Editing Wave–particle duality

{{Short description|Concept in quantum mechanics}}
{{Quantum mechanics|cTopic=Fundamental concepts}}
'''Wave–particle duality''' is the concept in [[quantum mechanics]] that fundamental entities of the universe, like [[photon]]s and [[electron]]s, exhibit [[particle]] or [[wave (physics)|wave]] properties according to the experimental circumstances.<ref name=Messiah>{{Cite book |last=Messiah |first=Albert |url=https://archive.org/details/quantummechanics0000mess/quantummechanics0000mess |title=Quantum Mechanics |date=1966 |publisher=North Holland, John Wiley & Sons |isbn=0486409244 |language=en}}</ref>{{rp|59}} It expresses the inability of the [[Classical physics|classical]] concepts such as particle or wave to fully describe the behavior of quantum objects.<ref name="FeynmanIII">{{Cite book |last1=Feynman |first1=Richard P. |title=Quantum Mechanics |url=https://www.feynmanlectures.caltech.edu/III_toc.html |last2=Leighton |first2=Robert B. |last3=Sands |first3=Matthew L. |date=2007 |publisher=Addison-Wesley |isbn=978-0-201-02118-9 |series=[[The Feynman Lectures on Physics]] |volume=3 |location=Reading/Mass. |author-link1=Richard Feynman |author-link2=Robert B. Leighton |author-link3=Matthew Sands}}</ref>{{rp|III:1-1}} During the 19th and early 20th centuries, [[light]] was found to behave as a wave then later was discovered to have a particle-like behavior, whereas electrons behaved like particles in early experiments then were later discovered to have wave-like behavior. The concept of duality arose to name these seeming contradictions.

== History ==

=== Wave-particle duality of light ===
In the late 17th century, Sir [[Isaac Newton]] had advocated that light was [[Corpuscular theory of light|corpuscular]] (particulate), but [[Christiaan Huygens]] took an opposing wave description. While Newton had favored a particle approach, he was the first to attempt to reconcile both wave and particle theories of light, and the only one in his time to consider both, thereby anticipating modern wave-particle duality.<ref>{{Cite book |last=Finkelstein |first=David Ritz |author-link=David Finkelstein |url=https://books.google.com/books?id=OvjsCAAAQBAJ&pg=PA156 |title=Quantum Relativity |date=1996 |publisher=Springer Berlin Heidelberg |isbn=978-3-642-64612-6 |location= |pages=156, 169–170 |language=en |doi=10.1007/978-3-642-60936-7}}</ref><ref>{{Cite book |last=Arianrhod |first=Robyn |author-link=Robyn Arianrhod |url=https://books.google.com/books?id=ODDwiGtK1RQC&pg=PA232 |title=Seduced by Logic: Émilie Du Châtelet, Mary Somerville and the Newtonian Revolution |date=2012 |publisher=Oxford University Press |isbn=978-0-19-993161-3 |location=New York |pages=232 |language=en}}</ref> [[Thomas Young (scientist)|Thomas Young]]'s [[Young's interference experiment|interference experiments]] in 1801, and [[François Arago]]'s detection of the [[Poisson spot]] in 1819, validated Huygens' wave models. However, the wave model was challenged in 1901 by [[Planck's law]] for [[black-body radiation]].<ref>{{Cite journal |last=Planck |first=Max |date=1901 |title=Ueber das Gesetz der Energieverteilung im Normalspectrum |journal=Annalen der Physik |language=de |volume=309 |issue=3 |pages=553–563 |doi=10.1002/andp.19013090310|doi-access=free }}</ref> [[Max Planck]] heuristically derived a formula for the observed spectrum by assuming that a hypothetical electrically charged [[Oscillation|oscillator]] in a cavity that contained black-body radiation could only change its [[energy]] in a minimal increment, ''E'', that was proportional to the frequency of its associated [[electromagnetic wave]]. In 1905 [[Albert Einstein]] interpreted the [[photoelectric effect]] also with discrete energies for photons.<ref>{{Cite book |last=Einstein |first=Albert |title=The collected papers of Albert Einstein. 3: The Swiss years: writings, 1909 - 1911: [English translation] |date=1993 |publisher=Princeton Univ. Pr |isbn=978-0-691-10250-4 |location=Princeton, NJ}}</ref> These both indicate particle behavior. Despite confirmation by various experimental observations, the [[photon]] theory (as it came to be called) remained controversial until [[Arthur Compton]] performed a [[Compton effect|series of experiments]] from 1922 to 1924 demonstrating the momentum of light.<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 |date=1989 |publisher=Dover Publ |isbn=978-0-486-26126-3 |edition=Repr |location=New York}}</ref>{{rp|211}} The experimental evidence of particle-like momentum and energy seemingly contradicted the earlier work demonstrating wave-like interference of light.

