partially "collapses" to a groundstate solution found by Moroz, Penrose, and Tod, with excess probability dispersing away. However, for a mass less than $\sim 1.14(\hbar^2/(G\sigma))^{1/3}$, the entire wave packet appears to spread like a free particle, albeit more slowly. It is argued that, on some scales (lower than the Planck scale), this theory predicts significant deviation from conventional (linear) quantum mechanics. However, owing to the difficulty of controlling quantum coherence on the one hand, and the weakness of gravity on the other, definitive experimental falsification poses a technologically formidable challenge.