View source for Physics:Quantum Matter wave

{{Short description|Quantum mechanical waves describing matter}}

{{Quantum book backlink|Wavefunctions and modes}}

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'''Matter waves''' are a central part of the theory of [[Physics:Quantum mechanics|quantum mechanics]], being half of wave–particle duality. At all scales where measurements have been practical, matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave.

The concept that matter behaves like a wave was proposed by French physicist [[Biography:Louis de Broglie|Louis de Broglie]] () in 1924, and so matter waves are also known as '''de Broglie waves'''.

The ''de Broglie wavelength'' is the wavelength, {{math|''λ''}}, associated with a particle with momentum {{math|''p''}} through the Planck constant, {{math|''h''}}:
<math display="block"> \lambda = \frac{h}{p}.</math>

Wave-like behavior of matter has been experimentally demonstrated, first for electrons in 1927 (independently by Davisson and Germer and [[Biography:George Paget Thomson|George Thomson]]) and later for other elementary particles, neutral atoms and molecules.

Matter waves have more complex velocity relations than solid objects and they also differ from electromagnetic waves (light). Collective matter waves are used to model phenomena in solid state physics; standing matter waves are used in molecular chemistry. 

Matter wave concepts are widely used in the study of materials where different wavelength and interaction characteristics of electrons, neutrons, and atoms are leveraged for advanced microscopy and diffraction technologies.
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[[File:Electron_buildup_movie_from_"Controlled_double-slit_electron_diffraction"_Roger_Bach_et_al_2013_New_J._Phys._15_033018.gif|thumb|280px|Matter wave in the Quantum Collection.]]
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== 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 history of wave and particle duality). In 1900, this division was questioned when, investigating the theory of black-body radiation, [[Biography:Max Planck|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 |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, [[Biography:Albert Einstein|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 photons. 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 Greek letter nu) and {{math|''λ''}} (lowercase Greek letter lambda) denote the frequency and wavelength of light respectively, {{math|''c''}} the speed of light, and {{math|''h''}} the Planck constant.<ref>[[Biography:Albert Einstein|Einstein, A.]] (1917). Zur Quantentheorie der Strahlung, ''Physicalische Zeitschrift'' '''18''': 121–128. Translated in 
{{cite book
 |last1=ter Haar |first1=D.
 |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 [[Biography: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 amplitude is blue, imaginary part is green. The probability (shown as the color 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>}}

[[Biography: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, electrons also have wave-like properties.
His thesis started from the hypothesis, "that to each portion of energy with a 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. |  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 frequency.)

To find the wavelength equivalent to a moving body, de Broglie<ref name="WhittakerII"/>{{rp|214}} set the 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</math>, 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 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 [[Biography:Peter Debye|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, [[Biography:Erwin Schrödinger|Erwin Schrödinger]] decided to find a proper three-dimensional wave equation for the electron. He was guided by [[Biography:William Rowan Hamilton|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 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 energy spectrum of hydrogen. Frequencies of solutions of the non-relativistic Schrödinger equation differ from de Broglie waves by the Compton frequency since the energy corresponding to the 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 |  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> [[Biography:Max Born|Max Born]] proposed that the modulus squared of the wavefunction is instead a 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, C. G. Darwin (grandson of the [[Biography:Charles Darwin|famous biologist Charles Darwin]]) explored 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 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 + \left( \frac{ht}{2\pi\sigma m} \right ) ^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 shown in the figure.

=== Experimental confirmation ===

In 1927, matter waves were first experimentally confirmed to occur in [[Biography:George Paget Thomson|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.<ref>{{Cite journal |last=Smart |first=Ashley G. |date=2016-09-29 |title=How to almost win the physics Nobel |url=https://pubs.aip.org/physicstoday/online/12115 |journal=Physics Today |language=en |volume=2016 |issue=9 |article-number=12115 |doi=10.1063/pt.5.9069|bibcode=2016PhT..2016i2115S |url-access=subscription }}</ref><ref name=EisbergResnick>
{{cite book |author1    = Eisberg, R.
 |author2    = Resnick, R.
 |year       = 1985
 |edition    = 2nd
 |chapter    = Chapter 3 – de Broglie's Postulate—Wavelike Properties of Particles
 |title      = Quantum Physics: of Atoms, Molecules, Solids, Nuclei, and Particles
 |publisher  = John Wiley & Sons
 |isbn       = 978-0-471-87373-0
 |chapter-url-access = registration
 |chapter-url        = https://archive.org/details/quantumphysicsof00eisb
}}</ref>{{rp|56}}
{{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.<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
 |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 | article-number=033018 | arxiv=1210.6243 | bibcode=2013NJPh...15c3018B | s2cid=832961 | url=https://iopscience.iop.org/article/10.1088/1367-2630/15/3/033018}}</ref>]]

==== Electrons ====
 In 1927 at Bell Labs, [[Biography:Clinton Davisson|Clinton Davisson]] and [[Biography:Lester Germer|Lester Germer]] fired slow-moving electrons at a crystalline 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 patterns predicted by Bragg for x-rays. 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 [[Biography:Hans Bethe|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 [[Physics:Quantum mechanics|quantum mechanics]]. Just as the photoelectric effect demonstrated the particle nature of light, these experiments showed the wave nature of matter.

