A thought experiment concerning determinism in quantum mechanics

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According to the uncertainty principle, when we measure a micro-object with a measuring device, we cannot predict what value the device will show. But if we knew exactly the wave function of this device, together with the wave function of the micro-object, could we exactly predict the result of the measurement?
According to the uncertainty principle, when we measure a micro-object with a measuring device, we cannot predict what value the device will show. But if we knew exactly the wave function of this device, together with the wave function of the micro-object, could we exactly predict the result of the measurement?
The question is, in other words, how modern quantum mechanics treats determinism. I’ve heard, that the Copenhagen interpretation is not-deterministic, while the many-worlds interpretation is deterministic. Can you help me understand these statements?
 

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According to the uncertainty principle, when we measure a micro-object with a measuring device, we cannot predict what value the device will show.
That's not the uncertainty principle. The uncertainty principle has to do with measurements of two non-commuting observables. It says nothing whatever about whether or not you can predict the result of a measurement of a single observable.

The fact that, if a quantum system is not in an eigenstate of the observable we are measuring, we cannot predict with certainty what the measurement result will be, but can only predict probabilities, is just the Born rule (which in addition tells you how to predict the probabilities).

if we knew exactly the wave function of this device, together with the wave function of the micro-object, could we exactly predict the result of the measurement?
No, because, as above, the thing that makes the prediction only probabilistic is that the quantum system is not in an eigenstate of the observable being measured. Knowing the measuring device's exact wave function does not change that.

I’ve heard, that the Copenhagen interpretation is not-deterministic, while the many-worlds interpretation is deterministic.
That's correct. The Copenhagen interpretation is non-deterministic because in this interpretation, only one result occurs for any measurement, and predicting that result can only be done probabilitistically for the reasons given above.

The MWI is deterministic because in this interpretation, all possible results occur for every measurement; that is the deterministic result of every measurement. This, of course, raises the question of what the "probabilities" that the Born rule talks about even mean, which is one of the critical issues many physicists see with the MWI.
 
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The usual Heisenberg-Robertson uncertainty principle in introductory textbooks is not about measurability of observables but about the possibility to prepare states. It says that it is in general not possible to prepare a system in a state, where two observables, whose representing self-adjoint operators do not commute ("incompatible observables"), take a determined value. E.g., it is impossible to prepare a particle in a state such that both position and momentum are determined very accurately. This possibility is constraint by the uncertainty relation between components of the position and momentum vectors in the same direction, ##\Delta x \Delta p \geq \hbar/2##.

This has nothing to do with our ability to measure either observable as accurately as we want (given enough expertise and resources to construct the necessary measurement devices, of course). Rather it is a property of the particle, described by the state it is prepared in.

The question about the disturbance of the system by measurement and the possibility or impossibility to meausure two incompatible observables is a much more complicated question and subject to ongoing research. A recent textbook on these issues is

Busch, P., Lahti, P., Pellonpää, J. P., & Ylinen, K. (2016). Quantum measurement (Vol. 23). Berlin: Springer.
 

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