Editing Matter wave (section)

== History ==
=== Background ===
At the end of the 19th century, light was thought to consist of waves of electromagnetic fields which propagated according to [[Maxwell's equations]], while matter was thought to consist of localized particles (see [[wave–particle duality#History|history of wave and particle duality]]). In 1900, this division was questioned when, investigating the theory of [[black-body radiation]], [[Max Planck]] proposed that the thermal energy of oscillating atoms is divided into discrete portions, or quanta.<ref>{{cite web|url=https://physicsworld.com/a/max-planck-the-reluctant-revolutionary/ |title=Max Planck: the reluctant revolutionary |first=Helge |last=Kragh |author-link=Helge Kragh |website=Physics World |date=2000-12-01 |access-date=2023-05-19}}</ref> Extending Planck's investigation in several ways, including its connection with the [[photoelectric effect]], [[Albert Einstein]] proposed in 1905 that light is also propagated and absorbed in quanta,<ref name="WhittakerII">{{cite book | last=Whittaker | first=Sir Edmund | title=A History of the Theories of Aether and Electricity | publisher=Courier Dover Publications | date=1989-01-01 | isbn=0-486-26126-3 | volume=2}}</ref>{{rp|87}} now called [[photon]]s. These quanta would have an energy given by the [[Planck–Einstein relation]]:
<math display="block">E = h\nu</math>
and a momentum vector <math>\mathbf{p}</math>
<math display="block">\left|\mathbf{p}\right| = p = \frac{E}{c} = \frac{h}{\lambda} ,</math>
where {{math|''ν''}} (lowercase [[Nu (letter)|Greek letter nu]]) and {{math|''λ''}} (lowercase [[Lambda|Greek letter lambda]]) denote the [[frequency]] and [[wavelength]] of the light, {{math|''c''}} the speed of light, and {{math|''h''}} the [[Planck constant]].<ref>[[Albert Einstein|Einstein, A.]] (1917). Zur Quantentheorie der Strahlung, ''Physicalische Zeitschrift'' '''18''': 121–128. Translated in 
{{cite book
 |last1=ter Haar |first1=D.
 |author-link=Dirk ter Haar
 |date=1967
 |pages=[https://archive.org/details/oldquantumtheory00haar/page/167 167–183]
 |title=The Old Quantum Theory
 |url=https://archive.org/details/oldquantumtheory00haar |url-access=registration |publisher=[[Pergamon Press]]
 |lccn=66029628
}}</ref> In the modern convention, frequency is symbolized by {{math|''f''}} as is done in the rest of this article. Einstein's postulate was verified experimentally<ref name="WhittakerII"/>{{rp|89}} by [[K. T. Compton]] and [[O. W. Richardson]]<ref name="Richardson Compton 1912 pp. 783–784">{{cite journal | last1=Richardson | first1=O. W. | last2=Compton | first2=Karl T. | title=The Photoelectric Effect | journal=Science | publisher=American Association for the Advancement of Science (AAAS) | volume=35 | issue=907 | date=1912-05-17 | issn=0036-8075 | doi=10.1126/science.35.907.783 | pages=783–784| pmid=17792421 | bibcode=1912Sci....35..783R | url=https://zenodo.org/record/1448080 }}</ref> and by A. L. Hughes<ref>Hughes, A. Ll. "XXXIII. The photo-electric effect of some compounds." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 24.141 (1912): 380–390.</ref> in 1912 then more carefully including a measurement of the [[Planck constant]] in 1916 by [[Robert Andrews Millikan|Robert Millikan]].<ref>{{cite journal
 |last1=Millikan
 |first1=R.
 |year=1916
 |title=A Direct Photoelectric Determination of Planck's "''h''"
 |journal=[[Physical Review]]
 |volume=7
 |issue=3
 |pages=355–388
 |bibcode=1916PhRv....7..355M
 |doi=10.1103/PhysRev.7.355
 |doi-access=free
 }}</ref>

=== De Broglie hypothesis ===
[[File:Propagation of a de broglie wave.svg|290px|right|thumb|Propagation of '''de Broglie waves''' in one dimension – real part of the [[complex number|complex]] amplitude is blue, imaginary part is green. The probability (shown as the color [[opacity (optics)|opacity]]) of finding the particle at a given point {{math|''x''}} is spread out like a waveform; there is no definite position of the particle. As the amplitude increases above zero the [[slope]] decreases, so the amplitude diminishes again, and vice versa. The result is an alternating amplitude: a wave. Top: [[plane wave]]. Bottom: [[wave packet]].]]