=== Wave-particle duality of matter ===
{{Main|Matter wave}}
The contradictory evidence from electrons arrived in the opposite order. Many experiments by [[J. J. Thomson]],<ref name="Whittaker2" />{{rp|I:361}} [[Robert Millikan]],<ref name="Whittaker2" />{{rp|I:89}} and [[Charles Thomson Rees Wilson|Charles Wilson]]<ref name="Whittaker2" />{{rp|I:4}}  among others had shown that free electrons had particle properties, for instance, the measurement of their mass by Thomson in 1897.<ref>{{Cite journal |last=Thomson |first=J. J. |date=1897 |title=XL. Cathode Rays |url=https://www.tandfonline.com/doi/full/10.1080/14786449708621070 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |language=en |volume=44 |issue=269 |pages=293–316 |doi=10.1080/14786449708621070 |issn=1941-5982|url-access=subscription }}</ref> In 1924, [[Louis de Broglie]] introduced his theory of [[electron]] waves in his PhD thesis ''Recherches sur la théorie des quanta''.<ref name="Broglie">{{cite web |last1=de Broglie |first1=Louis Victor |title=On the Theory of Quanta |url=https://fondationlouisdebroglie.org/LDB-oeuvres/De_Broglie_Kracklauer.pdf |access-date=25 February 2023 |website=Foundation of Louis de Broglie |edition=English translation by A.F. Kracklauer, 2004.}}</ref> He suggested that an electron around a nucleus could be thought of as being a [[standing wave]] and that electrons and all matter could be considered as waves. He merged the idea of thinking about them as particles, and of thinking of them as waves. He proposed that particles are bundles of waves ([[wave packet]]s) that move with a [[group velocity]] and have an [[Effective mass (solid-state physics)|effective mass]]. Both of these depend upon the energy, which in turn connects to the [[Wave vector|wavevector]] and the relativistic formulation of [[Albert Einstein]] a few years before.

Following de Broglie's proposal of wave–particle duality of electrons, in 1925 to 1926, [[Erwin Schrödinger]] developed the wave equation of motion for electrons. This rapidly became part of what was called by Schrödinger ''undulatory mechanics'',<ref name="Schroedinger">{{Cite journal |last=Schrödinger |first=E. |date=1926 |title=An Undulatory Theory of the Mechanics of Atoms and Molecules |url=https://link.aps.org/doi/10.1103/PhysRev.28.1049 |journal=Physical Review |language=en |volume=28 |issue=6 |pages=1049–1070 |bibcode=1926PhRv...28.1049S |doi=10.1103/PhysRev.28.1049 |issn=0031-899X|url-access=subscription }}</ref> now called the [[Schrödinger equation]] and also "wave mechanics".

