SWhat changed
Quantum uncertainty can be expressed as a relation for many pairs of observables. Robertson connected their standard deviations, which measure the spread of repeated outcomes, to the magnitude of the expected commutator: Delta A times Delta B is at least one-half of abs(<AB-BA>). The familiar position-momentum case becomes one example of this general relation, alongside angular-momentum components.
Scientific scope
Quantum uncertainty can be expressed as a relation for many pairs of observables. Robertson connected their standard deviations, which measure the spread of repeated outcomes, to the magnitude of the expected commutator: Delta A times Delta B is at least one-half of abs(<AB-BA>). The familiar position-momentum case becomes one example of this general relation, alongside angular-momentum components. The state must have suitable common operator domains and finite moments. The expected commutator can be zero in a particular state, so noncommuting operators do not automatically supply a strictly positive state-independent bound. The original gives a one-particle proof sketch for variables linear in momentum, uses integration by parts and discards surface terms; it is not a complete modern domain theorem for every unbounded operator. Heisenberg physical formulation, the earlier special-case proof cited to Weyl and Condon motivation remain distinct.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The state must have suitable common operator domains and finite moments. The expected commutator can be zero in a particular state, so noncommuting operators do not automatically supply a strictly positive state-independent bound. The original gives a one-particle proof sketch for variables linear in momentum, uses integration by parts and discards surface terms; it is not a complete modern domain theorem for every unbounded operator. Heisenberg physical formulation, the earlier special-case proof cited to Weyl and Condon motivation remain distinct.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
Provisional WRS assessment, edition 2026-10-23.5. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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