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    "title": "Energy variance bounds on quantum evolution",
    "summary": "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.",
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      "what_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.",
      "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.",
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        "description": "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.",
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          "citation": "The uncertainty relation between energy and time in non-relativistic quantum mechanics",
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      "statement": "# Energy variance bounds on quantum evolution\n\nMandelstam 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.\n\n## Scientific scope\n\nThe 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.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nL. Mandelstam; Ig. Tamm\n\nSource-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.\n\n\n\n## Original sources\n\n- [The uncertainty relation between energy and time in non-relativistic quantum mechanics](https://daarb.narod.ru/mandtamm/mt-eng.pdf)\n",
      "summary": "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.",
      "title": "Energy variance bounds on quantum evolution"
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    "what_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.",
    "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.",
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