Quantum cosmological backreactions. II. Purely homogeneous quantum cosmology

In this second paper in a series of four, we continue with the program of incorporating backreaction among the homogeneous and between the homogeneous and inhomogeneous degrees of freedom in quantum cosmological perturbation theory. The purpose of the present paper is to illustrate the formalism of space adiabatic perturbation theory for two simple quantum mechanical models, and to prove that backreaction indeed leads to additional correction terms in effective Hamiltonians that one would otherwise neglect in a crude Born-Oppenheimer approximation. The first model consists of a harmonic oscillator coupled to an anharmonic oscillator. The second model describes the coupling between a scalar matter field and gravity restricted to the purely homogeneous and isotropic sector. These results have potential phenomenological consequences for quantum cosmological models.