==== Neutrons ====
Neutrons, produced in nuclear reactors with kinetic energy of around {{val|1|u=MeV}}, 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, [[Biography:Ernest O. Wollan|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 [[Biography:Arthur Compton|Arthur Compton]], recognized the potential for applying thermal neutrons from the newly operational 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 [[Biography:Immanuel Estermann|Immanuel Estermann]] and [[Biography:Otto Stern|Otto Stern]] in 1930, when a Na beam was diffracted off a surface of NaCl.<ref>{{cite journal | last1 = Estermann | first1 = I. |  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 |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 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
 |article-number=170401 |journal=Physical Review Letters
 |bibcode=2009PhRvL.102q0401C
 |s2cid=19465661
}}</ref> 

==== Molecules ====
Recent experiments confirm the relations for molecules and even macromolecules 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 fullerenes.<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 |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=0|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 | article-number = 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 waves have been detected in 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 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 |article-number=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 |article-number=eabq3903 |doi=10.1126/sciadv.abq3903 |issn=2375-2548 |pmc=9812379 |pmid=36598973|arxiv=2204.01625 |bibcode=2023SciA....9.3903. }}</ref> and 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 |article-number=16 |doi=10.1007/s41365-023-01167-6 |bibcode=2023NuScT..34...16M |issn=2210-3147}}</ref>

== Traveling matter waves <span class="anchor" id="De Broglie relations"></span> ==
<!--This section is linked to by de Broglie relations -->
Waves have more complicated concepts for velocity than solid objects.
The simplest approach is to focus on the description in terms of plane matter waves for a free particle, that is a wave function described by
<math display="block">\psi (\mathbf{r}) = e^{ i \mathbf{k} \cdot \mathbf{r}-i \omega t },</math>
where <math>\mathbf{r}</math> is a position in real space, <math>\mathbf{k}</math>  is the wave vector in units of inverse meters, {{math|''ω''}} is the angular frequency with units of inverse time and <math>t</math> is time. (Here the physics definition for the wave vector is used, which is <math>2 \pi</math> times the wave vector used in crystallography, see wavevector.) The de Broglie equations relate the wavelength {{math|''λ''}} to the modulus of the momentum <math>|\mathbf{p}| = p</math>, and frequency {{math|''f''}} to the total energy {{math|''E''}} of a free particle as written above:<ref name="Resnick 1985">{{cite book |title=Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles |edition=2nd |first1=R. |last1=Resnick |first2=R. |last2=Eisberg |publisher=John Wiley & Sons |date=1985 |location=New York |isbn=978-0-471-87373-0 |url=https://archive.org/details/quantumphysicsof00eisb }}</ref>
<math display="block">\begin{align}
& \lambda = \frac {2 \pi}{|\mathbf{k}|} = \frac{h}{p}\\
& f = \frac{\omega}{2 \pi}= \frac{E}{h}
\end{align}</math>
where {{math|''h''}} is the Planck constant. The equations can also be written as
<math display="block">\begin{align}
& \mathbf{p} = \hbar \mathbf{k}\\
& E = \hbar \omega .\\
\end{align}</math>
Here, {{math|1=''ħ'' = ''h''/2''π''}} is the reduced Planck constant. The second equation is also referred to as the Planck–Einstein relation.

=== Group velocity ===
In the de Broglie hypothesis, the velocity of a particle equals the group velocity of the matter wave.<ref name="WhittakerII" />{{rp|214}}
In isotropic media or a vacuum the group velocity of a wave is defined by:
<math display="block"> \mathbf{v_g} = \frac{\partial \omega(\mathbf{k})}{\partial \mathbf{k}} </math>
The relationship between the angular frequency and wavevector is called the dispersion relationship. For the non-relativistic case this is:
<math display="block">\omega(\mathbf{k}) \approx \frac{m_0 c^2}{\hbar} + \frac{\hbar k^2}{2m_{0} }\,,</math>
where <math>m_0</math> is the rest mass. Applying the derivative gives the (non-relativistic) '''matter wave group velocity''':
<math display="block">\mathbf{v_g} = \frac{\hbar \mathbf{k}}{m_0} \, .</math>
For comparison, the group velocity of light, with a dispersion <math>\omega(k)=ck</math>, is the speed of light <math>c</math>.

As an alternative, using the relativistic dispersion relationship for matter waves
<math display="block"> \omega(\mathbf{k}) = \sqrt{k^2c^2 + \left(\frac{m_0c^2}{\hbar}\right)^2} \, ,</math>
then
<math display="block">\mathbf{v_g} = \frac{\mathbf{k}c^2}{\omega} \, . </math>
This relativistic form relates to the phase velocity as discussed below.

For non-isotropic media we use the Energy–momentum form instead:
<math display="block">\begin{align}
  \mathbf{v}_\mathrm{g} &= \frac{\partial \omega}{\partial \mathbf{k}} = \frac{\partial (E/\hbar)}{\partial (\mathbf{p} /\hbar)} = \frac{\partial E}{\partial \mathbf{p}} = \frac{\partial}{\partial \mathbf{p}} \left( \sqrt{p^2c^2+m_0^2c^4} \right)\\
    &= \frac{\mathbf{p} c^2}{\sqrt{p^2c^2 + m_0^2c^4}}\\
    &= \frac{\mathbf{p} c^2}{E} .
\end{align}</math>

But (see below), since the phase velocity is <math>\mathbf{v}_\mathrm{p} = E/\mathbf{p} = c^2/\mathbf{v}</math>, then
<math display="block">\begin{align}
 \mathbf{v}_\mathrm{g} &= \frac{\mathbf{p}c^2}{E}\\
    &= \frac{c^2}{\mathbf{v}_\mathrm{p}}\\
    &= \mathbf{v} ,
\end{align}</math>
where <math>\mathbf{v}</math> is the velocity of the center of mass of the particle, identical to the group velocity.