{{blockquote|When I conceived the first basic ideas of wave mechanics in 1923–1924, I was guided by the aim to perform a real physical synthesis, valid for all particles, of the coexistence of the wave and of the corpuscular aspects that Einstein had introduced for photons in his theory of light quanta in 1905.|de Broglie<ref>{{cite journal |first=Louis |last=de Broglie |title=The reinterpretation of wave mechanics |journal=Foundations of Physics |volume=1 |pages=5–15 |number=1 |year=1970|doi=10.1007/BF00708650 |bibcode=1970FoPh....1....5D |s2cid=122931010 }}</ref>}}

[[Louis de Broglie|De Broglie]], in his 1924 PhD thesis,<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> proposed that just as light has both wave-like and particle-like properties, [[electron]]s also have wave-like properties.
His thesis started from the hypothesis, "that to each portion of energy with a [[Invariant mass|proper mass]] {{math|''m''<sub>0</sub>}} one may associate a periodic phenomenon of the frequency {{math|''ν''<sub>0</sub>}}, such that one finds: {{math|1=''hν''<sub>0</sub> = ''m''<sub>0</sub>''c''<sup>2</sup>}}. The frequency {{math|''ν''<sub>0</sub>}} is to be measured, of course, in the rest frame of the energy packet. This hypothesis is the basis of our theory."<ref>{{cite journal | last1 = de Broglie | first1 = L. | author-link = Louis de Broglie | year = 1923 | title = Waves and quanta | journal = Nature | volume = 112 | issue = 2815| page = 540 | doi=10.1038/112540a0| bibcode = 1923Natur.112..540D| s2cid = 4082518 | doi-access = free }}</ref><ref name=Broglie />{{rp|p=8}}<ref name="Medicus">{{cite journal | last1 = Medicus | first1 = H.A. | year = 1974 | title = Fifty years of matter waves | journal = Physics Today | volume = 27 | issue = 2| pages = 38–45 | doi=10.1063/1.3128444| bibcode = 1974PhT....27b..38M}}</ref><ref name="MacKinnon">[http://scitation.aip.org/content/aapt/journal/ajp/44/11/10.1119/1.10583 MacKinnon, E. (1976). De Broglie's thesis: a critical retrospective, ''Am. J. Phys.'' '''44''': 1047–1055].</ref><ref>{{cite journal | last1 = Espinosa | first1 = J.M. | year = 1982 | title = Physical properties of de Broglie's phase waves | journal = Am. J. Phys. | volume = 50 | issue = 4| pages = 357–362 | doi=10.1119/1.12844| bibcode = 1982AmJPh..50..357E}}</ref><ref>{{cite journal | last1 = Brown | first1 = H.R. | last2 = Martins | year = 1984 | title = De Broglie's relativistic phase waves and wave groups | url = http://repositorio.unicamp.br/jspui/handle/REPOSIP/79307 | journal = Am. J. Phys. | volume = 52 | issue = 12 | pages = 1130–1140 | doi = 10.1119/1.13743 | bibcode = 1984AmJPh..52.1130B | access-date = 16 December 2019 | archive-date = 29 July 2020 | archive-url = https://web.archive.org/web/20200729040701/http://repositorio.unicamp.br/jspui/handle/REPOSIP/79307 | url-access = subscription }}</ref> (This frequency is also known as [[Compton wavelength|Compton frequency]].)

