Editing Uncertainty principle (section)

== Energy–time ==
{{anchor|Time–energy uncertainty relation}}
An energy–time uncertainty relation like
<math display="block"> \Delta E \Delta t \gtrsim \hbar/2,</math> has a long, controversial history; the meaning of <math>\Delta t</math> and <math>\Delta E</math> varies and different formulations have different arenas of validity.<ref name="Busch2002">{{Cite book |last=Busch |first=Paul |url=http://link.springer.com/10.1007/3-540-45846-8_3 |title=Time in Quantum Mechanics. Lecture Notes in Physics |date=2002 |publisher=Springer Berlin Heidelberg |isbn=978-3-540-43294-4 |editor-last=Muga |editor-first=J. G. |volume=72 |location=Berlin, Heidelberg |pages=69–98 |language=en |chapter=The Time-Energy Uncertainty Relation |doi=10.1007/3-540-45846-8_3 |editor-last2=Mayato |editor-first2=R. Sala |editor-last3=Egusquiza |editor-first3=I. L.}}</ref> However, one well-known application is both well established<ref>{{Cite book |last=Wigner |first=E. P. |chapter=On the Time–Energy Uncertainty Relation |date=1997 |editor-last=Wightman |editor-first=Arthur S. |title=Part I: Particles and Fields. Part II: Foundations of Quantum Mechanics |chapter-url=http://link.springer.com/10.1007/978-3-662-09203-3_58 |language=en |location=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |pages=538–548 |doi=10.1007/978-3-662-09203-3_58 |isbn=978-3-642-08179-8}}</ref><ref name=Hilgevoord/> and experimentally verified:<ref>{{Cite journal |last1=Lynch |first1=F. J. |last2=Holland |first2=R. E. |last3=Hamermesh |first3=M. |date=1960-10-15 |title=Time Dependence of Resonantly Filtered Gamma Rays from Fe 57 |url=https://link.aps.org/doi/10.1103/PhysRev.120.513 |journal=Physical Review |language=en |volume=120 |issue=2 |pages=513–520 |doi=10.1103/PhysRev.120.513 |issn=0031-899X}}</ref><ref>{{cite book
 | last = Frauenfelder |first=H.
 | year = 1962
 | title = The Mössbauer Effect
 | url = https://archive.org/details/mssbauereffec00frau | publisher = [[W. A. Benjamin]]
 | lccn = 61018181|page=66
}}</ref> the connection between the life-time of a resonance state, <math>\tau_{\sqrt{1/2}}</math> and its energy width <math>\Delta E</math>:
<math display=block>\tau_{\sqrt{1/2}} \Delta E = \pi\hbar/4.</math>
In particle-physics, widths from experimental fits to the [[Relativistic Breit–Wigner distribution|Breit–Wigner energy distribution]] are used to characterize the lifetime of quasi-stable or decaying states.<ref>{{Cite journal |last1=Bohm |first1=Arno R. |last2=Sato |first2=Yoshihiro |date=2005-04-28 |title=Relativistic resonances: Their masses, widths, lifetimes, superposition, and causal evolution |url=https://link.aps.org/doi/10.1103/PhysRevD.71.085018 |journal=Physical Review D |language=en |volume=71 |issue=8 |page=085018 |arxiv=hep-ph/0412106 |doi=10.1103/PhysRevD.71.085018 |bibcode=2005PhRvD..71h5018B |s2cid=119417992 |issn=1550-7998}}</ref>

An informal, heuristic meaning of the principle is the following:<ref>Karplus, Martin, and Porter, Richard Needham (1970). ''Atoms and Molecules''. California: Benjamin Cummings. p. 68 {{ISBN|978-0805352184}}. {{oclc|984466711}}</ref> A state that only exists for a short time cannot have a definite energy. To have a definite energy, the frequency of the state must be defined accurately, and this requires the state to hang around for many cycles, the reciprocal of the required accuracy. For example, in [[Electromagnetic spectroscopy|spectroscopy]], excited states have a finite lifetime. By the time–energy uncertainty principle, they do not have a definite energy, and, each time they decay, the energy they release is slightly different. The average energy of the outgoing photon has a peak at the theoretical energy of the state, but the distribution has a finite width called the [[Spectral linewidth|''natural linewidth'']]. Fast-decaying states have a broad linewidth, while slow-decaying states have a narrow linewidth.<ref>The broad linewidth of fast-decaying states makes it difficult to accurately measure the energy of the state, and researchers have even used detuned microwave cavities to slow down the decay rate, to get sharper peaks. {{Cite journal |last=Gabrielse |first=Gerald |author2=H. Dehmelt |title=Observation of Inhibited Spontaneous Emission |journal=Physical Review Letters |volume=55 |pages=67–70 |year=1985 |doi=10.1103/PhysRevLett.55.67 |pmid=10031682 |issue=1 |bibcode=1985PhRvL..55...67G}}</ref> The same linewidth effect also makes it difficult to specify the [[rest mass]] of unstable, fast-decaying particles in [[particle physics]]. The faster the [[particle decay]]s (the shorter its lifetime), the less certain is its mass (the larger the particle's [[Resonance (particle physics)|width]]).