=== Phase velocity ===
The phase velocity in isotropic media is defined as:
<math display="block">\mathbf{v_p} = \frac{\omega}{\mathbf{k}}</math>
Using the relativistic group velocity above:<ref name="WhittakerII"/>{{rp|p=215}}
<math display="block">\mathbf{v_p} = \frac{c^2 }{\mathbf{v_g}}</math>
This shows that <math>\mathbf{v_{p}}\cdot \mathbf{v_{g}}=c^2</math> as reported by R.W. Ditchburn in 1948 and J. L. Synge in 1952. Electromagnetic waves also obey  <math>\mathbf{v_{p}}\cdot \mathbf{v_{g}}=c^2</math>, as both <math>|\mathbf{v_p}|=c</math> and <math>|\mathbf{v_g}|=c</math>. Since for matter waves, <math>|\mathbf{v_g}| < c</math>, it follows that <math>|\mathbf{v_p}| > c</math>, but only the group velocity carries information. The superluminal phase velocity therefore does not violate special relativity, as it does not carry information.

For non-isotropic media, then
<math display="block">\mathbf{v}_\mathrm{p} = \frac{\omega}{\mathbf{k}} = \frac{E/\hbar}{\mathbf{p}/\hbar} = \frac{E}{\mathbf{p}}. </math>

Using the relativistic relations for energy and momentum yields
<math display="block">\mathbf{v}_\mathrm{p} = \frac{E}{\mathbf{p}} = \frac{m c^2}{m \mathbf{v}} = \frac{\gamma m_0 c^2}{\gamma m_0 \mathbf{v}} = \frac{c^2}{\mathbf{v}}.</math>
The variable <math>\mathbf{v}</math> can either be interpreted as the speed of the particle or the group velocity of the corresponding matter wave—the two are the same. Since the particle speed <math>|\mathbf{v}| < c </math> for any particle that has nonzero mass (according to special relativity), the phase velocity of matter waves always exceeds ''c'', i.e.,
<math display="block">| \mathbf{v}_\mathrm{p} | > c ,</math>
which approaches ''c'' when the particle speed is relativistic. The superluminal phase velocity does not violate special relativity, similar to the case above for non-isotropic media. See the article on ''Dispersion (optics)'' for further details.

=== Special relativity ===
Using two formulas from special relativity, one for the relativistic mass energy and one for the relativistic momentum
<math display="block">\begin{align}
E &= m c^2 = \gamma m_0 c^2 \\[1ex]
\mathbf{p} &= m\mathbf{v} = \gamma m_0 \mathbf{v}
\end{align} </math>
allows the equations for de Broglie wavelength and frequency to be written as
<math display="block">\begin{align}
&\lambda =\,\, \frac {h}{\gamma m_0v}\, =\, \frac {h}{m_0v}\,\,\, \sqrt{1 - \frac{v^2}{c^2}} \\[2.38ex]
& f = \frac{\gamma\,m_0 c^2}{h} = \frac {m_0 c^2}{h\sqrt{1 - \frac{v^2}{c^2}}} ,
\end{align}</math>
where <math>v=|\mathbf{v}|</math> is the velocity, <math>\gamma</math> the Lorentz factor, and <math>c</math> the speed of light in vacuum.<ref>{{cite book |title=Stationary states |first=Alan |last=Holden |publisher=Oxford University Press |date=1971 |location=New York |isbn=978-0-19-501497-6 }}</ref><ref>Williams, W.S.C. (2002). ''Introducing Special Relativity'', Taylor & Francis, London, {{ISBN|0-415-27761-2}}, p. 192.</ref> This shows that as the velocity of a particle approaches zero (rest) the de Broglie wavelength approaches infinity.

=== Four-vectors ===
Using four-vectors, the de Broglie relations form a single equation:
<math display="block">\mathbf{P}= \hbar\mathbf{K} ,</math>
which is frame-independent.
Likewise, the relation between group/particle velocity and phase velocity is given in frame-independent form by:
<math display="block">\mathbf{K} = \left(\frac{\omega_0}{c^2}\right)\mathbf{U} ,</math>
where
* Four-momentum <math>\mathbf{P} = \left(\frac{E}{c}, {\mathbf{p}} \right)</math>
* Four-wavevector <math>\mathbf{K} = \left(\frac{\omega}{c}, {\mathbf{k}} \right) </math>
* Four-velocity <math>\mathbf{U} = \gamma(c,{\mathbf{u}}) = \gamma(c,v_\mathrm{g} \hat{\mathbf{u}}) </math>

== General matter waves ==
The preceding sections refer specifically to free particles for which the wavefunctions are plane waves. There are significant numbers of other matter waves, which can be broadly split into three classes: single-particle matter waves, collective matter waves and standing waves.

