Talk:Complete set of commuting observables

AbsalomHicks (talk) 23:15, 3 November 2022 (UTC) Presumably the meaning of complete set of commuting observables is the following: measurement of all observables uniquely determines the state vector up to a scalar factor. The definition given:

"In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose common eigenvectors can be used as a basis to express any quantum state."

is not sufficient to guarantee this as a single selfadjoint operator with discrete spectrum satisfies the definition but the eigenvalues do not uniquely determine the eigenvector unless all eigenspaces have dimension one.

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).

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

You are wrong: The eigenvalues of a set of operators are a property of the operators themselves, unaffected by the value of the quantum state. 60.250.123.226 (talk) 18:37, 17 January 2022 (UTC)

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 similar problems.

Comment on the numerous problems
Note that one must take great care with the definition of "commuting observable" is when the observables are unbounded, because unbounded self-adjoint operators cannot have the entire Hilbert space as their domain and there are problems considering their products. This difficulty is handled in the Reed & Simon methods of mathematical physics series of 4 volumes by defining two unbounded self adjoint operators to commute when their spectral projections commute.

Regarding the sixth problem above, maybe another satisfactory definition of "Complete set of compatible observables" would be a set of observables S with the property that if A and B are observables commuting with the elements of S then A and B commute with each other. 60.250.123.228 (talk) 14:08, 19 January 2022 (UTC)