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| {{hatnote|The de Broglie relations redirect here.}}
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| {{About|the quantum mechanical concept [[wave-particle duality]]|the ordinary type of wave propagating through material media|Mechanical wave}}
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| {{quantum mechanics}}
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| In [[quantum mechanics]], the concept of '''matter waves''' or '''de Broglie waves''' {{IPAc-en|d|ə|ˈ|b|r|ɔɪ}} reflects the [[wave–particle duality]] of [[matter]]. The theory was proposed by [[Louis de Broglie]] in 1924 in his PhD thesis.<ref>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).</ref> The de Broglie relations show that the [[wavelength]] is [[inversely proportional]] to the [[momentum]] of a particle and is also called de Broglie wavelength. Also the [[frequency]] of matter waves, as deduced by de Broglie, is directly proportional to the [[total energy]] ''E'' (sum of its [[rest energy]] and the [[kinetic energy]]) of a particle.<ref name="Resnick 1985">{{cite book |title=Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles |edition=2nd |first=R. |last=Resnick |first2=R. |last2=Eisberg |publisher=John Wiley & Sons |year=1985 |location=New York |isbn=0-471-87373-X }}</ref>
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| ==Historical context==
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| [[File:Propagation of a de broglie wave.svg|290px|"290px"|right|thumb|Propagation of '''de Broglie waves''' in 1d – real part of the [[complex number|complex]] amplitude is blue, imaginary part is green. The probability (shown as the colour [[opacity (optics)|opacity]]) of finding the particle at a given point ''x'' is spread out like a waveform, there is no definite position of the particle. As the amplitude increases above zero the [[curvature]] decreases, so the amplitude decreases again, and vice versa – the result is an alternating amplitude: a wave. Top: [[plane wave]]. Bottom: [[wave packet]].]]
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| 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#Brief history of wave and particle viewpoints|history of wave and particle viewpoints]]). This division was challenged when, in his 1905 paper on the [[photoelectric effect]], [[Albert Einstein]] postulated that light was emitted and absorbed as localized packets, or "quanta" (now called [[photons]]). These quanta would have an energy
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| :<math>E=h\nu</math>
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| where <math>\scriptstyle \nu</math> is the frequency of the light and ''h'' is [[Planck’s constant]]. In the modern convention, frequency is symbolized by ''f'' as is done in the rest of this article. Einstein’s postulate was confirmed experimentally by [[Robert Millikan]] and [[Arthur Compton]] over the next two decades.
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| Thus it became apparent that light has both wave-like and particle-like properties. De Broglie, in his 1924 PhD thesis, sought to expand this wave-particle duality to all particles:
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| {{cquote|When I conceived the first basic ideas of wave mechanics in 1923–24, 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.|20px|20px|De Broglie<ref>Louis de Broglie [http://www.springerlink.com/content/n170347gr6h82147/ "The Reinterpretation of Wave Mechanics" Foundations of Physics, Vol. 1 No. 1 (1970)]</ref>}}
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| In 1926, [[Erwin Schrödinger]] published an equation describing how this matter wave should evolve—the matter wave equivalent of Maxwell’s equations—and used it to derive the energy spectrum of hydrogen. That same year [[Max Born]] published his now-standard interpretation that the square of the amplitude of the matter wave gives the probability to find the particle at a given place. This interpretation was in contrast to De Broglie’s own interpretation, in which the wave corresponds to the physical motion of a localized particle.
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| ==de Broglie relations<!--This section is linked to by [[de Broglie relations]]-->==
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| ===Quantum mechanics===
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| The [[de Broglie]] equations relate the [[wavelength]] ''λ'' to the [[momentum]] ''p'', and [[frequency]] ''f'' to the [[total energy]] ''E'' of a particle:<ref name="Resnick 1985"/>
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| {{Equation box 1
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| |indent=:
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| |equation = <math>\begin{align}
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| & \lambda = h/p\\
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| & f = E/h
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| \end{align}</math>
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| |cellpadding
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| |border
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| |border colour = #50C878
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| |background colour = #ECFCF4}}
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| where ''h'' is [[Planck's constant]]. The equation can be equivalently written as
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| {{Equation box 1
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| |indent=:
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| |equation=<math>\begin{align}
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| & p = \hbar k\\
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| & E = \hbar \omega\\
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| \end{align}</math>
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| |cellpadding
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| |border
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| |border colour = #0073CF
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| |background colour=#F5FFFA}}
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| using the definitions
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| * <math>\hbar=h/2\pi </math> is the reduced [[Planck's constant]] (also known as '''Dirac's constant''', pronounced "h-bar"),
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| * <math>k= 2\pi/\lambda</math> is the [[wavenumber#In wave equations|angular wavenumber]],
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| * <math>\omega=2\pi f</math> is the [[angular frequency]].