=== Single-particle matter waves ===
The more general description of matter waves corresponding to a single particle type (e.g. a single electron or neutron only) would have a form similar to
<math display="block">\psi (\mathbf{r}) = u(\mathbf{r},\mathbf{k})\exp(i\mathbf{k}\cdot \mathbf{r} - iE(\mathbf{k})t/\hbar)</math>
where now there is an additional spatial term <math>u(\mathbf{r},\mathbf{k})</math> in the front, and the energy has been written more generally as a function of the wave vector. The various terms given before still apply, although the energy is no longer always proportional to the wave vector squared. A common approach is to define an effective mass which in general is a tensor <math>m_{ij}^*</math> given by
<math display="block"> {m_{ij}^*}^{-1}  = \frac{1}{\hbar^2} \frac{\partial^2 E}{\partial k_i \partial k_j}</math>
so that in the simple case where all directions are the same the form is similar to that of a free wave above.<math display="block">E(\mathbf k) = \frac{\hbar^2 \mathbf k^2}{2 m^*}</math>In general the group velocity would be replaced by the probability current<ref name=Schiff>{{Cite book |last=Schiff |first=Leonard I. |title=Quantum mechanics |date=1987 |publisher=McGraw-Hill |isbn=978-0-07-085643-1 |edition=3. ed., 24. print |series=International series in pure and applied physics |location=New York}}</ref>
<math display="block">\mathbf{j}(\mathbf{r}) = \frac{\hbar}{2mi} \left( \psi^*(\mathbf{r}) \mathbf \nabla \psi(\mathbf{r}) - \psi(\mathbf{r}) \mathbf \nabla \psi^{*}(\mathbf{r}) \right) </math>
where <math>\nabla</math> is the del or gradient operator. The momentum would then be described using the kinetic momentum operator,<ref name="Schiff" />
<math display="block">\mathbf{p} = -i\hbar\nabla</math>
The wavelength is still described as the inverse of the modulus of the wavevector, although measurement is more complex. There are many cases where this approach is used to describe single-particle matter waves:
* Bloch wave, which form the basis of much of band structure as described in Ashcroft and Mermin, and are also used to describe the diffraction of high-energy electrons by solids.<ref>{{Cite book |last=Metherell |first=A. J. |title=Electron Microscopy in Materials Science |publisher=Commission of the European Communities |year=1972 |pages=397–552}}</ref><ref name="Peng"/>
* Waves with angular momentum such as electron vortex beams.<ref>{{Cite journal |last1=Verbeeck |first1=J. |last2=Tian |first2=H. |last3=Schattschneider |first3=P. |date=2010 |title=Production and application of electron vortex beams |url=https://www.nature.com/articles/nature09366 |journal=Nature |language=en |volume=467 |issue=7313 |pages=301–304 |doi=10.1038/nature09366 |pmid=20844532 |bibcode=2010Natur.467..301V |s2cid=2970408 |issn=1476-4687|url-access=subscription }}</ref>
* Evanescent waves, where the component of the wavevector in one direction is complex. These are common when matter waves are being reflected, particularly for grazing-incidence diffraction.

=== Collective matter waves ===
Other classes of matter waves involve more than one particle, so are called collective waves and are often quasiparticles. Many of these occur in solids – see Ashcroft and Mermin. Examples include:
* In solids, an '''electron quasiparticle''' is an electron where interactions with other electrons in the solid have been included. An electron quasiparticle has the same charge and spin as a "normal" (elementary particle) electron and, like a normal electron, it is a fermion. However, its effective mass can differ substantially from that of a normal electron.<ref name="Kaxiras">{{cite book|first1=Efthimios |last1=Kaxiras|title=Atomic and Electronic Structure of Solids|url=https://books.google.com/books?id=WTL_vgbWpHEC&pg=PA65|date=9 January 2003|publisher=Cambridge University Press|isbn=978-0-521-52339-4|pages=65–69 |}}</ref> Its electric field is also modified, as a result of electric field screening.
* A '''hole''' is a quasiparticle which can be thought of as a vacancy of an electron in a state; it is most commonly used in the context of empty states in the valence band of a semiconductor.<ref name="Kaxiras" /> A hole has the opposite charge of an electron.
* A '''polaron''' is a quasiparticle where an electron interacts with the polarization of nearby atoms.
* An '''exciton''' is an electron and hole pair which are bound together.
* A Cooper pair is two electrons bound together so they behave as a single matter wave.

=== Standing matter waves ===
[[File:InfiniteSquareWellAnimation.gif|thumb|200px|right|Some trajectories of a particle in a box according to Newton's laws of classical mechanics (A), and matter waves (B–F). In (B–F), the horizontal axis is position, and the vertical axis is the real part (blue) and imaginary part (red) of the wavefunction. The states (B,C,D) are energy eigenstates, but (E,F) are not.]]
The third class are matter waves which have a wavevector, a wavelength and vary with time, but have a zero group velocity or probability flux. The simplest of these, similar to the notation above would be
<math display="block">\cos(\mathbf{k}\cdot\mathbf{r} - \omega t)</math>
These occur as part of the particle in a box, and other cases such as in a ring. This can, and arguably should be, extended to many other cases. For instance, in early work de Broglie used the concept that an electron matter wave must be continuous in a ring to connect to the Bohr–Sommerfeld condition in the early approaches to quantum mechanics.<ref>{{Cite book |last=Jammer |first=Max |title=The conceptual development of quantum mechanics |date=1989 |publisher=Thomas publishers |isbn=978-0-88318-617-6 |edition=2nd |series=The history of modern physics |location=Los Angeles (Calif.)}}</ref> In that sense atomic orbitals around atoms, and also molecular orbitals are electron matter waves.<ref>{{Cite journal |last=Mulliken |first=Robert S. |date=1932 |title=Electronic Structures of Polyatomic Molecules and Valence. II. General Considerations |url=https://link.aps.org/doi/10.1103/PhysRev.41.49 |journal=Physical Review |volume=41 |issue=1 |pages=49–71 |doi=10.1103/PhysRev.41.49 |bibcode=1932PhRv...41...49M |url-access=subscription }}</ref><ref>{{Cite book |last=Griffiths |first=David J. |title=Introduction to quantum mechanics |date=1995 |publisher=Prentice Hall |isbn=978-0-13-124405-4 |location=Englewood Cliffs, NJ}}</ref><ref>{{Cite book |last=Levine |first=Ira N. |title=Quantum chemistry |date=2000 |publisher=Prentice Hall |isbn=978-0-13-685512-5 |edition=5th |location=Upper Saddle River, NJ}}</ref>

