Subsequent Measurements of two observables, compatible and incompatible pairs

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SUMMARY

The discussion centers on the measurement of quantum observables, specifically the implications of measuring compatible and incompatible pairs. When measuring a compatible observable \(\hat{B}\) after \(\hat{A}\), the state collapses to a defined state, ensuring that a subsequent measurement of \(\hat{A}\) returns the same result \(a\). Conversely, if \(\hat{A}\) and \(\hat{B}\) are incompatible, measuring \(\hat{B}\) results in the loss of information about \(\hat{A}\), leading to the possibility of obtaining different results upon re-measuring \(\hat{A}\).

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Homework Statement



On an arbitrary state, the observable \hat{A} is measured returning the result a. A compatible observable \hat{B} is then measured returning b.

If \hat{A} is then measured again, is the same result a obtained?
How about if \hat{A} and \hat{B} are not compatible observables?

Homework Equations



The Attempt at a Solution



To my understanding: for the compatible case, the second measurement of \hat{A} would return a since the state has collapsed to a (fully) defined state of a (complete) set of compatible observables; \hat{A} and \hat{B}.

However, if \hat{A} and \hat{B} are not compatible, upon measuring \hat{B}, the information of \hat{A} is lost, and so the second measurement of \hat{A} can return values other than a.

Is this correct?
Thanks in advance..
 
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Yes, that's correct.
 

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