Summary: It is shown how to obtain the simplest equations for the Mukhanov-Sasaki variables describing quantum linear scalar perturbations in the case of scalar fields without potential term. This was done through the implementation of canonical transformations at the classical level, and unitary transformations at the quantum level, without ever using any classical background equation, and it completes the simplification initiated in investigations by D. Langlois [Classical Quantum Gravity 11, No. 2, 389–407 (1994; Zbl 1222.83048)] and E. J. C. Pinho and N. Pinto-Neto [Phys. Rev. D 76, No. 2, Article ID 023506 (2007; Zbl 1222.83200)] for this case. These equations were then used to calculate the spectrum index \(n_s\) of quantum scalar perturbations of a nonsingular inflationary quantum background model, which starts at infinity past from flat space-time with Planckian size spacelike hypersurfaces, and inflates due to a quantum cosmological effect, until it makes an analytical graceful exit from this inflationary epoch to a decelerated classical stiff matter expansion phase. The result is \(n_s=3\), incompatible with observations.