Time in the Quantum Theory and the Uncertainty Relation for Time and Energy
Abstract
Because time does not appear in Schr\"odinger's equation as an operator but only as a parameter, the time-energy uncertainty relation must be formulated in a special way. This problem has in fact been studied by many authors and we give a summary of their treatments. We then criticize the main conclusion of these treatments; viz., that in a measurement of energy carried out in a time interval, $\ensuremath{\Delta}t$, there must be a minimum uncertainty in the transfer of energy to the observed system, given by $\ensuremath{\Delta}({E}^{\ensuremath{'}}\ensuremath{-}E)>~\frac{h}{\ensuremath{\Delta}t}$. We show that this conclusion is erroneous in two respects. First, it is not consistent with the general principles of the quantum theory, which require that all uncertainty relations be expressible in terms of the mathematical formalism, i.e., by means of operators, wave functions, etc. Secondly, the examples of measurement processes that were used to derive the above uncertainty relation are not general enough. We then develop a systematic presentation of our own point of view, with regard to the role of time in the quantum theory, and give a concrete example of a measurement process not satisfying the above uncertainty relation.
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Anandan and Aharanov extended the MT bound to time-dependent Hamiltonians using a geometric approach with the Fubini-Study metric.
“Several years later, Anandan and Aharanov [7] extended the MT bound to time-dependent Hamiltonians by using a geometric approach that exploits the Fubini-Study metric defined on the space of quantum pure states.”
From Generalized Geometric Quantum Speed Limits (2016) · cites this paper as [7]
The correct interpretation of the time-energy Heisenberg uncertainty principle is that a quantum state with spread in energy ΔE takes time at least Δt = π hbar/2ΔE to evolve to an orthogonal state.
“In particular, the correct interpretation of the time-energy Heisenberg uncertainty principle ∆ E ∆ t ≥ ¯ h is not that it takes time ∆ t to measure energy to an accuracy ∆ E (a fallacy that was put to rest by Aharonov and Bohm 23 -24 ) but rather that that a quantum state with spread in energy ∆ E takes time at least ∆ t = π ¯ h/ 2∆ E to evolve to an orthogonal (and hence distinguishable) state 2”
From Ultimate physical limits to computation (2000) · cites this paper as [23]
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Self-citation
- Authors
- Aharonov, Y., & Bohm, D.
- Venue
- Physical Review
- Publisher
- American Physical Society (APS)
- Published
- 1961-06-01
- Volume
- 122
- Issue
- 5
- Pages
- 1649-1658
- ISSN
- 0031-899X (ISSN-L)
- OpenAlex
- W2087435909
- Cited by
- 482
- References
- 6
- Citations by year
- 2026: 9 · 2025: 22 · 2024: 21 · 2023: 18 · 2022: 16 · 2021: 11 · 2020: 17 · 2019: 20 · 2018: 13 · 2017: 11 · 2016: 14 · 2015: 14 · 2014: 16 · 2013: 10 · 2012: 10
- FWCI
- 1.72
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Physical Review (publishedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Mathematics › Mathematical Physics › Spectral Theory in Mathematical Physics
- Keywords
- time-energy uncertainty relation, Schrödinger equation, energy measurement, measurement process, quantum measurement theory
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