The inflationary epoch at the beginning of the universe is commonly described within the framework of (linear) cosmological perturbation theory. The corresponding equation of motion for the gauge-invariant perturbations is the Mukhanov-Sasaki equation, which resembles a time-dependent harmonic oscillator. At first, we will consider a mechanical analogue of the Mukhanov-Sasaki equation and use the known Lewis-Riesenfeld invariant investigated by J. Hartley and J. Ray and the extended phase space formalism introduced by J. Struckmeier to analyse the system. These techniques allow to construct an extended canonical transformation that maps an explicitly time-dependent Hamiltonian into a time-independent one. The generators of this symplectic map can in turn be canonically quantised on the original part of the phase space, which is the constraint hypersurface of the extended theory, connecting to work done by M. Guasti and H. Moya-Cessa. Our further analysis leads us to a closed form of the time-evolution operator for the single-mode Mukhanov-Sasaki Hamiltonian, that is to the associated Dyson series. We will analyze the characteristic properties of this time-evolution operator and discuss whether it can be extended to the full Fock space. Finally we give an outlook towards possible applications of these techniques to inflationary quantum cosmology.