Physics:Quantum Wave–particle duality - Revision history

WikiHarold: Restore missing Quantum reference definitions

2026-05-24T00:31:34Z

Restore missing Quantum reference definitions

← Older revision		Revision as of 00:31, 24 May 2026
Line 54:		Line 54:

	== Electrons behaving as waves and particles ==		== Electrons behaving as waves and particles ==
	The electron double-slit experiment is one of the clearest demonstrations of wave–particle duality.<ref name="FeynmanIII" /><ref name="Bach2013" /> Electrons emitted toward two narrow slits form a diffraction pattern when one slit is open and an interference pattern when both slits are open. At high intensity the detector records a smooth pattern, but when the beam intensity is reduced so that electrons arrive one at a time, each detection appears as an individual localized dot. Over time, those dots accumulate into the same interference pattern predicted by wave mechanics.<ref name="Bach2013" />		The electron double-slit experiment is one of the clearest demonstrations of wave–particle duality.<ref name="FeynmanIII" /><ref name="Bach2013">{{cite journal \| last1=Bach \| first1=Roger \| last2=Pope \| first2=Damian \| last3=Liou \| first3=Sy-Hwang \| last4=Batelaan \| first4=Herman \| title=Controlled double-slit electron diffraction \| journal=New Journal of Physics \| publisher=IOP Publishing \| volume=15 \| issue=3 \| date=2013-03-13 \| issn=1367-2630 \| doi=10.1088/1367-2630/15/3/033018 \| page=033018 \| arxiv=1210.6243 \| bibcode=2013NJPh...15c3018B \| s2cid=832961 \| url=https://iopscience.iop.org/article/10.1088/1367-2630/15/3/033018 }}</ref> Electrons emitted toward two narrow slits form a diffraction pattern when one slit is open and an interference pattern when both slits are open. At high intensity the detector records a smooth pattern, but when the beam intensity is reduced so that electrons arrive one at a time, each detection appears as an individual localized dot. Over time, those dots accumulate into the same interference pattern predicted by wave mechanics.<ref name="Bach2013" />

	{{multiple image		{{multiple image
Line 101:		Line 101:
	{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}		{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}

	==References==		= References =
	{{reflist\|3}}		{{reflist\|3}}

Maintenance script: Normalize quantum page header order

2026-05-22T11:31:15Z

Normalize quantum page header order

← Older revision		Revision as of 11:31, 22 May 2026
Line 2:		Line 2:
	{{Quantum book backlink\|Conceptual and interpretations}}		{{Quantum book backlink\|Conceptual and interpretations}}
	{{Quantum article nav\|previous=Physics:Quantum Interpretations of quantum mechanics\|previous label=Interpretations of quantum mechanics\|next=Physics:Quantum Complementarity principle\|next label=Complementarity principle}}		{{Quantum article nav\|previous=Physics:Quantum Interpretations of quantum mechanics\|previous label=Interpretations of quantum mechanics\|next=Physics:Quantum Complementarity principle\|next label=Complementarity principle}}

	<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

Maintenance script: Clean Book label remnants and backlink spacing

2026-05-22T11:09:32Z

Clean Book label remnants and backlink spacing

← Older revision		Revision as of 11:09, 22 May 2026
Line 1:		Line 1:
	{{Short description\|Concept in quantum mechanics}}		{{Short description\|Concept in quantum mechanics}}

	{{Quantum book backlink\|Conceptual and interpretations}}		{{Quantum book backlink\|Conceptual and interpretations}}
	{{Quantum article nav\|previous=Physics:Quantum Interpretations of quantum mechanics\|previous label=Interpretations of quantum mechanics\|next=Physics:Quantum Complementarity principle\|next label=Complementarity principle}}		{{Quantum article nav\|previous=Physics:Quantum Interpretations of quantum mechanics\|previous label=Interpretations of quantum mechanics\|next=Physics:Quantum Complementarity principle\|next label=Complementarity principle}}

Maintenance script: Remove hidden BOM characters and direct Book label after Short description

2026-05-22T11:01:45Z

Remove hidden BOM characters and direct Book label after Short description

← Older revision		Revision as of 11:01, 22 May 2026
Line 1:		Line 1:
	{{Short description\|Concept in quantum mechanics}}~~Book I~~		{{Short description\|Concept in quantum mechanics}}

	{{Quantum book backlink\|Conceptual and interpretations}}		{{Quantum book backlink\|Conceptual and interpretations}}