To find the [[wavelength]] equivalent to a moving body, de Broglie<ref name="WhittakerII"/>{{rp|214}} set the [[Energy–momentum relation#Connection to E = mc2|total energy]] from [[special relativity]] for that body equal to {{math | ''h&nu;''}}:
<math display="block">E = \frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}} = h\nu</math>

(Modern physics no longer uses this form of the total energy; the [[energy–momentum relation]] has proven more useful.) De Broglie identified the velocity of the particle, {{math|''v''}}, with the wave [[group velocity]] in free space:
<math display="block"> v_\text{g} \equiv \frac{\partial \omega}{\partial k} = \frac{d\nu}{d(1/\lambda)} </math>

(The modern definition of group velocity uses angular frequency {{mvar|ω}} and wave number {{mvar|k}}). By applying the differentials to the energy equation and identifying the [[Momentum#Relativistic|relativistic momentum]]:
<math display="block"> p = \frac{mv}{\sqrt{1-\frac{v^2}{c^2}}} </math>

then integrating, de Broglie arrived at his formula for the relationship between the [[wavelength]], {{mvar|λ}}, associated with an electron and the modulus of its [[momentum]], {{math|''p''}}, through the [[Planck constant]], {{math|''h''}}:<ref>{{cite book |title=Introducing Quantum Theory |author1=McEvoy, J. P. |author2=Zarate, Oscar  |publisher=Totem Books |year=2004 |isbn=978-1-84046-577-8 |pages=110–114}}</ref>
<math display="block"> \lambda = \frac{h}{p}.</math>

=== Schrödinger's (matter) wave equation ===

Following up on de Broglie's ideas, physicist [[Peter Debye]] made an offhand comment that if particles behaved as waves, they should satisfy some sort of wave equation. Inspired by Debye's remark, [[Erwin Schrödinger]] decided to find a proper three-dimensional wave equation for the electron. He was guided by [[William Rowan Hamilton]]'s analogy between mechanics and optics (see [[Hamilton's optico-mechanical analogy]]), encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system – the trajectories of [[light rays]] become sharp tracks that obey [[Fermat's principle]], an analog of the [[principle of least action]].<ref>{{Cite book | last=Schrödinger | first=E. | year=1984 | title=Collected papers | publisher=Friedrich Vieweg und Sohn | isbn=978-3-7001-0573-2}} See the introduction to first 1926 paper.</ref>

In 1926, Schrödinger published the [[Schrödinger equation|wave equation that now bears his name]]<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 |doi=10.1103/PhysRev.28.1049 |bibcode=1926PhRv...28.1049S |issn=0031-899X|url-access=subscription }}</ref> – the matter wave analogue of [[Maxwell's equations]] – and used it to derive the [[Emission spectrum|energy spectrum]] of [[hydrogen]]. Frequencies of solutions of the non-relativistic Schrödinger equation differ from de Broglie waves by the [[Compton wavelength|Compton frequency]] since the energy corresponding to the [[Invariant mass|rest mass]] of a particle is not part of the non-relativistic Schrödinger equation. The Schrödinger equation describes the time evolution of a [[wavefunction]], a function that assigns a [[complex number]] to each point in space. Schrödinger tried to interpret the [[modulus squared]] of the wavefunction as a charge density. This approach was, however, unsuccessful.<ref name=Moore1992>{{cite book | last=Moore | first=W. J. | year=1992 | title=Schrödinger: Life and Thought | publisher=[[Cambridge University Press]] | isbn=978-0-521-43767-7|pages=219–220}}</ref><ref name="jammer1974">{{cite book | last=Jammer | first=Max | author-link=Max Jammer | title=Philosophy of Quantum Mechanics: The interpretations of quantum mechanics in historical perspective | url=https://archive.org/details/philosophyofquan0000jamm | url-access=registration | year=1974 | publisher=Wiley-Interscience | isbn=978-0-471-43958-5 |pages=24–25}}</ref><ref>{{Cite journal|last=Karam|first=Ricardo|date=June 2020| title=Schrödinger's original struggles with a complex wave function|url=http://aapt.scitation.org/doi/10.1119/10.0000852| journal=[[American Journal of Physics]] | language=en| volume=88| issue=6| pages=433–438| doi=10.1119/10.0000852| bibcode=2020AmJPh..88..433K |s2cid=219513834 |issn=0002-9505| url-access=subscription}}</ref> [[Max Born]] proposed that the modulus squared of the wavefunction is instead a [[Probability density function|probability density]], a successful proposal now known as the [[Born rule]].<ref name=Moore1992/>

