  We investigate a $D$ dimensional generalization of the Schroedinger-Newton
equations, which purport to describe quantum state reduction as resulting from
gravitational effects. For a single particle, the system is a combination of
the Schroedinger and Poisson equations modified so that the probability density
of the wavefunction is the source of the potential in the Schroedinger
equation. For spherically symmetric wavefunctions, a discrete set of energy
eigenvalue solutions emerges for dimensions $D<6$, accumulating at D=6.
Invoking Heisenberg's uncertainty principle to assign timescales of collapse
correspoding to each energy eigenvalue, we find that these timescales may vary
by many orders of magnitude depending on dimension. For example, the time taken
for the wavefunction of a free neutron in a spherically symmetric state to
collapse is many orders of magnitude longer than the age of the universe,
whereas for one confined to a box of picometer-sized cross-sectional dimensions
the collapse time is about two weeks.