Maintenance script: Clean Short description prefix and add Book I label

2026-05-22T10:46:49Z

Clean Short description prefix and add Book I label

← Older revision		Revision as of 10:46, 22 May 2026
Line 1:		Line 1:
	~~~~{{Short description\|Concept in quantum mechanics}}		{{Short description\|Concept in quantum mechanics}}Book I

	{{Quantum book backlink\|Conceptual and interpretations}}		{{Quantum book backlink\|Conceptual and interpretations}}

Maintenance script: Apply Quantum previous-next navigation

2026-05-20T12:22:52Z

Apply Quantum previous-next navigation

← Older revision		Revision as of 12:22, 20 May 2026
Line 1:		Line 1:
	{{Short description\|Concept in quantum mechanics}}		{{Short description\|Concept in quantum mechanics}}

	{{Quantum book backlink\|Conceptual and interpretations}}		{{Quantum book backlink\|Conceptual and interpretations}}
			{{Quantum article nav\|previous=Physics:Quantum Interpretations of quantum mechanics\|previous label=Interpretations of quantum mechanics\|next=Physics:Quantum Complementarity principle\|next label=Complementarity principle}}

	<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

Maintenance script: Clean missing Book I links and template output

2026-05-20T08:46:40Z

Clean missing Book I links and template output

← Older revision		Revision as of 08:46, 20 May 2026
Line 1:		Line 1:
	{{Short description\|Concept in quantum mechanics}}		{{Short description\|Concept in quantum mechanics}}

	{{Quantum book backlink\|Conceptual and interpretations}}		{{Quantum book backlink\|Conceptual and interpretations}}
Line 43:		Line 43:
	\| image1 = Rippletanksource1plus2superpositionBnW.png		\| image1 = Rippletanksource1plus2superpositionBnW.png
	\| alt1 = Wave interference in water due to two sources		\| alt1 = Wave interference in water due to two sources
	\| caption1 = ~~[[Physics:~~Wave interference~~\|Wave interference]]~~ in water from two sources.		\| caption1 = Wave interference in water from two sources.
	\| image2 = Inclinedthrow.gif		\| image2 = Inclinedthrow.gif
	\| caption2 = Classical trajectories for a mass thrown at an angle.		\| caption2 = Classical trajectories for a mass thrown at an angle.
Line 50:		Line 50:
	\| image4 = PositronDiscovery.png		\| image4 = PositronDiscovery.png
	\| alt4 = Curved arc in a cloud chamber		\| alt4 = Curved arc in a cloud chamber
	\| caption4 = A ~~[[Physics:Cloud chamber\|~~cloud chamber]] track of a ~~[[Physics:Positron\|~~positron]], showing particle-like motion.		\| caption4 = A cloud chamber track of a positron, showing particle-like motion.
	\| footer = '''Quantum systems exhibit both interference and localized detection events.'''		\| footer = '''Quantum systems exhibit both interference and localized detection events.'''
	}}		}}

Maintenance script: Clean Book I red links, intro, and image slots

2026-05-20T08:11:44Z

Clean Book I red links, intro, and image slots

Harold: Arrange page top as TOC lead image columns

2026-05-17T13:59:17Z

Arrange page top as TOC lead image columns

← Older revision		Revision as of 13:59, 17 May 2026
Line 2:		Line 2:

	{{Quantum book backlink\|Conceptual and interpretations}}		{{Quantum book backlink\|Conceptual and interpretations}}

			<div style="display:flex; gap:24px; align-items:flex-start; max-width:1200px;">

			<div style="width:280px;">
			__TOC__
			</div>

			<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
	'''Wave–particle duality''' is the concept in [[Physics:Quantum mechanics\|quantum mechanics]] that quantum entities exhibit particle-like or wave-like properties depending on the experimental circumstances.<ref name="Messiah">{{cite book \| last=Messiah \| first=Albert \| title=Quantum Mechanics \| publisher=North Holland, John Wiley & Sons \| date=1966 \| isbn=0486409244 \| url=https://archive.org/details/quantummechanics0000mess/quantummechanics0000mess }}</ref><ref name="FeynmanIII">{{cite book \| last1=Feynman \| first1=Richard P. \| last2=Leighton \| first2=Robert B. \| last3=Sands \| first3=Matthew L. \| title=The Feynman Lectures on Physics, Volume III: Quantum Mechanics \| publisher=Addison-Wesley \| date=2007 \| isbn=978-0-201-02118-9 \| url=https://www.feynmanlectures.caltech.edu/III_toc.html }}</ref> It expresses the failure of purely classical categories such as ‘‘particle’’ and ‘‘wave’’ to give a complete description of quantum objects.