Quantum Mechanics as such is not Lorentz-covariant and is not consistent with Special Relativity. Quantum Field Theory is. However, it is not consistent with General Relativity, nor it deals with non-inertial frames, accelerations, free fall, non-free fall and other.
Quantum Mechanics is formulated in Hamiltonian formalism. You have properties as momentum, position, energy and time. Then you have canonical commutation relations. They involve those properties that get modified under Lorentz transformations. But the commutator is not Lorentz-covariant. Moreover, naive treating the commutator as a tensor does not yield proper results. What you have to do is to take a different Hamiltonian in each frame. To take it bluntly, you have a whole set of different Quantum Mechanics theories, each for a different inertial frame. They describe physics in each frame properly, but they are not Lorentz-covariant. It is not enough to modify each tensor to move from one frame to another. You also have to take a different Hamiltonian and quantize each of them separately.
When you consider accelerations, it is even worse. In inertial frames one particle in one frame corresponds to one particle in another frame, it is just not Lorentz-covariant. With accelerating frames it is no longer the case. You can say, the very existence of a particle is acceleration-dependent. This is called the Unruh effect. In a frame accelerating relative to some inertial frame you will see some additional particles looking like a black-body radiation. That means, the transformation between accelerating frames is even more sophisticated. Transformation between particles in different inertial frames could be described as: Lorentz transformation of particle properties + some non-covariant correction. Between accelerating frames you can not do so, you can not even find the same particles to match. You will have to forget of particles and consider transformations of the spacetime as a whole. This topic is not well researched yet.
As with your thought experiment: one of your elevators (free-falling) reside in an inertial frame, while the second in an accelerating frame according to General Relativity. So you will see the Unruh effect in the second one. That means, not only their wave functions will evolve differently, but you will not even be able to find the matching wave functions to compare.