In 1926, [[Max Born]] gave a talk in an Oxford meeting about using the electron diffraction experiments to confirm the wave–particle duality of electrons. In his talk, Born cited experimental data from [[Clinton Davisson]] in 1923. It happened that Davisson also attended that talk. Davisson returned to his lab in the US to switch his experimental focus to test the wave property of electrons.<ref>{{Cite journal |last=Gehrenbeck |first=Richard K. |date=1978-01-01 |title=Electron diffraction: fifty years ago |url=https://doi.org/10.1063/1.3001830 |journal=Physics Today |volume=31 |issue=1 |pages=34–41 |doi=10.1063/1.3001830 |issn=0031-9228|url-access=subscription }}</ref>

In 1927, the wave nature of electrons was empirically confirmed by two experiments. The [[Davisson–Germer experiment]] at Bell Labs measured electrons scattered from [[Nickel|Ni metal]] surfaces.<ref>{{Cite journal | author=[[C. Davisson]] and [[L. H. Germer]] | s2cid=4104602 | title=The scattering of electrons by a single crystal of nickel | journal=Nature | year=1927 | volume=119 | pages=558–560 | doi=10.1038/119558a0|bibcode = 1927Natur.119..558D | issue=2998|ref=none | url=https://commons.wikimedia.org/wiki/File:The_Scattering_of_Electrons_by_a_Single_Crystal_of_Nickel.pdf}}</ref><ref name="DG12">{{Cite journal |last1=Davisson |first1=C. |last2=Germer |first2=L. H. |date=1927 |title=Diffraction of Electrons by a Crystal of Nickel |journal=Physical Review |volume=30 |issue=6 |pages=705–740 |bibcode=1927PhRv...30..705D |doi=10.1103/physrev.30.705 |issn=0031-899X |doi-access=free}}</ref><ref>{{Cite journal |last1=Davisson |first1=C. |last2=Germer |first2=L. H. |date=1927 |title=Diffraction of Electrons by a Crystal of Nickel |journal=Physical Review |language=en |volume=30 |issue=6 |pages=705–740 |bibcode=1927PhRv...30..705D |doi=10.1103/PhysRev.30.705 |issn=0031-899X |doi-access=free}}</ref><ref name="DG22">{{Cite journal |last1=Davisson |first1=C. J. |last2=Germer |first2=L. H. |date=1928 |title=Reflection of Electrons by a Crystal of Nickel |journal=Proceedings of the National Academy of Sciences |language=en |volume=14 |issue=4 |pages=317–322 |bibcode=1928PNAS...14..317D |doi=10.1073/pnas.14.4.317 |issn=0027-8424 |pmc=1085484 |pmid=16587341 |doi-access=free}}</ref><ref name=":02">{{Cite journal |last1=Davisson |first1=C. J. |last2=Germer |first2=L. H. |date=1928 |title=Reflection and Refraction of Electrons by a Crystal of Nickel |journal=Proceedings of the National Academy of Sciences |language=en |volume=14 |issue=8 |pages=619–627 |bibcode=1928PNAS...14..619D |doi=10.1073/pnas.14.8.619 |issn=0027-8424 |pmc=1085652 |pmid=16587378 |doi-access=free}}</ref> [[George Paget Thomson]] and Alexander Reid at Cambridge University scattered electrons through thin [[nickel]] films and observed concentric diffraction rings.<ref>{{Cite journal |last1=Thomson |first1=G. P. |last2=Reid |first2=A. |date=1927 |title=Diffraction of Cathode Rays by a Thin Film |journal=Nature |language=en |volume=119 |issue=3007 |pages=890 |bibcode=1927Natur.119Q.890T |doi=10.1038/119890a0 |issn=0028-0836 |s2cid=4122313 |doi-access=free}}</ref> Alexander Reid, who was Thomson's graduate student, performed the first experiments,<ref>{{Cite journal |last=Reid |first=Alexander |date=1928 |title=The diffraction of cathode rays by thin celluloid films |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |language=en |volume=119 |issue=783 |pages=663–667 |bibcode=1928RSPSA.119..663R |doi=10.1098/rspa.1928.0121 |issn=0950-1207 |s2cid=98311959 |doi-access=free}}</ref> but he died soon after in a motorcycle accident<ref>{{Cite journal |last=Navarro |first=Jaume |date=2010 |title=Electron diffraction chez Thomson: early responses to quantum physics in Britain |url=https://www.cambridge.org/core/product/identifier/S0007087410000026/type/journal_article |journal=The British Journal for the History of Science |language=en |volume=43 |issue=2 |pages=245–275 |doi=10.1017/S0007087410000026 |issn=0007-0874 |s2cid=171025814|url-access=subscription }}</ref> and is rarely mentioned. These experiments were rapidly followed by the first non-relativistic diffraction model for electrons by [[Hans Bethe]]<ref name="Bethe">{{Cite journal |last=Bethe |first=H. |date=1928 |title=Theorie der Beugung von Elektronen an Kristallen |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.19283921704 |journal=Annalen der Physik |language=de |volume=392 |issue=17 |pages=55–129 |doi=10.1002/andp.19283921704|bibcode=1928AnP...392...55B |url-access=subscription }}</ref> based upon the [[Schrödinger equation]], which is very close to how electron diffraction is now described. Significantly, Davisson and Germer noticed<ref name="DG22" /><ref name=":02" /> that their results could not be interpreted using a [[Bragg's law]] approach as the positions were systematically different; the approach of Bethe,<ref name="Bethe" /> which includes the refraction due to the average potential, yielded more accurate results. Davisson and Thomson were awarded the Nobel Prize in 1937 for experimental verification of wave property of electrons by diffraction experiments.<ref>{{Cite web |title=The Nobel Prize in Physics 1937 |url=https://www.nobelprize.org/prizes/physics/1937/summary/ |access-date=2024-03-18 |website=NobelPrize.org |language=en-US}}</ref> Similar crystal diffraction experiments were carried out by [[Otto Stern]] in the 1930s using beams of [[helium]] atoms and [[hydrogen]] molecules. These experiments further verified that wave behavior is not limited to electrons and is a general property of matter on a microscopic scale.