<ref name="Messiah" /><ref name="FeynmanIII" />		'''Wave–particle duality''' is the concept in [[Physics:Quantum mechanics\|quantum mechanics]] that quantum entities exhibit particle-like or wave-like properties depending on the experimental circumstances.<ref name="Messiah">{{cite book \| last=Messiah \| first=Albert \| title=Quantum Mechanics \| publisher=North Holland, John Wiley & Sons \| date=1966 \| isbn=0486409244 \| url=https://archive.org/details/quantummechanics0000mess/quantummechanics0000mess }}</ref><ref name="FeynmanIII">{{cite book \| last1=Feynman \| first1=Richard P. \| last2=Leighton \| first2=Robert B. \| last3=Sands \| first3=Matthew L. \| title=The Feynman Lectures on Physics, Volume III: Quantum Mechanics \| publisher=Addison-Wesley \| date=2007 \| isbn=978-0-201-02118-9 \| url=https://www.feynmanlectures.caltech.edu/III_toc.html }}</ref> It expresses the failure of purely classical categories such as ‘‘particle’’ and ‘‘wave’’ to give a complete description of quantum objects.<ref name="Messiah" /><ref name="FeynmanIII" />

	During the 19th and early 20th centuries, light was first understood as a wave and later shown to possess a particulate character, while electrons were initially treated as particles and later found to exhibit diffraction and interference. The term ''wave–particle duality'' arose to describe these apparently contradictory behaviors.		During the 19th and early 20th centuries, light was first understood as a wave and later shown to possess a particulate character, while electrons were initially treated as particles and later found to exhibit diffraction and interference. The term ''wave–particle duality'' arose to describe these apparently contradictory behaviors.
	[[File:Double slit yellow.jpg\|thumb\|~~400px~~\|Wave–particle duality ~~in an electron double-slit experiment: electrons arrive as localized impacts, yet build up an interference pattern characteristic of waves~~.<~~ref name="Bach2013"~~>{{cite journal \| last1=Bach \| first1=Roger \| last2=Pope \| first2=Damian \| last3=Liou \| first3=Sy-Hwang \| last4=Batelaan \| first4=Herman \| title=Controlled double-slit electron diffraction \| journal=New Journal of Physics \| publisher=IOP Publishing \| volume=15 \| issue=3 \| date=2013-03-13 \| issn=1367-2630 \| doi=10.1088/1367-2630/15/3/033018 \| page=033018 \| arxiv=1210.6243 \| bibcode=2013NJPh...15c3018B \| s2cid=832961 \| url=https://iopscience.iop.org/article/10.1088/1367-2630/15/3/033018 }}</~~ref~~>]]		</div>

			<div style="width:300px;">
			[[File:Double slit yellow.jpg\|thumb\|280px\|Quantum Wave–particle duality.]]
			</div>

			</div>

	== History ==		== History ==
	In the late 17th century, [[Biography:Isaac Newton\|Isaac Newton]] argued that light consisted of particles, whereas [[Biography:Christiaan Huygens\|Christiaan Huygens]] proposed a wave theory of light.<ref>{{cite book \| last=Huygens \| first=Christiaan \| title=Traité de la lumiere \| publisher=Pieter van der Aa \| location=Leiden \| date=1690 }}</ref> In the early 19th century, [[Biography:Thomas Young (scientist)\|Thomas Young]]'s interference experiments and [[Biography:François Arago\|François Arago]]'s observation of the Poisson spot strongly supported the wave picture of light.		In the late 17th century, [[Biography:Isaac Newton\|Isaac Newton]] argued that light consisted of particles, whereas [[Biography:Christiaan Huygens\|Christiaan Huygens]] proposed a wave theory of light.<ref>{{cite book \| last=Huygens \| first=Christiaan \| title=Traité de la lumiere \| publisher=Pieter van der Aa \| location=Leiden \| date=1690 }}</ref> In the early 19th century, [[Biography:Thomas Young (scientist)\|Thomas Young]]'s interference experiments and [[Biography:François Arago\|François Arago]]'s observation of the Poisson spot strongly supported the wave picture of light.

imported>WikiHarold: Repair Quantum Collection B backlink template

