Uncertainty principle & simultaneous measurement

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The discussion centers on the interpretation of the uncertainty principle in quantum mechanics, particularly regarding simultaneous measurements of observables A and B. It critiques the standard interpretation of the uncertainty relation σ(A)σ(B) ≥ 1/2 |μ([A,B])|, suggesting it needs reevaluation in light of simultaneous measurability. The paper referenced questions von Neumann's proof that asserts if A and B are simultaneously measurable, then [A,B]=0, highlighting the assumption that every observable corresponds to an operator. Counter-examples are presented to challenge the consistency of quantum mechanics postulates, particularly regarding the strong superposition principle. The conversation emphasizes the importance of accurately understanding the uncertainty principle, distinguishing between measurement accuracy and the inherent limitations of simultaneous observability in quantum systems.
  • #31
bhobba said:
Basically yes.

It's an operator O = ∑ yi |bi><bi|, yi real. It means the resolution of the identity |bi><bi| describes some observation such that the probability of outcome i depends only on the |bi><bi|. Given any resolution of the identity we can in principle find an observation that describes it. The yi are the arbitrary real numbers associated with each outcome and the operator from that association is O = ∑ yi |bi><bi|. Via the spectral theorem the two are in 1-1 correspondence meaning each implies the other.

In fact, via Gleason the above implies the Born rule.

For foundational issues its easier to consider finite spaces and extend it via the RHS formalism. That way subtle issues of what exactly is a Hermitian operator is avoided.

Thanks
Bill

Don't superselection rules imply that not all Hermitian operators are observables? For example, if O is an operator whose eigenstates are superpositions of states with different total charges, then those eigenstates are not physically realizable, and O is not observable.
 

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