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{{Short description|Fundamental limit on simultaneous measurement precision in quantum systems}}

{{Quantum book backlink|Conceptual and interpretations}}
'''Quantum Uncertainty principle''' is a foundational concept in [[Physics:Quantum mechanics|quantum mechanics]] stating that certain pairs of physical properties—most notably position and momentum—cannot be simultaneously measured with arbitrary precision. The more precisely one observable is known, the less precisely the conjugate observable can be determined.

First introduced by [[Biography:Werner Heisenberg|Werner Heisenberg]] in 1927,<ref name="Heisenberg1927">{{Cite journal |last=Heisenberg |first=W. |title=Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik |journal=Zeitschrift für Physik |year=1927 |volume=43 |pages=172–198 |doi=10.1007/BF01397280}}</ref> the principle reflects an intrinsic property of quantum systems rather than a limitation of measurement technology.<ref name="Sen2014">{{Cite journal |last=Sen |first=D. |title=The Uncertainty relations in quantum mechanics |journal=Current Science |volume=107 |issue=2 |year=2014 |pages=203–218}}</ref>
[[File:Quantum_uncertainty_principle_yellow.png|thumb|400px|Conceptual visualization of the uncertainty principle: increasing localization in position leads to spreading in momentum space, reflecting fundamental quantum limits.]]

== Mathematical formulation ==

The most well-known form of the uncertainty principle relates the standard deviations of position and momentum:

<math>\sigma_x \sigma_p \ge \frac{\hbar}{2}</math>

This inequality was formally derived by [[Biography:Earle Hesse Kennard|Earle Kennard]]<ref name="Kennard">{{cite journal |last=Kennard |first=E. H. |title=Zur Quantenmechanik einfacher Bewegungstypen |journal=Zeitschrift für Physik |year=1927 |volume=44 |pages=326–352 |doi=10.1007/BF01391200}}</ref> and later generalized by [[Biography:Hermann Weyl|Hermann Weyl]].<ref name="Weyl1928">{{cite book |last=Weyl |first=H. |title=Gruppentheorie und Quantenmechanik |year=1928}}</ref>

More generally, for any pair of observables represented by operators <math>\hat{A}</math> and <math>\hat{B}</math>, the Robertson relation holds:

<math>\sigma_A \sigma_B \ge \frac{1}{2} |\langle[\hat{A},\hat{B}]\rangle|</math><ref name="Robertson1929">{{cite journal |last=Robertson |first=H. P. |title=The Uncertainty Principle |journal=Physical Review |year=1929 |volume=34 |pages=163–164 |doi=10.1103/PhysRev.34.163}}</ref>

== Physical interpretation ==

The uncertainty principle arises from the wave-like nature of quantum objects. A particle is described by a [[Physics:Wave function|wave function]] <math>\psi(x)</math>, whose spatial localization and momentum distribution are related through the [[Fourier transform]].<ref name="Bialynicki2009">{{cite journal |last1=Bialynicki-Birula |first1=I. |last2=Bialynicka-Birula |first2=Z. |title=Why photons cannot be sharply localized |journal=Physical Review A |year=2009 |volume=79 |doi=10.1103/PhysRevA.79.032112}}</ref>

A sharply localized wave packet requires a superposition of many momentum components, leading to large momentum uncertainty. Conversely, a well-defined momentum corresponds to a delocalized position.

This relationship is mathematically expressed through conjugate variables such as position and momentum, linked via the de Broglie relation:

<math>p = \hbar k</math>

== Operator formulation ==

In [[Physics:Matrix mechanics|matrix mechanics]], observables are represented by operators. The uncertainty principle follows from their non-commutativity:

<math>[\hat{x},\hat{p}] = i\hbar</math>

This implies that no quantum state can be an eigenstate of both position and momentum simultaneously.<ref name="Cohen1996">{{cite book |author=Cohen-Tannoudji, C. |title=Quantum Mechanics |publisher=Wiley |year=1996}}</ref>

== Energy–time uncertainty ==

A related but distinct relation exists between energy and time:

<math>\Delta E \Delta t \gtrsim \frac{\hbar}{2}</math>

This form does not arise from operator non-commutativity but reflects limits on processes such as the lifetime of unstable states and spectral linewidths.<ref name="Busch2002">{{cite book |last=Busch |first=P. |title=Time in Quantum Mechanics |year=2002}}</ref>

For example, short-lived excited states exhibit broad energy distributions, while long-lived states have sharply defined energies.

== Generalizations ==

The uncertainty principle has been extended in multiple directions:

* '''Robertson–Schrödinger relation''' includes correlations between observables<ref name="Schrodinger1930">{{cite journal |last=Schrödinger |first=E. |title=Zum Heisenbergschen Unschärfeprinzip |year=1930}}</ref>
* '''Entropic uncertainty relations''' use information entropy instead of variance<ref name="BBM">{{cite journal |last1=Bialynicki-Birula |first1=I. |last2=Mycielski |first2=J. |title=Uncertainty Relations for Information Entropy |journal=Communications in Mathematical Physics |year=1975}}</ref>
* '''Maccone–Pati relations''' provide stronger bounds for incompatible observables<ref>{{cite journal |last1=Maccone |first1=L. |last2=Pati |first2=A. K. |title=Stronger Uncertainty Relations |journal=Physical Review Letters |year=2014}}</ref>

These formulations highlight that uncertainty is a fundamental structural feature of quantum theory.

== Physical meaning ==

The uncertainty principle is often misunderstood as a limitation of measurement. In modern quantum theory, it is understood as an intrinsic property of quantum systems arising from their wave nature.<ref>{{cite journal |last=Rozema |first=L. A. |title=Violation of Heisenberg's Measurement–Disturbance Relationship |journal=Physical Review Letters |year=2012}}</ref>

It is closely related to the concept of [[Physics:Complementarity|complementarity]], where different experimental setups reveal mutually exclusive aspects of a system.

== Applications ==

The uncertainty principle underlies many physical phenomena:

* Spectral linewidths in [[Physics:Spectroscopy|spectroscopy]]
* Stability of atoms (preventing electron collapse)
* Quantum tunneling and zero-point energy
* Limits in precision measurements and [[Quantum metrology]]

It is also central to modern technologies such as interferometry and quantum information systems.<ref>{{cite journal |last=Caves |first=C. |title=Quantum-mechanical noise in an interferometer |journal=Physical Review D |year=1981}}</ref>

== See also ==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

==References==
{{reflist|3}}
{{Author|Harold Foppele}}
{{DEFAULTSORT:Quantum Uncertainty Principle}}
[[Category:Quantum mechanics]]
[[Category:Fundamental principles]]
[[Category:Mathematical physics]]

{{Sourceattribution|Physics:Quantum Uncertainty principle|1}}

Physics:Quantum Uncertainty principle - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template