== Classical waves and particles ==
Before proceeding further, it is critical to introduce some definitions of waves and particles both in a classical sense and in quantum mechanics.  Waves and particles are two very different models for physical systems, each with an exceptionally large range of application. Classical waves obey the [[wave equation]]; they have continuous values at many points in space that vary with time; their spatial extent can vary with time due to [[diffraction]], and they display [[wave interference]]. Physical systems exhibiting wave behavior and described by the mathematics of wave equations include [[water waves]], [[seismic waves]], [[sound waves]], [[radio waves]], and more.

Classical particles obey [[classical mechanics]]; they have some [[center of mass]] and extent; they follow [[trajectories]] characterized by [[Position (geometry)|positions]] and [[velocities]] that vary over time; in the absence of [[forces]] their trajectories are straight lines. [[Stars]], [[planets]], [[spacecraft]], [[tennis balls]], [[bullets]], [[sand grain]]s: particle models work across a huge scale. Unlike waves, particles do not exhibit interference.

{{multiple image
| header            = Classical waves interfere. Particles follow trajectories.
| align             = center
| perrow            = 2
| total_width       = 500
| image_style       = border:none;
| image1            = Rippletanksource1plus2superpositionBnW.png
| alt1              = Wave interference in water due to two sources marked as red points on the left
| caption1          = [[Wave interference]] in water due to two sources marked as red points on the left.
| image2            = Inclinedthrow.gif|thumb|400px
| caption2          = Classical [[trajectories]] for a mass thrown at an angle of 70°, at different speeds.
| image3            = BachEtAl Interference.png
| caption3          = Line trace for a two-slit electron interference pattern. Compare to a slice through the image of the water wave pattern above.
| image4            = PositronDiscovery.png
| alt4              = Curved arc shows a cloud chamber trajectory of a positron.
| caption4          = Curved arc shows a [[cloud chamber]] trajectory of a [[positron]] acting like a particle.
| footer            = '''Both interference and trajectories are observed in quantum systems'''
}}
Some experiments on quantum systems show wave-like interference and diffraction; some experiments show particle-like collisions.

