Time dilation in QM

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Lets say I have 2 elevators And I have the same experiment in both. Now the first elevator goes into free fall in a constant gravitational field. Now a little bit later the second elevator goes into free fall. Now as the second elevator goes into free fall its wave function is the same as the other elevator. Even though their is a relative speed between them and their both in free fall do their wave functions evolve the same as time passes?
 

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The wave functions of the systems inside evolve with their proper time as seen in the elevators, and with the time of the observers for other observers. You can use ordinary clocks instead of isolated quantum systems and perform the same experiment. Quantum field theory includes special relativity, and while we do not know how to combine it with general relativity in a mathematical way, the time dilation is still true.
 
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interesting thanks for your answer. So for the moving observer they would have a different wave function that they calculate.
 
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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.
 

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