2026-05-08T19:47:21Z

Repair Quantum Collection B backlink template

← Older revision	Revision as of 19:47, 8 May 2026
(No difference)

← Older revision		Revision as of 11:09, 22 May 2026
Line 1:		Line 1:
	{{Short description\|Concept in quantum mechanics}}		{{Short description\|Concept in quantum mechanics}}

	{{Quantum book backlink\|Conceptual and interpretations}}		{{Quantum book backlink\|Conceptual and interpretations}}
	{{Quantum article nav\|previous=Physics:Quantum Interpretations of quantum mechanics\|previous label=Interpretations of quantum mechanics\|next=Physics:Quantum Complementarity principle\|next label=Complementarity principle}}		{{Quantum article nav\|previous=Physics:Quantum Interpretations of quantum mechanics\|previous label=Interpretations of quantum mechanics\|next=Physics:Quantum Complementarity principle\|next label=Complementarity principle}}

← Older revision		Revision as of 08:11, 20 May 2026
Line 1:		Line 1:
	{{Short description\|Concept in quantum mechanics}}		{{Short description\|Concept in quantum mechanics}}

	{{Quantum book backlink\|Conceptual and interpretations}}		{{Quantum book backlink\|Conceptual and interpretations}}
Line 22:		Line 22:

	== History ==		== History ==
	In the late 17th century, ~~[[Biography:~~Isaac Newton~~\|Isaac Newton]]~~ argued that light consisted of particles, whereas ~~[[Biography:Christiaan Huygens\|~~Christiaan Huygens]] proposed a wave theory of light.<ref>{{cite book \| last=Huygens \| first=Christiaan \| title=Traité de la lumiere \| publisher=Pieter van der Aa \| location=Leiden \| date=1690 }}</ref> In the early 19th century, ~~[[Biography:Thomas Young (scientist)\|~~Thomas Young]]'s interference experiments and ~~[[Biography:François Arago\|~~François Arago]]'s observation of the Poisson spot strongly supported the wave picture of light.		In the late 17th century, Isaac Newton argued that light consisted of particles, whereas Christiaan Huygens proposed a wave theory of light.<ref>{{cite book \| last=Huygens \| first=Christiaan \| title=Traité de la lumiere \| publisher=Pieter van der Aa \| location=Leiden \| date=1690 }}</ref> In the early 19th century, Thomas Young's interference experiments and François Arago's observation of the Poisson spot strongly supported the wave picture of light.

	That picture was challenged by ~~[[Physics:~~Planck's law~~\|Planck's law]]~~ for ~~[[Astronomy:Black-body radiation\|~~black-body radiation]], which required energy exchange in discrete quanta.<ref name="Planck1901">{{cite journal \| last=Planck \| first=Max \| title=Ueber das Gesetz der Energieverteilung im Normalspectrum \| journal=Annalen der Physik \| volume=309 \| issue=3 \| pages=553–563 \| date=1901 \| language=de \| doi=10.1002/andp.19013090310 \| doi-access=free }}</ref> In 1905, [[Biography:Albert Einstein\|Albert Einstein]] interpreted the ~~[[Physics:Photoelectric effect\|~~photoelectric effect]] using discrete light quanta, later called photons.<ref name="Einstein1905">{{cite journal \| last=Einstein \| first=Albert \| title=Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt \| journal=Annalen der Physik \| volume=17 \| issue=6 \| pages=132–148 \| date=1905 \| doi=10.1002/andp.19053220607 \| doi-access=free }}</ref> The case for particle-like light was greatly strengthened by ~~[[Biography:Arthur Compton\|~~Arthur Compton]]'s measurements of photon momentum in scattering experiments.<ref name="Whittaker2">{{cite book \| last=Whittaker \| first=Edmund T. \| title=A history of the theories of aether & electricity. 2: The modern theories, 1900–1926 \| publisher=Dover Publications \| location=New York \| date=1989 \| isbn=978-0-486-26126-3 }}</ref>		That picture was challenged by Planck's law for black-body radiation, which required energy exchange in discrete quanta.<ref name="Planck1901">{{cite journal \| last=Planck \| first=Max \| title=Ueber das Gesetz der Energieverteilung im Normalspectrum \| journal=Annalen der Physik \| volume=309 \| issue=3 \| pages=553–563 \| date=1901 \| language=de \| doi=10.1002/andp.19013090310 \| doi-access=free }}</ref> In 1905, [[Biography:Albert Einstein\|Albert Einstein]] interpreted the photoelectric effect using discrete light quanta, later called photons.