[[File:Guassian Dispersion.gif|180 px|thumb|right|Position space probability density of an initially Gaussian state moving in one dimension at minimally uncertain, constant momentum in free space]]
The following year, 1927, [[Charles Galton Darwin|C. G. Darwin]] (grandson of the [[Charles Darwin|famous biologist]]) explored [[Schrödinger equation|Schrödinger's equation]] in several idealized scenarios.<ref>Darwin, Charles Galton. "Free motion in the wave mechanics." Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 117.776 (1927): 258–293.</ref> For an unbound electron in free space he worked out the propagation of the wave, assuming an initial [[Wave packet#Gaussian wave packets in quantum mechanics|Gaussian wave packet]]. Darwin showed that at time <math>t</math> later the position <math>x</math> of the packet traveling at velocity <math>v</math> would be
<math display=block>x_0 + vt \pm \sqrt{\sigma^2 + (ht/2\pi\sigma m)^2}</math>
where <math>\sigma</math> is the uncertainty in the initial position. This position uncertainty creates uncertainty in velocity (the extra second term in the square root) consistent with [[Heisenberg]]'s [[uncertainty relation]] The wave packet spreads out as show in the figure.

=== Experimental confirmation ===

In 1927, matter waves were first experimentally confirmed to occur in [[George Paget Thomson]] and Alexander Reid's diffraction experiment<ref name=GPTdiff>{{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 |page=890 |doi=10.1038/119890a0 |bibcode=1927Natur.119Q.890T |s2cid=4122313 |issn=0028-0836|doi-access=free }}</ref> and the [[Davisson–Germer experiment]],<ref name="DG1">{{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 |doi=10.1103/physrev.30.705 |bibcode=1927PhRv...30..705D |issn=0031-899X|doi-access=free }}</ref><ref name="DG2">{{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 |doi=10.1073/pnas.14.4.317 |issn=0027-8424 |pmc=1085484 |pmid=16587341|bibcode=1928PNAS...14..317D |doi-access=free }}</ref> both for electrons.
{{multiple image
| total_width       = 600
| align             = center
| image1            = Original electron diffraction camera used by G P Thomson.jpg
| caption1          = Original electron diffraction camera made and used by Nobel laureate G P Thomson and his student Alexander Reid in 1925
| image2            = A565 G P Thomson Electron Diffraction.jpg
| caption2          = Example original electron diffraction photograph from the laboratory of G. P. Thomson, recorded 1925–1927
}}

The de Broglie hypothesis and the existence of matter waves has been confirmed for other elementary particles, neutral atoms and even molecules have been shown to be wave-like.<ref>{{Cite journal|last1=Arndt |first1=Markus |last2=Hornberger |first2=Klaus |date=April 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 |doi=10.1038/nphys2863 |issn=1745-2473|arxiv=1410.0270 |bibcode=2014NatPh..10..271A |s2cid=56438353 }}</ref>

The first electron wave interference patterns directly demonstrating [[wave–particle duality]] used electron biprisms<ref>Merli, P. G., G. F. Missiroli, and G. Pozzi. "On the statistical aspect of electron interference phenomena." American Journal of Physics 44 (1976): 306</ref><ref name="Tonomura Endo Matsuda Kawasaki 1989 pp. 117–120">{{cite journal | last1=Tonomura | first1=A. | last2=Endo | first2=J. | last3=Matsuda | first3=T. | last4=Kawasaki | first4=T. | last5=Ezawa | first5=H. | title=Demonstration of single-electron buildup of an interference pattern | journal=American Journal of Physics | publisher=American Association of Physics Teachers (AAPT) | volume=57 | issue=2 | year=1989 | issn=0002-9505 | doi=10.1119/1.16104 | pages=117–120| bibcode=1989AmJPh..57..117T }}</ref> (essentially a wire placed in an electron microscope) and measured single electrons building up the diffraction pattern.
A close copy of the famous [[double-slit experiment]]<ref name="BornAndWolf">
{{cite book
 |last1=Born |first1=M. |author1-link=Max Born
 |last2=Wolf |first2=E. |author2-link=Emil Wolf
 |year=1999
 |title=[[Principles of Optics]]
 |publisher=[[Cambridge University Press]]
 |isbn=978-0-521-64222-4
}}</ref>{{rp|p=260}} using electrons through physical apertures gave the movie shown.<ref name="Bach Pope Liou Batelaan 2013 p=033018"></ref>
[[File:Electron buildup movie from "Controlled double-slit electron diffraction" Roger Bach et al 2013 New J. Phys. 15 033018.gif|center|thumb|200x200px|Matter wave [[double slit diffraction]] pattern building up electron by electron. Each white dot represents a single electron hitting a detector; with a statistically large number of electrons interference fringes appear.<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>]]

