A thought experiment concerning determinism in quantum mechanics

In summary, the uncertainty principle states that when measuring a micro-object, we cannot predict the exact result due to the particle not being in an eigenstate. Knowing the exact wave function of the measuring device does not change this. The Copenhagen interpretation is non-deterministic while the many-worlds interpretation is deterministic, leading to the question of the meaning of probabilities in the MWI. The Heisenberg-Robertson uncertainty principle is about the impossibility of preparing a system in a state where two incompatible observables have a determined value, not about our ability to measure them accurately. The disturbance of a system by measurement and the possibility of measuring two incompatible observables is a complex and ongoing research topic.
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Spathi
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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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Spathi said:
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).

Spathi said:
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.

Spathi said:
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.
 

Related to A thought experiment concerning determinism in quantum mechanics

1. What is a thought experiment concerning determinism in quantum mechanics?

A thought experiment concerning determinism in quantum mechanics is a hypothetical scenario used to explore the concept of determinism in the context of quantum mechanics. It involves imagining a situation in which all the variables and factors that determine the outcome of a quantum event are known, and then considering whether the outcome can still be determined with certainty.

2. How does this thought experiment relate to the concept of determinism?

This thought experiment relates to determinism because it challenges the idea that the outcome of a quantum event is completely random and unpredictable. It suggests that if all the variables and factors are known, then the outcome can be determined with certainty, which goes against the principle of indeterminacy in quantum mechanics.

3. What are some arguments for determinism in quantum mechanics?

Some arguments for determinism in quantum mechanics include the idea that there may be hidden variables that determine the outcome of quantum events, and that our current understanding of quantum mechanics is incomplete. Additionally, some scientists argue that the apparent randomness in quantum events may be due to our limited ability to measure and observe them.

4. Are there any counterarguments to determinism in quantum mechanics?

Yes, there are counterarguments to determinism in quantum mechanics. One of the main counterarguments is the principle of indeterminacy, which states that it is impossible to know both the position and momentum of a particle with absolute certainty. This suggests that there are inherent limits to our ability to determine the outcome of quantum events.

5. What are the implications of this thought experiment for our understanding of quantum mechanics?

This thought experiment has significant implications for our understanding of quantum mechanics. It challenges the traditional interpretation of quantum mechanics as a fundamentally probabilistic theory and raises questions about the nature of reality and the role of observation in shaping it. It also highlights the need for further research and exploration in this field to fully understand the nature of quantum events.

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