  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.