Quantum systems obey wave equations that predict particle probability distributions. These particles are associated with discrete values called [[quantum|quanta]] for properties such as [[Spin (physics)|spin]], [[electric charge]] and [[magnetic moment]]. These particles arrive one at time, randomly, but build up a pattern. The probability that experiments will measure particles at a point in space is the square of a complex-number valued wave. Experiments can be designed to exhibit diffraction and interference of the [[probability amplitude]].<ref name=Messiah/> Thus statistically large numbers of the random particle appearances can display wave-like properties. Similar equations govern collective excitations called [[quasiparticle]]s.

== Electrons behaving as waves and particles ==
{{See also|Double-slit experiment}}
The electron double slit experiment is a textbook demonstration of wave-particle duality.<ref name=FeynmanIII/> A modern version of the experiment is shown schematically in the figure below.

[[File:Roger Bach et al 2013 New J. Phys. 15 033018 Nj458349f1.jpg|Left half: schematic setup for electron double-slit experiment with masking; inset micrographs of slits and mask; Right half: results for slit 1, slit 2 and both slits open.<ref name="Bach Pope Liou Batelaan 2013 p=033018"/>|center|thumb|554x554px]]

Electrons from the source hit a wall with two thin slits. A mask behind the slits can expose either one or open to expose both slits. The results for high electron intensity are shown on the right, first for each slit individually, then with both slits open. With either slit open there is a smooth intensity variation due to diffraction. When both slits are open the intensity oscillates, characteristic of wave interference.

Having observed wave behavior, now change the experiment, lowering the intensity of the electron source until only one or two are detected per second, appearing as individual particles, dots in the video. As shown in the movie clip below, the dots on the detector seem at first to be random. After some time a pattern emerges, eventually forming an alternating sequence of light and dark bands.

{{multiple image
| align             = center
| perrow            = 1
| total_width       = 320
| image_style       = border:none;
| image1            = Roger Bach et al 2013 New J. Phys. 15 033018 Figure 3 cropped to top frame.jpg
| alt1              = Electron diffraction pattern
| image2            = Electron_buildup_movie_from_"Controlled_double-slit_electron_diffraction"_Roger_Bach_et_al_2013_New_J._Phys._15_033018.gif
| alt2              = Dots slowly filling an interference pattern.
| caption2          = Experimental electron double slit diffraction pattern.<ref name="Bach Pope Liou Batelaan 2013 p=033018">{{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> Across the middle of the image at the top the intensity alternates from high to low showing interference in the signal from the two slits. Bottom: movie of the pattern build up dot by dot. '''Click on the thumbnail to enlarge the movie.'''
}}

The experiment shows wave interference revealed a single particle at a time—quantum mechanical electrons display both wave and particle behavior. Similar results have been shown for atoms and even large molecules.<ref>{{Cite journal |last1=Arndt |first1=Markus |last2=Hornberger |first2=Klaus |date=2014 |title=Testing the limits of quantum mechanical superpositions |url=https://www.nature.com/articles/nphys2863 |journal=Nature Physics |language=en |volume=10 |issue=4 |pages=271–277 |arxiv=1410.0270v1 |doi=10.1038/nphys2863 |bibcode=2014NatPh..10..271A |s2cid=56438353 |issn=1745-2473}}</ref>