==== Electrons ====
{{Further|Davisson–Germer experiment|Electron diffraction}}
In 1927 at Bell Labs, [[Clinton Davisson]] and [[Lester Germer]] [[Davisson–Germer experiment|fired]] slow-moving [[electron]]s at a [[crystal]]line [[nickel]] target.<ref name="DG1" /><ref name="DG2" /> The diffracted electron intensity was measured, and was determined to have a similar angular dependence to [[diffraction|diffraction patterns]] predicted by [[William Lawrence Bragg|Bragg]] for [[x-ray]]s. At the same time George Paget Thomson and Alexander Reid at the University of Aberdeen were independently firing electrons at thin celluloid foils and later metal films, observing rings which can be similarly interpreted.<ref name=GPTdiff/> (Alexander Reid, who was Thomson's graduate student, performed the first experiments 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 |s2cid=171025814 |issn=0007-0874|url-access=subscription }}</ref> and is rarely mentioned.) Before the acceptance of the de Broglie hypothesis, diffraction was a property that was thought to be exhibited only by waves. Therefore, the presence of any [[diffraction]] effects by matter demonstrated the wave-like nature of matter.<ref>Mauro Dardo, ''Nobel Laureates and Twentieth-Century Physics'', Cambridge University Press 2004, pp. 156–157</ref> The matter wave interpretation was placed onto a solid foundation in 1928 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> who solved the [[Schrödinger equation]],<ref name="Schroedinger"/> showing how this could explain the experimental results. His approach is similar to what is used in modern [[electron diffraction]] approaches.<ref name="Cowley95">{{Cite book |last=John M. |first=Cowley |title=Diffraction physics |date=1995 |publisher=Elsevier |isbn=0-444-82218-6 |oclc=247191522}}</ref><ref name="Peng">{{Cite book |last1=Peng |first1=L.-M. |title=High energy electron diffraction and microscopy |date=2011 |publisher=Oxford University Press |first2=S. L.| last2=Dudarev | first3=M. J. |last3=Whelan |isbn=978-0-19-960224-7 |location=Oxford |oclc=656767858}}</ref>

This was a pivotal result in the development of [[quantum mechanics]]. Just as the [[photoelectric effect]] demonstrated the particle nature of light, these experiments showed the wave nature of matter.