<ref name="Einstein1905">{{cite journal \| last=Einstein \| first=Albert \| title=Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt \| journal=Annalen der Physik \| volume=17 \| issue=6 \| pages=132–148 \| date=1905 \| doi=10.1002/andp.19053220607 \| doi-access=free }}</ref> The case for particle-like light was greatly strengthened by Arthur Compton's measurements of photon momentum in scattering experiments.<ref name="Whittaker2">{{cite book \| last=Whittaker \| first=Edmund T. \| title=A history of the theories of aether & electricity. 2: The modern theories, 1900–1926 \| publisher=Dover Publications \| location=New York \| date=1989 \| isbn=978-0-486-26126-3 }}</ref>

	For electrons, the historical sequence was reversed. Their charge-to-mass ratio and trajectories in electromagnetic fields supported a particle picture.<ref name="Thomson1897">{{cite journal \| last=Thomson \| first=J. J. \| title=XL. Cathode Rays \| journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science \| volume=44 \| issue=269 \| pages=293–316 \| date=1897 \| doi=10.1080/14786449708621070 }}</ref> In 1924, ~~[[Biography:~~Louis de Broglie~~\|Louis de Broglie]]~~ proposed that matter has an associated wavelength, relating a particle’s momentum to a wave description.<ref name="Broglie">{{cite web \| last=de Broglie \| first=Louis Victor \| title=On the Theory of Quanta \| url=https://fondationlouisdebroglie.org/LDB-oeuvres/De_Broglie_Kracklauer.pdf \| website=Foundation of Louis de Broglie }}</ref> This idea rapidly led to wave mechanics and the ~~[[Physics:Schrödinger equation\|~~Schrödinger equation]].<ref name="Schrodinger1926">{{cite journal \| last=Schrödinger \| first=E. \| title=An Undulatory Theory of the Mechanics of Atoms and Molecules \| journal=Physical Review \| volume=28 \| issue=6 \| pages=1049–1070 \| date=1926 \| doi=10.1103/PhysRev.28.1049 \| bibcode=1926PhRv...28.1049S }}</ref>		For electrons, the historical sequence was reversed. Their charge-to-mass ratio and trajectories in electromagnetic fields supported a particle picture.<ref name="Thomson1897">{{cite journal \| last=Thomson \| first=J. J. \| title=XL. Cathode Rays \| journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science \| volume=44 \| issue=269 \| pages=293–316 \| date=1897 \| doi=10.1080/14786449708621070 }}</ref> In 1924, Louis de Broglie proposed that matter has an associated wavelength, relating a particle’s momentum to a wave description.<ref name="Broglie">{{cite web \| last=de Broglie \| first=Louis Victor \| title=On the Theory of Quanta \| url=https://fondationlouisdebroglie.org/LDB-oeuvres/De_Broglie_Kracklauer.pdf \| website=Foundation of Louis de Broglie }}</ref> This idea rapidly led to wave mechanics and the Schrödinger equation.<ref name="Schrodinger1926">{{cite journal \| last=Schrödinger \| first=E. \| title=An Undulatory Theory of the Mechanics of Atoms and Molecules \| journal=Physical Review \| volume=28 \| issue=6 \| pages=1049–1070 \| date=1926 \| doi=10.1103/PhysRev.28.1049 \| bibcode=1926PhRv...28.1049S }}</ref>

	Experimental confirmation soon followed in the ~~[[Physics:~~Davisson–Germer experiment~~\|Davisson–Germer experiment]]~~ and in the diffraction experiments of ~~[[Biography:George Paget Thomson\|~~George Paget Thomson]] and Alexander Reid, showing that electrons produce diffraction patterns characteristic of waves.<ref name="DG1927">{{cite journal \| last1=Davisson \| first1=C. \| last2=Germer \| first2=L. H. \| title=Diffraction of Electrons by a Crystal of Nickel \| journal=Physical Review \| volume=30 \| issue=6 \| pages=705–740 \| date=1927 \| doi=10.1103/PhysRev.30.705 \| doi-access=free }}</ref><ref name="ThomsonReid1927">{{cite journal \| last1=Thomson \| first1=G. P. \| last2=Reid \| first2=A. \| title=Diffraction of Cathode Rays by a Thin Film \| journal=Nature \| volume=119 \| issue=3007 \| pages=890 \| date=1927 \| doi=10.1038/119890a0 \| doi-access=free }}</ref>		Experimental confirmation soon followed in the Davisson–Germer experiment and in the diffraction experiments of George Paget Thomson and Alexander Reid, showing that electrons produce diffraction patterns characteristic of waves.