== Matter waves vs. electromagnetic waves (light) ==
[[Biography:Erwin Schrödinger|Schrödinger]] applied Hamilton's optico-mechanical analogy to develop his wave mechanics for subatomic particles.<ref>{{Cite book |last=Schrödinger |first=Erwin |title=Collected papers on wave mechanics: together with his Four lectures on wave mechanics |date=2001 |publisher=AMS Chelsea Publishing, American Mathematical Society |isbn=978-0-8218-3524-1 |edition=Third (augmented) edition, New York 1982 |location=Providence, Rhode Island |translator-last=Shearer |translator-first=J. F. |translator-last2=Deans |translator-first2=Winifred Margaret}}</ref>{{rp|p=xi}} Consequently, wave solutions to the Schrödinger equation share many properties with  results of light wave optics. In particular, Kirchhoff's diffraction formula works well for electron optics<ref name="BornAndWolf" />{{rp|p=745}} and for atomic optics.<ref name=adams>{{Cite journal |last1=Adams |first1=C.S |last2=Sigel |first2=M |last3=Mlynek |first3=J |date=1994-05-01 |title=Atom optics |journal=Physics Reports |language=en |volume=240 |issue=3 |pages=143–210 |doi=10.1016/0370-1573(94)90066-3|bibcode=1994PhR...240..143A |doi-access=free }}</ref> The approximation works well as long as the electric fields change more slowly than the de Broglie wavelength. Macroscopic apparatus fulfill this condition; slow electrons moving in solids do not.

Beyond the equations of motion, other aspects of matter wave optics differ from the corresponding light optics cases.

'''Sensitivity of matter waves to environmental condition.'''
Many examples of electromagnetic (light) diffraction occur in air under many environmental conditions. Obviously visible light interacts weakly with air molecules. By contrast, strongly interacting particles like slow electrons and molecules require vacuum: the matter wave properties rapidly fade when they are exposed to even low pressures of gas.<ref name="Schlosshauer">{{Cite journal |last=Schlosshauer |first=Maximilian |date=2019-10-01 |title=Quantum decoherence |url=https://linkinghub.elsevier.com/retrieve/pii/S0370157319303084 |journal=Physics Reports |language=en |volume=831 |pages=1–57 |arxiv=1911.06282 |doi=10.1016/j.physrep.2019.10.001|bibcode=2019PhR...831....1S |s2cid=208006050 }}</ref> With special apparatus, high velocity electrons can be used to study liquids and gases. Neutrons, an important exception, interact primarily by collisions with nuclei, and thus travel several hundred feet in air.<ref>{{Cite web |last=Pynn |first=Roger |date=1990-07-01 |title=Neutron Scattering – A Primer |url=https://www.ncnr.nist.gov/summerschool/ss16/pdf/NeutronScatteringPrimer.pdf |access-date=2023-06-24 |website=National Institute of Standards and Technology}}</ref>

'''Dispersion.''' Light waves of all frequencies travel at the same speed of light while matter wave velocity varies strongly with frequency. The relationship between frequency (proportional to energy) and wavenumber or velocity (proportional to momentum) is called a dispersion relation. Light waves in a vacuum have linear dispersion relation between frequency: <math>\omega = ck</math>. For matter waves the relation is non-linear:
<math display=block>\omega(k) \approx \frac{m_0 c^2}{\hbar} + \frac{\hbar k^2}{2m_{0} }\,.</math>
This non-relativistic matter wave dispersion relation says the frequency in vacuum varies with wavenumber (<math>k=1/\lambda</math>) in  two parts: a constant part due to the de Broglie frequency of the rest mass (<math>\hbar \omega_0 = m_{0}c^2</math>) and a quadratic part due to kinetic energy. The quadratic term causes rapid spreading of wave packets of matter waves.

'''Coherence''' The visibility of diffraction features using an optical theory approach depends on the beam coherence,<ref name="BornAndWolf" /> which at the quantum level is equivalent to a density matrix approach.<ref>{{Cite journal |last=Fano |first=U. |date=1957 |title=Description of States in Quantum Mechanics by Density Matrix and Operator Techniques |url=https://link.aps.org/doi/10.1103/RevModPhys.29.74 |journal=Reviews of Modern Physics |language=en |volume=29 |issue=1 |pages=74–93 |doi=10.1103/RevModPhys.29.74 |bibcode=1957RvMP...29...74F |issn=0034-6861|url-access=subscription }}</ref><ref>{{Citation |last=Hall |first=Brian C. |title=Systems and Subsystems, Multiple Particles |date=2013 |url=https://link.springer.com/10.1007/978-1-4614-7116-5_19 |work=Quantum Theory for Mathematicians |series=Graduate Texts in Mathematics |volume=267 |pages=419–440 |access-date=2023-08-13 |place=New York, NY |publisher=Springer New York |language=en |doi=10.1007/978-1-4614-7116-5_19 |isbn=978-1-4614-7115-8|url-access=subscription }}</ref> As with light, transverse coherence (across the direction of propagation) can be increased by collimation. Electron optical systems use stabilized high voltage to give a narrow energy spread in combination with collimating (parallelizing) lenses and pointed filament sources to achieve good coherence.<ref>{{Cite book |last1=Hawkes |first1=Peter W. |title=Electron optics and electron microscopy |last2=Hawkes |first2=P. W. |date=1972 |publisher=Taylor & Francis |isbn=978-0-85066-056-2 |location=London | page = 117}}</ref> Because light at all frequencies travels the same velocity, longitudinal and temporal coherence are linked; in matter waves these are independent. For example, for atoms, velocity (energy) selection controls longitudinal coherence and pulsing or chopping controls temporal coherence.<ref name=adams />{{rp|p=154}}