==== Neutrons ====
{{Also| Neutron diffraction}}
[[Neutron]]s, produced in [[nuclear reactors]] with kinetic energy of around {{val|1|u=MeV}}, [[Neutron temperature#Thermal|thermalize]] to around {{val|0.025|u=eV}} as they scatter from light atoms. The resulting de&nbsp;Broglie wavelength (around {{val|180|ul=pm}}) matches interatomic spacing and neutrons scatter strongly from hydrogen atoms. Consequently, neutron matter waves are used in [[crystallography]], especially for biological materials.<ref>{{Cite journal |last1=Blakeley |first1=Matthew P |last2=Langan |first2=Paul |last3=Niimura |first3=Nobuo |last4=Podjarny |first4=Alberto |date=2008-10-01 |title=Neutron crystallography: opportunities, challenges, and limitations |journal=Current Opinion in Structural Biology |series=Carbohydrates and glycoconjugates / Biophysical methods |volume=18 |issue=5 |pages=593–600 |doi=10.1016/j.sbi.2008.06.009 |issn=0959-440X |pmc=2586829 |pmid=18656544}}</ref> Neutrons were discovered in the early 1930s, and their diffraction was observed in 1936.<ref>{{Cite journal |last1=Mason |first1=T. E. |last2=Gawne |first2=T. J. |last3=Nagler |first3=S. E. |last4=Nestor |first4=M. B. |last5=Carpenter |first5=J. M. |date=2013-01-01 |title=The early development of neutron diffraction: science in the wings of the Manhattan Project |url=https://journals.iucr.org/a/issues/2013/01/00/wl5168/index.html |journal=Acta Crystallographica Section A: Foundations of Crystallography |language=en |volume=69 |issue=1 |pages=37–44 |doi=10.1107/S0108767312036021 |issn=0108-7673 |pmc=3526866 |pmid=23250059}}</ref> In 1944, [[Ernest O. Wollan]], with a background in X-ray scattering from his PhD work<ref name=PhysicsTodayObit>
{{cite journal
 |last1=Snell |first1=A. H.
 |last2=Wilkinson |first2=M. K.
 |last3=Koehler |first3=W. C.
 |year=1984
 |title=Ernest Omar Wollan
 |journal=[[Physics Today]]
 |volume=37 |issue=11 |page=120
 |bibcode=1984PhT....37k.120S
 |doi=10.1063/1.2915947
 |doi-access=free
}}</ref> under [[Arthur Compton]], recognized the potential for applying thermal neutrons from the newly operational [[X-10 Graphite Reactor|X-10 nuclear reactor]] to [[crystallography]]. Joined by [[Clifford G. Shull]], they developed<ref>
{{cite book
 |last=Shull |first=C. G.
 |date=1997
 |chapter=Early Development of Neutron Scattering
 |chapter-url=https://www.nobelprize.org/nobel_prizes/physics/laureates/1994/shull-lecture.pdf
 |archive-url=https://web.archive.org/web/20170519024556/https://www.nobelprize.org/nobel_prizes/physics/laureates/1994/shull-lecture.pdf
 |archive-date=2017-05-19
 |editor-last=Ekspong |editor-first=G.
 |title=Nobel Lectures, Physics 1991–1995
 |pages=145–154
 |publisher=[[World Scientific Publishing]]
}}</ref> [[neutron diffraction]] throughout the 1940s. 
In the 1970s, a [[neutron interferometer]] demonstrated the action of [[gravity]] in relation to wave–particle duality.<ref>{{Cite journal |doi = 10.1103/PhysRevLett.34.1472|title = Observation of Gravitationally Induced Quantum Interference |url=https://www.rpi.edu/dept/phys/Courses/PHYS6510/PhysRevLett.34.1472.pdf |year = 1975|last1 = Colella|first1 = R.|last2 = Overhauser|first2 = A. W.|last3 = Werner|first3 = S. A.|journal = Physical Review Letters|volume = 34|issue = 23|pages = 1472–1474|bibcode = 1975PhRvL..34.1472C}}</ref> The double-slit experiment was performed using neutrons in 1988.<ref>{{Cite journal |last1=Zeilinger |first1=Anton |last2=Gähler |first2=Roland |last3=Shull |first3=C. G. |last4=Treimer |first4=Wolfgang |last5=Mampe |first5=Walter |date=1988-10-01 |title=Single- and double-slit diffraction of neutrons |url=https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.60.1067 |journal=Reviews of Modern Physics |volume=60 |issue=4 |pages=1067–1073 |doi=10.1103/RevModPhys.60.1067|bibcode=1988RvMP...60.1067Z }}</ref>

