  The Schr\"odinger-Newton equation has gained attention in the recent past as
a nonlinear modification of the Schr\"odinger equation due to a gravitational
self-interaction. Such a modification is expected from a fundamentally
semi-classical theory of gravity, and can therefore be considered a test case
for the necessity of the quantisation of the gravitational field. Here we
provide a thorough study of the effects of the Schr\"odinger-Newton equation
for a micron-sized sphere trapped in a harmonic oscillator potential. We
discuss both the effect on the energy eigenstates and the dynamical behaviour
of squeezed states, covering the experimentally relevant parameter regimes.