'''Optically shaped matter waves'''
Optical manipulation of matter plays a critical role in matter wave optics: "Light waves can act as refractive, reflective, and absorptive structures for matter waves, just as glass interacts with light waves."<ref>{{Cite journal |last1=Cronin |first1=Alexander D. |last2=Schmiedmayer |first2=Jörg |last3=Pritchard |first3=David E. |date=2009-07-28 |title=Optics and interferometry with atoms and molecules |url=https://hdl.handle.net/1721.1/52372 |journal=Reviews of Modern Physics |language=en |volume=81 |issue=3 |pages=1051–1129 |doi=10.1103/RevModPhys.81.1051 |arxiv=0712.3703 |bibcode=2009RvMP...81.1051C |hdl=1721.1/52372 |s2cid=28009912 |issn=0034-6861}}</ref> Laser light momentum transfer can cool matter particles and alter the internal excitation state of atoms.<ref>{{cite journal | doi=10.1126/sciadv.abq2659 | title=Optical manipulation of matter waves | date=2022 | last1=Akbari | first1=Kamran | last2=Di Giulio | first2=Valerio | last3=García De Abajo | first3=F. Javier | journal=Science Advances | volume=8 | issue=42 | article-number=eabq2659 | pmid=36260664 | pmc=12419127 | arxiv=2203.07257 | bibcode=2022SciA....8.2659A }}</ref>

'''Multi-particle experiments'''
While single-particle free-space optical and matter wave equations are identical, multiparticle systems like coincidence experiments are not.<ref>{{Cite journal |last1=Brukner |first1=Časlav |last2=Zeilinger |first2=Anton |date=1997-10-06 |title=Nonequivalence between Stationary Matter Wave Optics and Stationary Light Optics |url=https://link.aps.org/doi/10.1103/PhysRevLett.79.2599 |journal=Physical Review Letters |language=en |volume=79 |issue=14 |pages=2599–2603 |doi=10.1103/PhysRevLett.79.2599 |bibcode=1997PhRvL..79.2599B |issn=0031-9007|url-access=subscription }}</ref>

== Applications of matter waves ==
The following subsections provide links to pages describing applications of matter waves as probes of materials or of fundamental quantum properties. In most cases these involve some method of producing travelling matter waves which initially have the simple form <math>\exp(i \mathbf{k}\cdot \mathbf{r} - i\omega t)</math>, then using these to probe materials.

As shown in the table below, matter wave mass ranges over 6 orders of magnitude and energy over 9 orders but the wavelengths are all in the picometre range, comparable to atomic spacings. (Atomic diameters range from 62 to 520&nbsp;pm, and the typical length of a carbon–carbon single bond is 154&nbsp;pm.) Reaching longer wavelengths requires special techniques like laser cooling to reach lower energies; shorter wavelengths make diffraction effects more difficult to discern.<ref name="Adams Sigel Mlynek 1994 pp. 143–210" /> Therefore, many applications focus on material structures, in parallel with applications of electromagnetic waves, especially X-rays. Unlike light, matter wave particles may have mass, electric charge, magnetic moments, and internal structure, presenting new challenges and opportunities.

