For me this question becomes really apparent when we make a transformation e.g. in a H-atom from the proton frame of reference to the electron frame of reference. The whole point of QM is that the electron cannot be treated as a dot somewhere in space but that it has an associated wavefunction and only acquires a certain position when I make an appropriate measurement. So how can I transform to the electron frame of reference making the origin to be the position of the electron if I don't know it?

In a wider sense how can I define a coordinate system in quantum mechanics? The usual "take a corner of the lab as the origin and the edges of the walls as axes" surely doesn't work because these all consist of particles that don't have well-defined positions.

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    $\begingroup$ You can arbitrary define the origin. The approach of taking the corner of the lab as origin works pretty well. If you are unsatisfied with the particles of the wall, you can just define the origin to be the average position of the wavefunction of your electron. $\endgroup$ – Rishabh Jain Dec 4 '17 at 11:37

The electron doesn't have a position so it makes no sense to talk about a frame with the origin at the position of the electron.

You can choose a frame in which the expectation value of the electron momentum is zero i.e. a frame in which the electron is on average stationary. Then shift the origin of this frame to make the expectation value of the electron position is zero. That would be a plausible candidate for the frame of the electron. This would of course coincide with the frame of the proton.

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