<ref name="DG1927">{{cite journal \| last1=Davisson \| first1=C. \| last2=Germer \| first2=L. H. \| title=Diffraction of Electrons by a Crystal of Nickel \| journal=Physical Review \| volume=30 \| issue=6 \| pages=705–740 \| date=1927 \| doi=10.1103/PhysRev.30.705 \| doi-access=free }}</ref><ref name="ThomsonReid1927">{{cite journal \| last1=Thomson \| first1=G. P. \| last2=Reid \| first2=A. \| title=Diffraction of Cathode Rays by a Thin Film \| journal=Nature \| volume=119 \| issue=3007 \| pages=890 \| date=1927 \| doi=10.1038/119890a0 \| doi-access=free }}</ref>

	== Classical waves and particles ==		== Classical waves and particles ==
Line 72:		Line 72:

	== Observing photons as particles ==		== Observing photons as particles ==
	For photons, the historical path ran in the opposite direction. Light was long understood as a wave, but the ~~[[Physics:Photoelectric effect\|~~photoelectric effect]] showed that energy transfer from light to matter occurs in discrete units.<ref name="Whittaker1">{{cite book \| last=Whittaker \| first=E. T. \| title=A History of the Theories of Aether and Electricity: From the Age of Descartes to the Close of the Nineteenth Century \| publisher=Longman, Green and Co. \| date=1910 }}</ref><ref>{{cite journal \| last=Wheaton \| first=Bruce R. \| title=Philipp Lenard and the Photoelectric Effect, 1889-1911 \| journal=Historical Studies in the Physical Sciences \| volume=9 \| pages=299–322 \| date=1978 \| doi=10.2307/27757381 \| jstor=27757381 }}</ref> Einstein proposed that light of frequency <math>f</math> is absorbed or emitted in quanta of energy		For photons, the historical path ran in the opposite direction. Light was long understood as a wave, but the photoelectric effect showed that energy transfer from light to matter occurs in discrete units.<ref name="Whittaker1">{{cite book \| last=Whittaker \| first=E. T. \| title=A History of the Theories of Aether and Electricity: From the Age of Descartes to the Close of the Nineteenth Century \| publisher=Longman, Green and Co. \| date=1910 }}</ref><ref>{{cite journal \| last=Wheaton \| first=Bruce R. \| title=Philipp Lenard and the Photoelectric Effect, 1889-1911 \| journal=Historical Studies in the Physical Sciences \| volume=9 \| pages=299–322 \| date=1978 \| doi=10.2307/27757381 \| jstor=27757381 }}</ref> Einstein proposed that light of frequency <math>f</math> is absorbed or emitted in quanta of energy

	:<math>E = hf</math>		:<math>E = hf</math>

	where <math>h</math> is the ~~[[Physics:~~Planck constant~~\|Planck constant]]~~.<ref name="Einstein1905" /> This relation explains why electrons are emitted only when the photon frequency exceeds a threshold set by the material’s ~~[[Physics:Work function\|~~work function]]. Increasing the light intensity below that threshold does not liberate electrons, whereas increasing the photon frequency above threshold increases the maximum kinetic energy of emitted electrons.		where <math>h</math> is the Planck constant.<ref name="Einstein1905" /> This relation explains why electrons are emitted only when the photon frequency exceeds a threshold set by the material’s work function. Increasing the light intensity below that threshold does not liberate electrons, whereas increasing the photon frequency above threshold increases the maximum kinetic energy of emitted electrons.

	Further evidence for photon-like behavior came from ~~[[Physics:~~Compton scattering~~\|Compton scattering]]~~, where X-rays exchange momentum with electrons in a way consistent with particle collisions.<ref name="Whittaker2" /> Thus light exhibits interference and diffraction characteristic of waves, while also carrying discrete energy and momentum characteristic of particles.		Further evidence for photon-like behavior came from Compton scattering, where X-rays exchange momentum with electrons in a way consistent with particle collisions.<ref name="Whittaker2" /> Thus light exhibits interference and diffraction characteristic of waves, while also carrying discrete energy and momentum characteristic of particles.

