  We propose and study a semi-classical cosmological system akin to the
Newton-Schr\"odinger equation where matter field evolution is determined by
time dependent Schr\"odinger equation. The resulting dynamics is one where the
scale factor self-consistently informs the quantum evolution of the scalar
field wave function via the Friedmann equation. We study various potentials,
and show that for each case, inflation arises naturally, without fine-tuning,
for arbitrary initial wave functions for the scalar field. Furthermore, due to
zero-point quantum fluctuations of the scalar field, the equation of state
$P=-\rho$ arises dynamically at late times for all initial states and
potentials.