==== Atoms ====
Interference of atom matter waves was first observed by [[Immanuel Estermann]] and [[Otto Stern]] in 1930, when a Na beam was diffracted off a surface of NaCl.<ref>{{cite journal | last1 = Estermann | first1 = I. | author-link2 = Otto Stern | last2 = Stern | first2 = Otto | year = 1930 | title =  Beugung von Molekularstrahlen| journal = Z. Phys. | volume = 61 | issue = 1–2| page = 95 | doi=10.1007/bf01340293|bibcode = 1930ZPhy...61...95E | s2cid = 121757478 }}</ref> The short de Broglie wavelength of atoms prevented progress for many years until two technological breakthroughs revived interest: [[microlithography]] allowing precise small devices and [[laser cooling]] allowing atoms to be slowed, increasing their de Broglie wavelength.<ref name="Adams Sigel Mlynek 1994 pp. 143–210">{{cite journal | last1=Adams | first1=C.S | last2=Sigel | first2=M | last3=Mlynek | first3=J | title=Atom optics | journal=Physics Reports | publisher=Elsevier BV | volume=240 | issue=3 | year=1994 | issn=0370-1573 | doi=10.1016/0370-1573(94)90066-3 | pages=143–210| bibcode=1994PhR...240..143A | doi-access=free }}</ref> The double-slit experiment on atoms was performed in 1991.<ref>{{Cite journal |last1=Carnal |first1=O. |last2=Mlynek |first2=J. |date=1991-05-27 |title=Young's double-slit experiment with atoms: A simple atom interferometer |url=https://pubmed.ncbi.nlm.nih.gov/10043591/ |journal=Physical Review Letters |volume=66 |issue=21 |pages=2689–2692 |doi=10.1103/PhysRevLett.66.2689 |issn=1079-7114 |pmid=10043591|bibcode=1991PhRvL..66.2689C }}</ref>

Advances in [[laser cooling]] allowed cooling of neutral atoms down to nanokelvin temperatures. At these temperatures, the de Broglie wavelengths come into the micrometre range. Using [[Bragg's law|Bragg diffraction]] of atoms and a Ramsey interferometry technique, the de Broglie wavelength of cold [[sodium]] atoms was explicitly measured and found to be consistent with the temperature measured by a different method.<ref name="Cla">
{{cite journal
 |author=Pierre Cladé
 |author2=Changhyun Ryu |author3=Anand Ramanathan |author4=Kristian Helmerson |author5=William D. Phillips
 |title=Observation of a 2D Bose Gas: From thermal to quasi-condensate to superfluid
 |date=2008
 |arxiv=0805.3519
 |doi=10.1103/PhysRevLett.102.170401
 |pmid=19518764 |volume=102
 |issue=17
 |page=170401 |journal=Physical Review Letters
 |bibcode=2009PhRvL.102q0401C
 |s2cid=19465661
}}</ref> 

==== Molecules ====
Recent experiments confirm the relations for molecules and even [[macromolecule]]s that otherwise might be supposed too large to undergo quantum mechanical effects. In 1999, a research team in [[Vienna]] demonstrated diffraction for molecules as large as [[fullerene]]s.<ref name="Arndt 680–682">{{cite journal| title=Wave–particle duality of C<sub>60</sub>| first=M.| last=Arndt| author2=O. Nairz |author3=J. Voss-Andreae |author3-link=Julian Voss-Andreae |author4=C. Keller |author5=G. van der Zouw |author6=A. Zeilinger |author6-link=Anton Zeilinger | journal=Nature| volume=401| issue=6754| pages=680–682|date=14 October 1999| pmid=18494170| doi=10.1038/44348| bibcode=1999Natur.401..680A| s2cid=4424892}}</ref> The researchers calculated a de Broglie wavelength of the most probable C<sub>60</sub> velocity as {{val|2.5|ul=pm}}.
More recent experiments prove the quantum nature of molecules made of 810 atoms and with a mass of {{val|10123|ul=Da}}.<ref>{{Cite journal |last1=Eibenberger|first1=Sandra |last2=Gerlich|first2=Stefan |last3=Arndt|first3=Markus |last4=Mayor|first4=Marcel |last5=Tüxen|first5=Jens |date=14 August 2013 |title=Matter–wave interference of particles selected from a molecular library with masses exceeding {{val|10000|u=amu}} |journal=Physical Chemistry Chemical Physics |language=en |volume=15|issue=35 |pages=14696–700 |doi=10.1039/c3cp51500a |pmid=23900710 |issn=1463-9084 |arxiv=1310.8343 |bibcode=2013PCCP...1514696E |s2cid=3944699 }}</ref> As of 2019, this has been pushed to molecules of {{val|25000|u=Da}}.<ref>{{Cite web |url=https://phys.org/news/2019-09-atoms-quantum-superposition.html |title=2000 atoms in two places at once: A new record in quantum superposition |website=phys.org |language=en-us |access-date=2019-09-25 }}</ref>