===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"/>

===Mandelstam–Tamm===
In 1945, [[Leonid Mandelstam]] and [[Igor Tamm]] derived a non-relativistic ''time–energy uncertainty relation'' as follows.<ref>L. I. Mandelstam, I. E. Tamm, [http://daarb.narod.ru/mandtamm/index-eng.html ''The uncertainty relation between energy and time in nonrelativistic quantum mechanics''] {{Webarchive|url=https://web.archive.org/web/20190607131054/http://daarb.narod.ru/mandtamm/index-eng.html |date=2019-06-07 }}, 1945.</ref><ref name="Busch2002"/> From Heisenberg mechanics, the generalized [[Ehrenfest theorem]] for an observable ''B'' without explicit time dependence, represented by a self-adjoint operator <math>\hat B</math> relates time dependence of the average value of <math>\hat B</math> to the average of its commutator with the Hamiltonian:

<math display=block> \frac{d\langle \hat{B} \rangle}{dt} = \frac{i}{\hbar}\langle [\hat{H},\hat{B}]\rangle. </math>

The value of <math>\langle [\hat{H},\hat{B}]\rangle</math> is then substituted in the [[#Robertson–Schrödinger_uncertainty_relations|Robertson uncertainty relation]] for the energy operator <math>\hat H</math> and <math>\hat B</math>:
<math display=block> \sigma_H\sigma_B \geq \left|\frac{1}{2i} \langle[ \hat{H}, \hat{B}] \rangle\right|, </math>
giving
<math display="block"> \sigma_H \frac{\sigma_B}{\left| \frac{d\langle \hat B \rangle}{dt}\right |} \ge \frac{\hbar}{2}</math>
(whenever the denominator is nonzero).
While this is a universal result, it depends upon the observable chosen and that the deviations <math>\sigma_H</math> and <math>\sigma_B</math> are computed for a particular state.
Identifying <math>\Delta E \equiv \sigma_E </math> and the characteristic time
<math display="block">\tau_B \equiv \frac{\sigma_B}{\left| \frac{d\langle \hat B \rangle}{dt}\right |}</math>
gives an energy–time relationship <math>\Delta E \tau_B \ge \frac{\hbar}{2}.</math>
Although <math>\tau_B</math> has the dimension of time, it is different from the time parameter ''t'' that enters the [[Schrödinger equation]]. This <math>\tau_B</math> can be interpreted as time for which the expectation value of the observable, <math>\langle \hat B \rangle,</math> changes by an amount equal to one standard deviation.<ref>{{Cite book |last=Naber |first=Gregory L. |url=https://books.google.com/books?id=kARGEAAAQBAJ |title=Quantum Mechanics: An Introduction to the Physical Background and Mathematical Structure |year=2021 |publisher=Walter de Gruyter GmbH & Co KG |isbn=978-3-11-075194-9 |pages=230 |language=en |access-date=2024-01-20 |archive-date=2024-02-23 |archive-url=https://web.archive.org/web/20240223155539/https://books.google.com/books?id=kARGEAAAQBAJ |url-status=live }}</ref>
Examples:
* The time a free quantum particle passes a point in space is more uncertain as the energy of the state is more precisely controlled: <math>\Delta T = \hbar/2\Delta E.</math> Since the time spread is related to the particle position spread and the energy spread is related to the momentum spread, this relation is directly related to position–momentum uncertainty.<ref name="GriffithsSchroeter2018" />{{rp|144}}
* A [[Delta particle]], a quasistable composite of quarks related to protons and neutrons, has a lifetime of 10<sup>−23</sup>&nbsp;s, so its measured [[Mass–energy equivalence| mass equivalent to energy]], 1232&nbsp;MeV/''c''<sup>2</sup>, varies by ±120&nbsp;MeV/''c''<sup>2</sup>; this variation is intrinsic and not caused by measurement errors.<ref name="GriffithsSchroeter2018" />{{rp|144}}
* Two energy states <math>\psi_{1,2}</math> with energies <math>E_{1,2},</math> superimposed to create a composite state
:<math display="block">\Psi(x,t) = a\psi_1(x) e^{-iE_1t/h} + b\psi_2(x) e^{-iE_2t/h}.</math>
:The probability amplitude of this state has a time-dependent interference term:
:<math display="block">|\Psi(x,t)|^2 = a^2|\psi_1(x)|^2 + b^2|\psi_2(x)|^2 + 2ab\cos(\frac{E_2 - E_1}{\hbar}t).</math>
:The oscillation period varies inversely with the energy difference: <math>\tau = 2\pi\hbar/(E_2 - E_1)</math>.<ref name="GriffithsSchroeter2018" />{{rp|144}}
Each example has a different meaning for the time uncertainty, according to the observable and state used.

===Quantum field theory===

Some formulations of [[quantum field theory]] uses temporary electron–positron pairs in its calculations called [[virtual particles]]. The mass-energy and lifetime of these particles are related by the energy–time uncertainty relation. The energy of a quantum systems is not known with enough precision to limit their behavior to a single, simple history. Thus the influence of ''all histories'' must be incorporated into quantum calculations, including those with much greater or much less energy than the mean of the measured/calculated energy distribution.

The energy–time uncertainty principle does not temporarily violate [[conservation of energy]]; it does not imply that energy can be "borrowed" from the universe as long as it is "returned" within a short amount of time.<ref name="GriffithsSchroeter2018" />{{rp|145}} The energy of the universe is not an exactly known parameter at all times.<ref name=Sen2014/> When events transpire at very short time intervals, there is uncertainty in the energy of these events.