	[[File:Photoelectric_effect_in_a_solid_-_diagram.svg\|thumb\|400px\|The photoelectric effect demonstrates the quantized transfer of energy from light to matter.]]		[[File:Photoelectric_effect_in_a_solid_-_diagram.svg\|thumb\|400px\|The photoelectric effect demonstrates the quantized transfer of energy from light to matter.]]
Line 85:		Line 85:

	== Duality with other properties ==		== Duality with other properties ==
	Wave–particle duality is part of a broader quantum pattern in which classical attributes do not remain cleanly separated. ~~[[Physics:~~Angular momentum~~\|Angular momentum]]~~, for example, is often pictured as a particle-like quantity, but electromagnetic waves can carry orbital angular momentum when arranged in vortex beams.<ref>{{cite journal \| last1=Allen \| first1=L. \| last2=Beijersbergen \| first2=M. W. \| last3=Spreeuw \| first3=R. J. C. \| last4=Woerdman \| first4=J. P. \| title=Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes \| journal=Physical Review A \| volume=45 \| issue=11 \| pages=8185–8189 \| date=1992 \| doi=10.1103/PhysRevA.45.8185 \| pmid=9906912 }}</ref> Likewise, electron waves can be prepared with large orbital angular momentum and still display diffraction, interference, and magnetic interactions associated with particle properties.<ref>{{cite journal \| last1=Verbeeck \| first1=J. \| last2=Tian \| first2=H. \| last3=Schattschneider \| first3=P. \| title=Production and application of electron vortex beams \| journal=Nature \| volume=467 \| issue=7313 \| pages=301–304 \| date=2010 \| doi=10.1038/nature09366 \| pmid=20844532 \| s2cid=2970408 }}</ref><ref>{{cite journal \| last1=Tavabi \| first1=A. H. \| last2=Rosi \| first2=P. \| last3=Roncaglia \| first3=A. \| last4=Rotunno \| first4=E. \| last5=Beleggia \| first5=M. \| last6=Lu \| first6=P.-H. \| last7=Belsito \| first7=L. \| last8=Pozzi \| first8=G. \| last9=Frabboni \| first9=S. \| last10=Tiemeijer \| first10=P. \| last11=Dunin-Borkowski \| first11=R. E. \| last12=Grillo \| first12=V. \| title=Generation of electron vortex beams with over 1000 orbital angular momentum quanta using a tunable electrostatic spiral phase plate \| journal=Applied Physics Letters \| volume=121 \| issue=7 \| date=2022 \| doi=10.1063/5.0093411 \| arxiv=2203.00477 \| s2cid=247187983 }}</ref>		Wave–particle duality is part of a broader quantum pattern in which classical attributes do not remain cleanly separated. Angular momentum, for example, is often pictured as a particle-like quantity, but electromagnetic waves can carry orbital angular momentum when arranged in vortex beams.<ref>{{cite journal \| last1=Allen \| first1=L. \| last2=Beijersbergen \| first2=M. W. \| last3=Spreeuw \| first3=R. J. C. \| last4=Woerdman \| first4=J. P. \| title=Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes \| journal=Physical Review A \| volume=45 \| issue=11 \| pages=8185–8189 \| date=1992 \| doi=10.1103/PhysRevA.45.8185 \| pmid=9906912 }}</ref> Likewise, electron waves can be prepared with large orbital angular momentum and still display diffraction, interference, and magnetic interactions associated with particle properties.<ref>{{cite journal \| last1=Verbeeck \| first1=J. \| last2=Tian \| first2=H. \| last3=Schattschneider \| first3=P. \| title=Production and application of electron vortex beams \| journal=Nature \| volume=467 \| issue=7313 \| pages=301–304 \| date=2010 \| doi=10.1038/nature09366 \| pmid=20844532 \| s2cid=2970408 }}</ref><ref>{{cite journal \| last1=Tavabi \| first1=A. H. \| last2=Rosi \| first2=P. \| last3=Roncaglia \| first3=A. \| last4=Rotunno \| first4=E. \| last5=Beleggia \| first5=M. \| last6=Lu \| first6=P.-H. \| last7=Belsito \| first7=L. \| last8=Pozzi \| first8=G. \| last9=Frabboni \| first9=S. \| last10=Tiemeijer \| first10=P. \| last11=Dunin-Borkowski \| first11=R. E. \| last12=Grillo \| first12=V. \| title=Generation of electron vortex beams with over 1000 orbital angular momentum quanta using a tunable electrostatic spiral phase plate \| journal=Applied Physics Letters \| volume=121 \| issue=7 \| date=2022 \| doi=10.1063/5.0093411 \| arxiv=2203.00477 \| s2cid=247187983 }}</ref>