In these experiments the build-up of such interference patterns could be recorded in real time and with single molecule sensitivity.<ref name="Nano-20120325">{{cite journal |author=Juffmann, Thomas|title=Real-time single-molecule imaging of quantum interference |journal=Nature Nanotechnology |volume=7 |issue=5 |pages=297–300 |date=25 March 2012 |display-authors=etal|doi=10.1038/nnano.2012.34 |pmid=22447163 |arxiv=1402.1867 |bibcode=2012NatNa...7..297J |s2cid=5918772 }}</ref>
Large molecules are already so complex that they give experimental access to some aspects of the quantum-classical interface, i.e., to certain [[decoherence]] mechanisms.<ref>{{cite journal | first = Klaus | last = Hornberger | author2 = Stefan Uttenthaler | author3 = Björn Brezger | author4 = Lucia Hackermüller | author5 = Markus Arndt | author6 = Anton Zeilinger | year = 2003 | title = Observation of Collisional Decoherence in Interferometry | journal = Phys. Rev. Lett. | volume = 90 | pages = 160401 | doi = 10.1103/PhysRevLett.90.160401 | pmid = 12731960 | issue = 16 | bibcode = 2003PhRvL..90p0401H | arxiv = quant-ph/0303093 | s2cid = 31057272 }}</ref><ref>{{cite journal | first = Lucia | last = Hackermüller |author2=Klaus Hornberger |author3=Björn Brezger |author4=Anton Zeilinger |author5=Markus Arndt | year = 2004 | title = Decoherence of matter waves by thermal emission of radiation| journal = Nature | volume = 427 | pages = 711–714 | doi = 10.1038/nature02276 | pmid = 14973478 | issue = 6976 |arxiv = quant-ph/0402146 |bibcode = 2004Natur.427..711H | s2cid = 3482856 }}</ref>

==== Others ====
Matter wave was detected in [[Van der Waals molecule|van der Waals molecules]],<ref>{{Cite journal |last1=Schöllkopf |first1=Wieland |last2=Toennies |first2=J. Peter |date=1994-11-25 |title=Nondestructive Mass Selection of Small van der Waals Clusters |url=https://www.science.org/doi/10.1126/science.266.5189.1345 |journal=Science |language=en |volume=266 |issue=5189 |pages=1345–1348 |doi=10.1126/science.266.5189.1345 |pmid=17772840 |bibcode=1994Sci...266.1345S |issn=0036-8075|url-access=subscription }}</ref> [[Rho meson|rho mesons]],<ref>{{Cite journal |last=Ma |first=Yu-Gang |date=2023-01-30 |title=New type of double-slit interference experiment at Fermi scale |url=https://link.springer.com/article/10.1007/s41365-023-01167-6 |journal=Nuclear Science and Techniques |language=en |volume=34 |issue=1 |pages=16 |doi=10.1007/s41365-023-01167-6 |bibcode=2023NuScT..34...16M |issn=2210-3147}}</ref><ref>{{Cite journal |last=STAR Collaboration |date=2023-01-06 |title=Tomography of ultrarelativistic nuclei with polarized photon-gluon collisions |journal=Science Advances |language=en |volume=9 |issue=1 |pages=eabq3903 |doi=10.1126/sciadv.abq3903 |issn=2375-2548 |pmc=9812379 |pmid=36598973|arxiv=2204.01625 |bibcode=2023SciA....9.3903. }}</ref> [[Bose–Einstein condensate|Bose-Einstein condensate]].<ref>{{Cite journal |last=Ma |first=Yu-Gang |date=2023-01-30 |title=New type of double-slit interference experiment at Fermi scale |url=https://link.springer.com/article/10.1007/s41365-023-01167-6 |journal=Nuclear Science and Techniques |language=en |volume=34 |issue=1 |pages=16 |doi=10.1007/s41365-023-01167-6 |bibcode=2023NuScT..34...16M |issn=2210-3147}}</ref>