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We propose and study a semi-classical cosmological system akin to theNewton-Schr\"odinger equation where matter field evolution is determined bytime dependent Schr\"odinger equation. The resulting dynamics is one where thescale factor self-consistently informs the quantum evolution of the scalarfield 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 tozero-point quantum fluctuations of the scalar field, the equation of state$P=-\rho$ arises dynamically at late times for all initial states andpotentials.

Semiclassical cosmology with back reaction: the Friedmann-Schrodinger equation and inflation