SWhat changed
Mandelstam and Tamm related the energy spread of an isolated quantum system to the rate at which an observable changes: ΔH ΔR ≥ (ℏ/2)|d⟨R⟩/dt|. Applying this to the projector onto the initial pure state gives a survival probability of at least cos²(ΔH t/ℏ) up to πℏ/(2ΔH), hence a lower bound on the time to reach an orthogonal state. They also extended the observable inequality to statistical mixtures.
Scientific scope
Mandelstam and Tamm related the energy spread of an isolated quantum system to the rate at which an observable changes: ΔH ΔR ≥ (ℏ/2)|d⟨R⟩/dt|. Applying this to the projector onto the initial pure state gives a survival probability of at least cos²(ΔH t/ℏ) up to πℏ/(2ΔH), hence a lower bound on the time to reach an orthogonal state. They also extended the observable inequality to statistical mixtures. The result uses a time-independent Hamiltonian and an observable without explicit time dependence, with finite relevant variances and well-defined derivatives. Time is an evolution parameter, not a universal self-adjoint operator conjugate to energy; no universal minimum energy-measurement duration is asserted. The cosine bound is restricted to its stated time interval. Energy uncertainty is standard deviation, not the mean energy above the ground state. Not a universal time operator, energy-measurement-duration law, mean-energy speed limit, arbitrary driving or a cosine lower bound after its permitted first quarter-period. Robertson inequality is an inherited input. S.Frenkel is the printed translator, not a scientific coauthor.
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 result uses a time-independent Hamiltonian and an observable without explicit time dependence, with finite relevant variances and well-defined derivatives. Time is an evolution parameter, not a universal self-adjoint operator conjugate to energy; no universal minimum energy-measurement duration is asserted. The cosine bound is restricted to its stated time interval. Energy uncertainty is standard deviation, not the mean energy above the ground state. Not a universal time operator, energy-measurement-duration law, mean-energy speed limit, arbitrary driving or a cosine lower bound after its permitted first quarter-period. Robertson inequality is an inherited input. S.Frenkel is the printed translator, not a scientific coauthor.
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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