	[[File:~~Focused Laguerre-Gaussian beam~~.~~webm~~\|thumb\|400px\|A focused Laguerre–Gaussian beam carrying orbital angular momentum.]]		[[File:Quantum_book1_wave_particle_duality_yellow.png\|thumb\|400px\|A focused Laguerre–Gaussian beam carrying orbital angular momentum.]]

	== Which-slit experiments ==		== Which-slit experiments ==
	In “which-slit” or “which-way” experiments, detectors are placed so that the path taken by a quantum particle can be inferred. Quantum mechanics predicts that when path information becomes available, the interference pattern disappears because coherence between the alternatives is lost.<ref name="FeynmanIII" /> Many realizations of this idea, including delayed-choice experiments, confirm the same principle: path information and interference visibility are mutually constrained.<ref>{{cite journal \| last1=Ma \| first1=Xiao-song \| last2=Kofler \| first2=Johannes \| last3=Zeilinger \| first3=Anton \| title=Delayed-choice gedanken experiments and their realizations \| journal=Reviews of Modern Physics \| volume=88 \| issue=1 \| page=015005 \| date=2016 \| doi=10.1103/RevModPhys.88.015005 \| arxiv=1407.2930 \| s2cid=34901303 }}</ref>		In “which-slit” or “which-way” experiments, detectors are placed so that the path taken by a quantum particle can be inferred. Quantum mechanics predicts that when path information becomes available, the interference pattern disappears because coherence between the alternatives is lost.<ref name="FeynmanIII" /> Many realizations of this idea, including delayed-choice experiments, confirm the same principle: path information and interference visibility are mutually constrained.<ref>{{cite journal \| last1=Ma \| first1=Xiao-song \| last2=Kofler \| first2=Johannes \| last3=Zeilinger \| first3=Anton \| title=Delayed-choice gedanken experiments and their realizations \| journal=Reviews of Modern Physics \| volume=88 \| issue=1 \| page=015005 \| date=2016 \| doi=10.1103/RevModPhys.88.015005 \| arxiv=1407.2930 \| s2cid=34901303 }}</ref>

	A standard optical realization uses a ~~[[Physics:~~Mach–Zehnder interferometer~~\|Mach–Zehnder interferometer]]~~. A beam is split into two paths and later recombined. When the second beam splitter is present, the outputs depend on interference between the two paths. When it is removed, each output reveals path information and the interference effect vanishes.<ref>{{cite journal \| last1=Schneider \| first1=Mark B. \| last2=LaPuma \| first2=Indhira A. \| title=A simple experiment for discussion of quantum interference and which-way measurement \| journal=American Journal of Physics \| volume=70 \| issue=3 \| pages=266–271 \| date=2002 \| doi=10.1119/1.1450558 \| url=https://digital.grinnell.edu/islandora/object/grinnell%3A47/datastream/OBJ/download/A_Simple_Experiment_for_Discussion_of_Quantum_Interference_and_Which-Way_Measurement.pdf }}</ref>		A standard optical realization uses a Mach–Zehnder interferometer. A beam is split into two paths and later recombined. When the second beam splitter is present, the outputs depend on interference between the two paths. When it is removed, each output reveals path information and the interference effect vanishes.<ref>{{cite journal \| last1=Schneider \| first1=Mark B. \| last2=LaPuma \| first2=Indhira A. \| title=A simple experiment for discussion of quantum interference and which-way measurement \| journal=American Journal of Physics \| volume=70 \| issue=3 \| pages=266–271 \| date=2002 \| doi=10.1119/1.1450558 \| url=https://digital.grinnell.edu/islandora/object/grinnell%3A47/datastream/OBJ/download/A_Simple_Experiment_for_Discussion_of_Quantum_Interference_and_Which-Way_Measurement.pdf }}</ref>

	[[File:Mach Zehnder interferometer schematic diagram.jpg\|thumb\|400px\|A Mach–Zehnder interferometer illustrates the relation between path information and interference.]]		[[File:Mach Zehnder interferometer schematic diagram.jpg\|thumb\|400px\|A Mach–Zehnder interferometer illustrates the relation between path information and interference.]]