{| class="wikitable"
|+ Various matter wave wavelengths
|-
! matter !! style=text-align:right | mass !! style=text-align:right | kinetic energy !! style=text-align:right | wavelength !! reference
|-
|Electron ||style=text-align:right | 1/1823 Da ||style=text-align:right |{{val|54|ul=eV}} || style=text-align:right | {{val|167|ul=pm}} ||Davisson–Germer experiment
|-
|Electron ||style=text-align:right | 1/1823 Da ||style=text-align:right |{{val|5|e=4|ul=eV}} || style=text-align:right | {{val|5|ul=pm}} ||Tonomura et al.<ref>
 {{cite journal
  |author1=Tonomura, Akira
  |author2=Endo, J.
  |author3=Matsuda, T.
  |author4=Kawasaki, T.
  |author5= Ezawa, H.
  |title=Demonstration of single-electron buildup of an interference pattern
  |journal=American Journal of Physics
  |volume=57 |issue=2 |year=1989 |pages=117–120
  |doi=10.1119/1.16104
  |bibcode=1989AmJPh..57..117T
  |url=https://pubs.aip.org/aapt/ajp/article-abstract/57/2/117/1040594/Demonstration-of-single-electron-buildup-of-an
 |url-access=subscription
  }}</ref>
|-
|He atom, H2 molecule
|style=text-align:right |{{val|4|ul=Da}}
|
|style=text-align:right |{{val|50|ul=pm}}
|Estermann and Stern<ref>{{Cite journal |last1=Estermann |first1=I. |last2=Stern |first2=O. |date=1930-01-01 |title=Beugung von Molekularstrahlen |journal=Zeitschrift für Physik |language=de |volume=61 |issue=1 |pages=95–125 |doi=10.1007/BF01340293 |bibcode=1930ZPhy...61...95E |s2cid=121757478 |issn=0044-3328}}</ref>
|-
|Neutron|| style="text-align:right" | {{val|1|ul=Da}}|| style="text-align:right" |{{val|0.025|ul=eV}}|| style="text-align:right" | {{val|181|ul=pm}}||Wollan and Shull<ref>
 {{cite journal
  | title = The Diffraction of Neutrons by Crystalline Powders
  | author1 = Wollan, E. O.
  | author2 = Shull, C. G.
  | journal = Physical Review
  | volume = 73 | issue = 8 | pages = 830–841
  | year = 1948 | publisher = American Physical Society
  | doi = 10.1103/PhysRev.73.830 | bibcode = 1948PhRv...73..830W
  | hdl = 2027/mdp.39015086506584
  | url = https://link.aps.org/doi/10.1103/PhysRev.73.830
  | hdl-access = free
 }}</ref>
|-
|Sodium atom
|style=text-align:right |{{val|23|ul=Da}}
|
|style=text-align:right |{{val|20|ul=pm}}
|Moskowitz et al.<ref>{{Cite journal |last1=Moskowitz |first1=Philip E. |last2=Gould |first2=Phillip L. |last3=Atlas |first3=Susan R. |last4=Pritchard |first4=David E. |date=1983-08-01 |title=Diffraction of an Atomic Beam by Standing-Wave Radiation |url=https://link.aps.org/doi/10.1103/PhysRevLett.51.370 |journal=Physical Review Letters |volume=51 |issue=5 |pages=370–373 |doi=10.1103/PhysRevLett.51.370|bibcode=1983PhRvL..51..370M |url-access=subscription }}</ref>
|-
|Helium || style="text-align:right" | {{val|4|ul=Da}}|| style="text-align:right" |{{val|0.065|ul=eV}}|| style="text-align:right" | {{val|56|ul=pm}}||Grisenti et al.<ref>
 {{cite journal
  |author1=Grisenti, R. E. |author2=W. Schöllkopf |author3=J. P. Toennies |author4=J. R. Manson |author5=T. A. Savas |author6=Henry I. Smith
  |title=He-atom diffraction from nanostructure transmission gratings: The role of imperfections
  |journal=Physical Review A
  |volume=61
  |issue=3
  |year=2000
  |article-number=033608
  |doi=10.1103/PhysRevA.61.033608
  |bibcode=2000PhRvA..61c3608G
 }}</ref>
|-
|Na<sub>2</sub> || style="text-align:right" | {{val|23|ul=Da}} || style="text-align:right" |{{val|0.00017|ul=eV}} || style="text-align:right" | {{val|459|ul=pm}} ||Chapman et al.<ref>
 {{cite journal
  |author1=Chapman, Michael S. |author2=Christopher R. Ekstrom |author3=Troy D. Hammond
  |author4=Richard A. Rubenstein |author5=Jörg Schmiedmayer |author6=Stefan Wehinger| author7=David E. Pritchard
  |title=Optics and interferometry with Na<sub>2</sub> molecules
  |journal=Physical Review Letters
  |volume=74 |issue=24
  |year=1995
  |pages=4783–4786 |doi=10.1103/PhysRevLett.74.4783 |pmid=10058598 |bibcode=1995PhRvL..74.4783C
 }}</ref>
|-
|C<sub>60</sub> fullerene
|style=text-align:right |{{val|720|ul=Da}}
|style=text-align:right |{{val|0.2|ul=eV}}
|style=text-align:right |{{val|5|ul=pm}}
|Arndt et al.<ref name="Arndt 680–682"/>
|-
|C<sub>70</sub> fullerene || style="text-align:right" | {{val|841|ul=Da}} || style="text-align:right" |{{val|0.2|ul=eV}} || style="text-align:right" | {{val|2|ul=pm}} ||Brezger et al.<ref>
 {{cite journal
  |journal=Physical Review Letters
  |title=Matter–Wave Interferometer for Large Molecules
  |last1=Brezger |first1=B.
  |author2=Hackermüller, L.
  |author3=Uttenthaler, S. |author4=Petschinka, J. |author5=Arndt, M. |author6=Zeilinger, A.
  |volume=88 |issue=10 |article-number=100404
  |date=February 2002
  |doi=10.1103/PhysRevLett.88.100404
  |url=http://homepage.univie.ac.at/Lucia.Hackermueller/unsereArtikel/Brezger2002a.pdf
  |format=reprint
  |access-date=2007-04-30|pmid=11909334|bibcode=2002PhRvL..88j0404B
  |arxiv=quant-ph/0202158|s2cid=19793304
  |url-status=live
  |archive-url=https://web.archive.org/web/20070813092642/http://homepage.univie.ac.at/Lucia.Hackermueller/unsereArtikel/Brezger2002a.pdf|archive-date=2007-08-13
 }}</ref>
|-
|polypeptide, Gramicidin A
|style=text-align:right |{{val|1860|ul=Da}}
|
|style=text-align:right |{{val|360|ul=fm}}
|Shayeghi et al.<ref>{{Cite journal |last1=Shayeghi |first1=A. |last2=Rieser |first2=P. |last3=Richter |first3=G. |last4=Sezer |first4=U. |last5=Rodewald |first5=J. H. |last6=Geyer |first6=P. |last7=Martinez |first7=T. J. |last8=Arndt |first8=M. |date=2020-03-19 |title=Matter-wave interference of a native polypeptide |journal=Nature Communications |language=en |volume=11 |issue=1 |page=1447 |doi=10.1038/s41467-020-15280-2 |issn=2041-1723 |pmc=7081299 |pmid=32193414|arxiv=1910.14538 |bibcode=2020NatCo..11.1447S }}</ref>
|-
|functionalized oligoporphyrins
|style=text-align:right |{{val|25000|ul=Da}}
|style=text-align:right |{{val|17|ul=eV}}
|style=text-align:right |{{val|53|ul=fm}}
|Fein et al.<ref name=fein1242>{{Cite journal |last1=Fein |first1=Yaakov Y. |last2=Geyer |first2=Philipp |last3=Zwick |first3=Patrick |last4=Kiałka |first4=Filip |last5=Pedalino |first5=Sebastian |last6=Mayor |first6=Marcel |last7=Gerlich |first7=Stefan |last8=Arndt |first8=Markus |date=December 2019 |title=Quantum superposition of molecules beyond 25 kDa |url=https://www.nature.com/articles/s41567-019-0663-9 |journal=Nature Physics |language=en |volume=15 |issue=12 |pages=1242–1245 |doi=10.1038/s41567-019-0663-9 |bibcode=2019NatPh..15.1242F |s2cid=256703296 |issn=1745-2481|url-access=subscription }}</ref>
|}

=== Electrons ===
Electron diffraction patterns emerge when energetic electrons reflect or penetrate ordered solids; analysis of the patterns leads to models of the atomic arrangement in the solids.

They are used for imaging from the micron to atomic scale using electron microscopes, in transmission, using scanning, and for surfaces at low energies.

