# I Eigenstates and repeated measurements

#### jlcd

A measurement X collapses the wave function randomly into an eigenstate of X. Then if a different measurement Y is made the wave function will randomly collapse into an eigenstate of Y. So for example if you measure position, the wave function will collapse into a narrow peak. Now if you measure momentum, the wave function will collapse into a spread out wave.

Are all eigenstates like that? Or are there examples of eigenstates where if you make repeated measurements, it stays the same?

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#### Nugatory

Mentor
Only if you make repeated measurements of observables that commute with one another. Position and momentum do not commute, which is why the example you describe above behaves the way it does.

An added complication is that although measuring an observable leaves the system in an eigenstate of that operator, the system will only stay in that state if the operator commutes with the Hamiltonian.

#### jlcd

Only if you make repeated measurements of observables that commute with one another. Position and momentum do not commute, which is why the example you describe above behaves the way it does.

An added complication is that although measuring an observable leaves the system in an eigenstate of that operator, the system will only stay in that state if the operator commutes with the Hamiltonian.
If you make repeated measurements of observables that commute with one another and the operator commutes with the Hamiltonian. Would it remain the same state if there was no collapse? I was thinking how to analyze it using only unitary dynamics formalism (without collapse).

#### jlcd

To rephrase it. In interpretations like Quantum Darwinism that has unitary only dynamics (without outright collapse). If you make repeated measurements of observables that commute with one another and the operator commutes with the Hamiltonian. Would it remain the same state or not? I guess not and this is why Zurek has to use the ideas of fragments, right? In his theory, states are the primitives and not observations.

Anyone got a clue?

#### jlcd

I guess the answer is the same. That is, in both collapse and unitary only interpretations, one can keep a quantum system frozen in an eigenstate of an observable by repeatedly making the observation, often enough so that significant quantum state evolution has no time to happen before the state is "reset" back (almost certainly) to the nearest eigenstate?

And the difference between collapse and unitary only interpretations is in large object or systems such that without collapse, there will be no objective or classical world? This is why Zurek has to proposed fragments from the einselected pointed states that spread into the enviroment? Without this. We can see both cat but you would see it as male while I can see it as female or different colors? (at least theoretically in a universe without collapse and no einselected pointer states?)

Can someone help. Thank you.

#### PeterDonis

Mentor
I guess the answer is the same.
Different interpretations of QM all agree on all experimental predictions. So all interpretations will agree on what happens in any given experiment. The only thing they disagree on is what story to tell about what happens--what is going on "behind the scenes" where we can't observe.

one can keep a quantum system frozen in an eigenstate of an observable by repeatedly making the observation, often enough so that significant quantum state evolution has no time to happen before the state is "reset" back (almost certainly) to the nearest eigenstate?
Yes, this has been verified in experiments. It's called the "quantum Zeno effect":

"Eigenstates and repeated measurements"

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