  In this paper it is shown how to obtain, without ever using the background
classical equations of motion, a simple second order Hamiltonian involving the
Mukhanov-Sasaki variable describing quantum linear scalar perturbations for the
case of scalar fields with arbitrary potentials and arbitrary spacelike
hyper-surfaces. It is a generalization of previous works, where the scalar
field potential was absent and the spacelike hypersurfaces were flat. This was
possible due to the implementation of a new method, together with the
Faddeev-Jackiw procedure for the constraint reduction. The resulting
Hamiltonian can then be used to study the evolution of quantum cosmological
perturbations in quantum backgrounds.
