  This is the second paper in a series of four in which we use space adiabatic
methods in order to incorporate backreactions among the homogeneous and between
the homogeneous and inhomogeneous degrees of freedom in quantum cosmological
perturbation theory.
  The purpose of the present paper is twofold. On the one hand, it illustrates
the formalism of space adiabatic perturbation theory (SAPT) for two simple
quantum mechanical toy models. On the other, it proves the main point, namely
that backreactions lead to additional correction terms in effective
Hamiltonians that one would otherwise neglect in a crude Born-Oppenheimer
approximation.
  The first model that we consider is a harmonic oscillator coupled to an
anharmonic oscillator. We chose it because it displays many similarities with
the more interesting second model describing the coupling between an inflaton
and gravity restricted to the purely homogeneous and isotropic sector. These
results have potential phenomenological consequences in particular for quantum
cosmological theories describing big bounces such as Loop Quantum Cosmology
(LQC).