The measurements of the energy they lose in electron energy loss spectroscopy provides information about the chemistry and electronic structure of materials. Beams of electrons also lead to characteristic X-rays in energy dispersive spectroscopy which can produce information about chemical content at the nanoscale.

Quantum tunneling explains how electrons escape from metals in an electrostatic field at energies less than classical predictions allow: the matter wave penetrates of the work function barrier in the metal.

Scanning tunneling microscope leverages quantum tunneling to image the top atomic layer of solid surfaces.

Electron holography, the electron matter wave analog of optical holography, probes the electric and magnetic fields in thin films.

=== Neutrons ===
Neutron diffraction complements x-ray diffraction through the different scattering cross sections and sensitivity to magnetism.

Small-angle neutron scattering provides way to obtain structure of disordered systems that is sensitivity to light elements, isotopes and magnetic moments.

Neutron reflectometry is a neutron diffraction technique for measuring the structure of thin films.

=== Neutral atoms ===
Atom interferometers, similar to optical interferometers, measure the difference in phase between atomic matter waves along different paths.

Atom optics mimic many light optic devices, including mirrors, atom focusing zone plates.

Scanning helium microscopy uses He atom waves to image solid structures non-destructively.

[[Physics:Quantum reflection|Quantum reflection]] uses matter wave behavior to explain grazing angle atomic reflection, the basis of some atomic mirrors.

[[Physics:Quantum decoherence#Experimental observations|Quantum decoherence]] measurements rely on Rb atom wave interference.

=== Molecules ===
[[Physics:Quantum superposition#Experiments and applications|Quantum superposition]] revealed by interference of matter waves from large molecules probes the limits of wave–particle duality and quantum macroscopicity.<ref name=fein1242/><ref>{{Cite journal |last1=Nimmrichter |first1=Stefan |last2=Hornberger |first2=Klaus |date=2013-04-18 |title=Macroscopicity of Mechanical Quantum Superposition States |url=https://link.aps.org/doi/10.1103/PhysRevLett.110.160403 |journal=Physical Review Letters |volume=110 |issue=16 |article-number=160403 |doi=10.1103/PhysRevLett.110.160403|pmid=23679586 |arxiv=1205.3447 |bibcode=2013PhRvL.110p0403N |s2cid=12088376 }}</ref>

Matter-wave interfererometers generate nanostructures on molecular beams that can be read with nanometer accuracy and therefore be used for highly sensitive force measurements, from which one can deduce a plethora of properties of individualized complex molecules.<ref>{{Citation |last1=Gerlich |first1=Stefan |title=Otto Stern's Legacy in Quantum Optics: Matter Waves and Deflectometry |date=2021 |work=Molecular Beams in Physics and Chemistry: From Otto Stern's Pioneering Exploits to Present-Day Feats |pages=547–573 |editor-last=Friedrich |editor-first=Bretislav |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-030-63963-1_24 |isbn=978-3-030-63963-1 |last2=Fein |first2=Yaakov Y. |last3=Shayeghi |first3=Armin |last4=Köhler |first4=Valentin |last5=Mayor |first5=Marcel |last6=Arndt |first6=Markus |editor2-last=Schmidt-Böcking |editor2-first=Horst|doi-access=free |bibcode=2021mbpc.book..547G }}</ref>

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

== References ==
{{Reflist}}

== Further reading ==
* L. de Broglie, ''Recherches sur la théorie des quanta'' (Researches on the quantum theory), Thesis (Paris), 1924; L. de Broglie, ''Ann. Phys.'' (Paris) '''3''', 22 (1925). [https://fondationlouisdebroglie.org/LDB-oeuvres/De_Broglie_Kracklauer.pdf English translation by A.F. Kracklauer.]
* [https://www.nobelprize.org/uploads/2018/06/broglie-lecture.pdf Broglie, Louis de, ''The wave nature of the electron'' Nobel Lecture, 12, 1929]
* Tipler, Paul A. and Ralph A. Llewellyn (2003). ''Modern Physics''. 4th ed. New York; W. H. Freeman and Co. {{ISBN|0-7167-4345-0}}. pp.&nbsp;203–4, 222–3, 236.
* {{cite book |first=Steven S. |last=Zumdahl |title=Chemical Principles |edition=5th |date=2005 |location=Boston |publisher=Houghton Mifflin |isbn=978-0-618-37206-5 |url=https://archive.org/details/chemicalprincipl00zumd }}
* {{Cite journal |journal=Reviews of Modern Physics |volume=81 |doi=10.1103/RevModPhys.81.1051 |last1=Cronin |first1=Alexander D. |last2=Schmiedmayer |first2=Jörg |last3=Pritchard |first3=David E. |date=28 July 2009 |title=Optics and interferometry with atoms and molecules |issue=3 |pages=1051–1129 |arxiv=0712.3703 |bibcode=2009RvMP...81.1051C |hdl=1721.1/52372 |url=http://www.atomwave.org/rmparticle/RMPLAO.pdf |archive-url=https://web.archive.org/web/20110719220930/http://www.atomwave.org/rmparticle/RMPLAO.pdf |archive-date=2011-07-19 |via=Atomwave}}
* [https://arxiv.org/abs/1005.4534 "Scientific Papers Presented to Max Born on his retirement from the Tait Chair of Natural Philosophy in the University of Edinburgh"], 1953 (Oliver and Boyd)

== External links ==
* {{cite web |last=Bowley|first=Roger |title=de Broglie Waves |url=http://www.sixtysymbols.com/videos/debroglie.htm |work=Sixty Symbols |publisher=[[Biography:Brady Haran|Brady Haran]] for the University of Nottingham}}
{{DEFAULTSORT:Matter Wave}}

{{Sourceattribution|Matter wave}}