== Observing photons as particles ==
[[File:Photoelectric_effect_in_a_solid_-_diagram.svg|thumb|Photoelectric effect in a solid]]
{{Main|Photoelectric effect|Compton scattering}}
While electrons were thought to be particles until their wave properties were discovered, for photons it was the opposite. In 1887, [[Heinrich Hertz]] observed that when light with sufficient frequency hits a metallic surface, the surface emits [[cathode rays]], what are now called electrons.<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 |year=1910 |publisher=Longman, Green and Co.}}</ref>{{rp|399}} In 1902, [[Philipp Lenard]] discovered that the maximum possible energy of an ejected electron is unrelated to its [[Intensity (physics)|intensity]].<ref>{{cite journal |last=Wheaton |first=Bruce R. |year=1978 |title=Philipp Lenard and the Photoelectric Effect, 1889-1911 |journal=Historical Studies in the Physical Sciences |volume=9 |pages=299–322 |doi=10.2307/27757381 |jstor=27757381}}</ref> This observation is at odds with classical electromagnetism, which predicts that the electron's energy should be proportional to the intensity of the incident radiation.<ref name="Hawking2">{{cite journal |last1=Hawking |first1=Stephen |url=https://fb2bookfree.com/science/831-the-universe-in-a-nutshell.html |title=The Universe in a Nutshell |date=November 6, 2001 |publisher=Bantam Spectra |others=Impey, C.D. |journal=Physics Today |isbn=978-0553802023 |volume=55 |issue=4 |publication-date=April 2002 |page=80~ |language=en |doi=10.1063/1.1480788 |author-link1=Stephen Hawking |doi-access=free |s2cid-access=free |access-date=December 14, 2020 |orig-date=November 5, 2001 |archive-url=https://web.archive.org/web/20200921192954/https://fb2bookfree.com/science/831-the-universe-in-a-nutshell.html |archive-date=September 21, 2020 |via=Random House Audiobooks |s2cid=120382028}} [[iarchive:StephenHawkingTheUniverseInANutshellBookFi|Alt URL]].</ref>{{rp|24}} In 1905, [[Albert Einstein]] suggested that the energy of the light must occur a finite number of energy quanta.<ref name="Eistein-photoelectric2">{{cite journal |last=Einstein |first=Albert |author-link=Albert Einstein |year=1905 |title=Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt |bibcode-access=free |journal=Annalen der Physik |volume=17 |issue=6 |pages=132–48 |bibcode=1905AnP...322..132E |doi=10.1002/andp.19053220607 |doi-access=free |postscript=, }} translated into English as {{cite book |url-status=dead |chapter-url=http://lorentz.phl.jhu.edu/AnnusMirabilis/AeReserveArticles/eins_lq.pdf |archive-url=https://web.archive.org/web/20090611234106/http://lorentz.phl.jhu.edu/AnnusMirabilis/AeReserveArticles/eins_lq.pdf|archive-date=11 June 2009 |chapter=On a Heuristic Point of View about the Creation and Conversion of Light |title=The Old Quantum Theory |first1=A |last1=Einstein }}  The term "photon" was introduced in 1926.</ref> He postulated that electrons can receive energy from an electromagnetic field only in discrete units (quanta or photons): an amount of [[energy]] ''E'' that was related to the [[frequency]] ''f'' of the light by

: <math>E=hf</math>

[[File:Compton-scattering.svg|right|thumb|200x200px|A photon of wavelength <math>\lambda</math> comes in from the left, collides with a target at rest, and a new photon of wavelength <math>\lambda'</math> emerges at an angle <math>\theta</math>. The target recoils, and the photons have provided momentum to the target.]]
where ''h'' is the [[Planck constant]] (6.626×10<sup>−34</sup> J⋅s). Only photons of a high enough frequency (above a certain ''threshold'' value which, when multiplied by the Planck constant, is the [[work function]]) could knock an electron free. For example, photons of blue light had sufficient energy to free an electron from the metal he used, but photons of red light did not. One photon of light above the threshold frequency could release only one electron; the higher the frequency of a photon, the higher the kinetic energy of the emitted electron, but no amount of light below the threshold frequency could release an electron. Despite confirmation by various experimental observations, the [[photon]] theory (as it came to be called later) remained controversial until [[Arthur Compton]] performed a [[Compton effect|series of experiments]] from 1922 to 1924 demonstrating the momentum of light.<ref name="Whittaker2" />{{rp|211}}

Both discrete (quantized) energies and also momentum are, classically, particle attributes. There are many other examples where photons display particle-type properties, for instance in [[solar sail]]s, where sunlight could propel a space vehicle and [[laser cooling]] where the momentum is used to slow down (cool) atoms. These are a different aspect of wave-particle duality.

