Editing Uncertainty principle (section)

===Time in quantum mechanics===
The concept of "time" in quantum mechanics offers many challenges.<ref name=HilgevoordConfusion/> There is no quantum theory of time measurement; relativity is both fundamental to time and difficult to include in quantum mechanics.<ref name="Busch2002"/> While position and momentum are associated with a single particle, time is a system property: it has no operator needed for the Robertson–Schrödinger relation.<ref name=Sen2014/> The mathematical treatment of stable and unstable quantum systems differ.<ref>{{Cite journal |last=Bohm |first=Arno |date=January 2011 |title=Resonances/decaying states and the mathematics of quantum physics |url=https://linkinghub.elsevier.com/retrieve/pii/S0034487711600189 |journal=Reports on Mathematical Physics |language=en |volume=67 |issue=3 |pages=279–303 |doi=10.1016/S0034-4877(11)60018-9 |bibcode=2011RpMP...67..279B |access-date=2024-01-24 |archive-date=2023-12-04 |archive-url=https://web.archive.org/web/20231204062259/https://linkinghub.elsevier.com/retrieve/pii/S0034487711600189 |url-status=live }}</ref> These factors combine to make energy–time uncertainty principles controversial.

Three notions of "time" can be distinguished:<ref name="Busch2002"/> external, intrinsic, and observable. External or laboratory time is seen by the experimenter; intrinsic time is inferred by changes in dynamic variables, like the hands of a clock or the motion of a free particle; observable time concerns time as an observable, the measurement of time-separated events.

An external-time energy–time uncertainty principle might say that measuring the energy of a quantum system to an accuracy <math>\Delta E</math> requires a time interval <math>\Delta t > h/\Delta E</math>.<ref name=Hilgevoord>{{Cite journal |last=Hilgevoord |first=Jan |date=1996-12-01 |title=The uncertainty principle for energy and time |url=https://pubs.aip.org/ajp/article/64/12/1451/1054748/The-uncertainty-principle-for-energy-and-time |journal=American Journal of Physics |language=en |volume=64 |issue=12 |pages=1451–1456 |doi=10.1119/1.18410 |bibcode=1996AmJPh..64.1451H |issn=0002-9505 |access-date=2023-11-12 |archive-date=2024-02-23 |archive-url=https://web.archive.org/web/20240223155750/https://pubs.aip.org/aapt/ajp/article-abstract/64/12/1451/1054748/The-uncertainty-principle-for-energy-and-time?redirectedFrom=fulltext |url-status=live }}</ref> However, [[Yakir Aharonov]] and [[David Bohm]]<ref>{{Cite journal |url=http://148.216.10.84/archivoshistoricosMQ/ModernaHist/Aharonov%20a.pdf |title=Time in the Quantum Theory and the Uncertainty Relation for Time and Energy |journal=Physical Review |volume=122 |issue=5 |date=June 1, 1961 |first1=Y. |last1=Aharonov |first2=D. |last2=Bohm |pages=1649–1658 |doi=10.1103/PhysRev.122.1649 |bibcode=1961PhRv..122.1649A |access-date=2012-01-21 |archive-date=2014-01-09 |archive-url=https://web.archive.org/web/20140109081758/http://148.216.10.84/archivoshistoricosMQ/ModernaHist/Aharonov%20a.pdf |url-status=dead }}</ref><ref name="Busch2002"/> have shown that, in some quantum systems, energy can be measured accurately within an arbitrarily short time: external-time uncertainty principles are not universal.

Intrinsic time is the basis for several formulations of energy–time uncertainty relations, including the Mandelstam–Tamm relation discussed in the next section. A physical system with an intrinsic time closely matching the external laboratory time is called a "clock".<ref name=HilgevoordConfusion>{{Cite journal |last=Hilgevoord |first=Jan |date=March 2005 |title=Time in quantum mechanics: a story of confusion |url=https://linkinghub.elsevier.com/retrieve/pii/S1355219804000760 |journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics |language=en |volume=36 |issue=1 |pages=29–60 |doi=10.1016/j.shpsb.2004.10.002 |bibcode=2005SHPMP..36...29H |access-date=2024-01-28 |archive-date=2022-10-23 |archive-url=https://web.archive.org/web/20221023233903/https://linkinghub.elsevier.com/retrieve/pii/S1355219804000760 |url-status=live }}</ref>{{rp|31}}

Observable time, measuring time between two events, remains a challenge for quantum theories; some progress has been made using [[POVM| positive operator-valued measure]] concepts.<ref name="Busch2002"/>