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Cited by 141Quantum Mechanics and ApplicationsSpectral Theory in Mathematical PhysicsQuantum chaos and dynamical systems

Generalized time-energy uncertainty relations and bounds on lifetimes of resonances

Pfeifer, Peter, Fröhlich, Jürg·1995·Reviews of Modern Physics
DOI 10.1103/revmodphys.67.759
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Abstract

A precise form of the quantum-mechanical time-energy uncertainty relation is derived. For any given initial state (density operator), time-dependent Hamiltonian, and subspace of reference states, it gives upper and lower bounds for the probability of finding the system in a state in that subspace at a later or earlier time. The bounds involve only the initial data, the energy uncertainty in the initial state, and the energy uncertainty in the reference subspace. They describe how fast the state enters or leaves the reference subspace. They are exact if, but not only if, the initial state or the projection onto the reference subspace commutes with the Hamiltonian. The basic tool used in the proof is a simple inequality for expectation values of commutators, which generalizes the usual uncertainty relation. By introducing suitable comparison dynamics (trial propagators), the bounds can be made arbitrarily tight. They represent a time-dependent variational principle, in terms of trial propagators, which provides explicit error estimates and reproduces the exact time evolution when one varies over all trial propagators. As illustrations, we derive accurate lower bounds on the escape time of a particle out of a potential well modeling a quantum dot, and the total time before which a ${\mathrm{He}}^{+}$ ion moving in a uniform magnetic field loses its electron.

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  • The quantum speed limit time gives the fundamental limit on the rate of evolution of quantum systems.

    “The quantum speed limit time gives the fundamental limit on the rate of evolution of quantum systems [20, 21, 22].”

    From Energy–time uncertainty relation for driven quantum systems (2013) · cites this paper as [21]

  • The bound is attained if and only if the evolution occurs on a geodesic.

    “The above discussion makes it clear that this bound is attained if and only if the evolution occurs on a geodesic, which is, incidentally, the same condition for attainability of the Mandelstam-Tamm bound for unitary processes with time-independent Hamiltonians [14].”

    From Quantum Speed Limit for Physical Processes (2013) · cites this paper as [14]

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Authors
Pfeifer, P., & Fröhlich, J.
Affiliations
École Polytechnique (FR)
ORCID
Venue
Reviews of Modern Physics
Publisher
American Physical Society (APS)
Published
1995-10-01
Volume
67
Issue
4
Pages
759-779
ISSN
  • 0034-6861 (ISSN-L)
  • 1539-0756
DOI
10.1103/revmodphys.67.759
OpenAlex
W2001290092
Cited by
141
References
49
Citations by year
2026: 1 · 2025: 4 · 2024: 8 · 2023: 10 · 2022: 12 · 2021: 9 · 2020: 17 · 2019: 6 · 2018: 5 · 2017: 6 · 2016: 11 · 2015: 11 · 2014: 3 · 2013: 5 · 2012: 1
FWCI
0.97
Type
journal article
Language
en
Open access
closed
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Reviews of Modern Physics (publishedVersion)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Mathematics › Mathematical Physics › Spectral Theory in Mathematical Physics
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Quantum chaos and dynamical systems
Keywords
time-energy uncertainty relation, resonance lifetimes, time-dependent Hamiltonian, time-dependent variational principle, escape time, quantum dots, He+ ions, energy uncertainty

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