Physics:Quantum Uncertainty principle - Revision history

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	== See also ==		== See also ==
	* [[Physics:Quantum Matrix mechanics]]
	* [[Physics:Quantum Commutator]]
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	== See also ==		== See also ==
			* [[Physics:Quantum Matrix mechanics]]
			* [[Physics:Quantum Commutator]]
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	{{Short description\|Fundamental limit on simultaneous measurement precision in quantum systems}}~~Book I~~		{{Short description\|Fundamental limit on simultaneous measurement precision in quantum systems}}

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	'''Uncertainty principle''' ~~is a Book I topic in the Quantum Collection. Quantum~~ Uncertainty principle is a foundational concept in 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 Werner Heisenberg in 1927, the principle reflects an intrinsic property of quantum systems rather than a limitation of measurement technology. Quantum Uncertainty principle is a foundational concept in 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.		'''Uncertainty principle''' quantum Uncertainty principle is a foundational concept in 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 Werner Heisenberg in 1927, the principle reflects an intrinsic property of quantum systems rather than a limitation of measurement technology. Quantum Uncertainty principle is a foundational concept in 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. The most well-known form of the uncertainty principle relates the standard deviations of position and momentum:
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	'''~~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.		'''Uncertainty principle''' is a Book I topic in the Quantum Collection. Quantum Uncertainty principle is a foundational concept in 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 Werner Heisenberg in 1927, the principle reflects an intrinsic property of quantum systems rather than a limitation of measurement technology. Quantum Uncertainty principle is a foundational concept in 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>~~
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	<math>\sigma_x \sigma_p \ge \frac{\hbar}{2}</math>		<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>		This inequality was formally derived by 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 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:		More generally, for any pair of observables represented by operators <math>\hat{A}</math> and <math>\hat{B}</math>, the Robertson relation holds:
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	== Physical interpretation ==		== 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>		The uncertainty principle arises from the wave-like nature of quantum objects. A particle is described by a 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.		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.
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	== Operator formulation ==		== Operator formulation ==

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

	<math>[\hat{x},\hat{p}] = i\hbar</math>		<math>[\hat{x},\hat{p}] = i\hbar</math>
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	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>		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.		It is closely related to the concept of complementarity, where different experimental setups reveal mutually exclusive aspects of a system.

	== Applications ==		== Applications ==
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	The uncertainty principle underlies many physical phenomena:		The uncertainty principle underlies many physical phenomena:

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

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	'''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.		'''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>		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~~.]]		</div>

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			[[File:Quantum_uncertainty_principle_yellow.png\|thumb\|280px\|Quantum Uncertainty principle.]]
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	== Mathematical formulation ==		== Mathematical formulation ==

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