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Talk:Complete set of commuting observables

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FP Eblen (talk) 15:31, 2 June 2016 (UTC)The first sentence, "In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose eigenvalues completely specify the state of a system.[1]," is somewhat incorrect I believe since a CSCO does not necessarily completely specify the state of a system. Instead, a CSCO is complete in the sense that all degeneracies have been removed or broken. A CSCO specifies all that "can be known about a system," but can easily be an incomplete specification of the state since some parameters of the state may be obtainable, or specified, only by some non-commuting observable(s).[reply]

Recommend the first sentence reads:

"In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose eigenvalues completely specify all that can be known simultaneously about the state of a system."

Please correct me if I am wrong.

FP Eblen (talk) 15:31, 2 June 2016 (UTC)pateblen[reply]

Numerous Problems

In my opinion, the definition found here has numerous problems.

1) A process is described ("if not, we add one more compatible observable and continue the process till a CSCO is obtained") but it is not discussed whether the process terminates or converges.

2) Notions are employed which are not defined or fuzzy ("completely specify the state") but could and should be made more precise.

3) No connections to well known notions of completeness of sets in Hilbert spaces are made (and seem necessary, since here completeness is something different than there). At least remarks regarding topological and linear (basis) completeness should be made.

4) The notion in the literature most often is used to mean degeneration-free sets (and, rather imprecisely calls this "complete"), whereas here it is interpreted in the sense of basis-completeness of Hilbert spaces.

5) The hydrogen example demonstrates very clearly that and how the process does not yield a complete set but only a degeneration-free set: Spin is missing. The remark "disregarding spin ... the set forms a CSCO" is not a solution, since the definition and abstract treatment talks of "CSCO" and not of "CSCO ... disregarding this and that"

6) The presented material essentially only works for finite Hilbert spaces and for situations with discrete/finite spectrum. None of the interesting and complicated aspects are mentioned when these simplifying assumptions do not hold.

However, one must admit that most cited literature has at least similar problems.

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