== Which slit experiments<span class="anchor" id="Which way experiment"></span> ==
In a "which way" experiment, particle detectors are placed at the slits to determine which slit the electron traveled through. When these detectors are inserted, quantum mechanics predicts that the interference pattern disappears because the detected part of the electron wave has changed (loss of [[Coherence (physics)|coherence]]).<ref name=FeynmanIII/> Many similar [[Wheeler's delayed-choice experiment|proposals]] have been made and many have been converted into experiments and tried out.<ref name="MaKoflerZeilinger">{{Cite journal |last1=Ma |first1=Xiao-song |last2=Kofler |first2=Johannes |last3=Zeilinger |first3=Anton |date=2016-03-03 |title=Delayed-choice gedanken experiments and their realizations |url=https://link.aps.org/doi/10.1103/RevModPhys.88.015005 |journal=Reviews of Modern Physics |language=en |volume=88 |issue=1 |page=015005 |doi=10.1103/RevModPhys.88.015005 |issn=0034-6861|arxiv=1407.2930 |bibcode=2016RvMP...88a5005M |s2cid=34901303 }}</ref> Every single one shows the same result: as soon as electron trajectories are detected, interference disappears.

A simple example of these "which way" experiments uses a [[Mach–Zehnder interferometer]], a device based on lasers and mirrors sketched below.<ref name=SchneiderLaPuma>{{Cite journal |last1=Schneider |first1=Mark B. |last2=LaPuma |first2=Indhira A. |date=2002-03-01 |title=A simple experiment for discussion of quantum interference and which-way measurement |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 |journal=American Journal of Physics |language=en |volume=70 |issue=3 |pages=266–271 |doi=10.1119/1.1450558 |issn=0002-9505}}</ref>

[[File:Mach Zehnder interferometer schematic diagram.jpg|thumb|center|600px|Interferometer schematic diagram]]

A laser beam along the input port splits at a half-silvered mirror. Part of the beam continues straight, passes though a glass [[quarter wave plate|phase shifter]], then reflects downward. The other part of the beam reflects from the first mirror then turns at another mirror. The two beams meet at a second half-silvered beam splitter.

Each output port has a camera to record the results. The two beams show interference characteristic of wave propagation. If the laser intensity is turned sufficiently low, individual dots appear on the cameras, building up the pattern as in the electron example.<ref name=SchneiderLaPuma/>

The first beam-splitter mirror acts like double slits, but in the interferometer case we can remove the second beam splitter. Then the beam heading down ends up in output port 1: any photon particles on this path gets counted in that port. The beam going across the top ends up on output port 2. In either case the counts will track the photon trajectories. However, as soon as the second beam splitter is removed the interference pattern disappears.<ref name=SchneiderLaPuma/>

== See also ==
* {{annotated link| Basic concepts of quantum mechanics }}
* {{annotated link| Complementarity (physics)  }}
* [[Einstein's thought experiments]]
* [[Interpretations of quantum mechanics]]
* {{annotated link| Wheeler's delayed choice experiment }}
* [[Uncertainty principle]]
* [[Matter wave]]
* [[Corpuscular theory of light]]

== References ==
{{Reflist}}

== External links ==
* {{cite web |author=R. Nave |title=Wave–Particle Duality |url=http://hyperphysics.phy-astr.gsu.edu/hbase/mod1.html |access-date=December 12, 2005 |work=HyperPhysics |publisher=Georgia State University, Department of Physics and Astronomy }}
* {{cite web |url=https://www.aps.org/programs/outreach/physicsquest/wave-particle.cfm |title=Wave–particle duality |website=PhysicsQuest |publisher=[[American Physical Society]] |access-date=August 31, 2023}}
* {{cite web |url=https://perimeterinstitute.ca/quantum-101-quantum-science-explained |first=Katie |last=Mack |title=Quantum 101 – Quantum Science Explained |publisher=[[Perimeter Institute for Theoretical Physics]] |author-link=Katie Mack (astrophysicist) |access-date=August 31, 2023}}

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