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| In each pair, the second is also referred to as the [[Planck constant|Planck-Einstein relation]], since it was also proposed by [[Max Planck|Planck]] and [[Albert Einstein|Einstein]].
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| ===Special relativity===
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| Using the [[relativistic momentum]] formula from [[special relativity]]
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| :<math>p = \gamma m_0v </math>
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| allows the equations to be written as<ref>{{cite book |title=Stationary states |first=Alan |last=Holden |publisher=Oxford University Press |year=1971 |location=New York |isbn=0-19-501497-9 }}</ref>
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| :<math>\begin{align}&\lambda = \frac {h}{\gamma m_0v} = \frac {h}{m_0v} \sqrt{1 - \frac{v^2}{c^2}}\\
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| & f = \frac{\gamma\,m_0c^2}{h} = \frac {m_0c^2}{h\sqrt{1 - \frac{v^2}{c^2}}}
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| \end{align}</math>
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| where ''m''<sub>0</sub> is the particle's [[rest mass]], ''v'' is the particle's [[velocity]], γ is the [[Lorentz factor]], and ''c'' is the [[speed of light]] in a vacuum. See below for details of the derivation of the de Broglie relations. Group velocity (equal to the particle's speed) should not be confused with [[phase velocity]] (equal to the product of the particle's frequency and its wavelength). In the case of a non-[[dispersion relation|dispersive medium]], they happen to be equal, but otherwise they are not.
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| {{cleanup merge|Group velocity|Phase velocity}}
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| ====Group velocity====
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| [[Albert Einstein]] first explained the [[wave–particle duality]] of light in 1905. [[Louis de Broglie]] hypothesized that any particle should also exhibit such a duality. The velocity of a particle, he concluded then (but may be questioned today, see above), should always equal the group velocity of the corresponding wave. De Broglie deduced that if the duality equations already known for light were the same for any particle, then his hypothesis would hold. This means that
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| :<math>v_g = \frac{\partial \omega}{\partial k} = \frac{\partial (E/\hbar)}{\partial (p/\hbar)} = \frac{\partial E}{\partial p}</math>
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| where ''E'' is the [[total energy]] of the particle, ''p'' is its [[momentum]], ''ħ'' is the [[reduced Planck constant]]. For a free non-relativistic particle it follows that
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| :<math>\begin{align}
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| v_g &= \frac{\partial E}{\partial p} = \frac{\partial}{\partial p} \left( \frac{1}{2}\frac{p^2}{m} \right),\\
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| &= \frac{p}{m},\\
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| &= v.
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| \end{align}</math>
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| where ''<math>m</math>'' is the [[mass]] of the particle and ''v'' its velocity.
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| Also in [[special relativity]] we find that
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| :<math>\begin{align}
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| v_g &= \frac{\partial E}{\partial p} = \frac{\partial}{\partial p} \left( \sqrt{p^2c^2+m^2c^4} \right),\\
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| &= \frac{pc^2}{\sqrt{p^2c^2 + m^2c^4}},\\
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| &= \frac{p}{m\sqrt{\left(\frac{p}{mc}\right)^2+1}},\\
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| &= \frac{p}{m\gamma},\\
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| &= \frac{mv\gamma}{m\gamma},\\
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| &= v.
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| \end{align}</math>
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| where ''m'' is the [[mass]] of the particle, ''c'' is the [[speed of light]] in a vacuum, <math>\gamma</math> is the [[Lorentz factor]], and ''v'' is the velocity of the particle regardless of wave behavior.
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| Group velocity (equal to an electron's speed) should not be confused with [[phase velocity]] (equal to the product of the electron's frequency multiplied by its wavelength).
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| Both in relativistic and non-relativistic quantum physics, we can identify the group velocity of a particle's wave function with the particle velocity. [[Quantum mechanics]] has very accurately demonstrated this hypothesis, and the relation has been shown explicitly for particles as large as [[molecules]].{{Citation needed|date=April 2008}}
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| ====Phase velocity====
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| In [[quantum mechanics]], particles also behave as waves with [[complex number|complex]] phases. By the de Broglie hypothesis, we see that
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| :<math>v_\mathrm{p} = \frac{\omega}{k} = \frac{E/\hbar}{p/\hbar} = \frac{E}{p}. </math> | |
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| Using [[special relativity|relativistic]] relations for energy and momentum, we have
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| :<math>v_\mathrm{p} = \frac{E}{p} = \frac{\gamma m c^2}{\gamma m v} = \frac{c^2}{v} = \frac{c}{\beta}</math>
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| where ''E'' is the [[total energy]] of the particle (i.e. [[rest energy]] plus [[kinetic energy]] in [[kinematic]] sense), ''p'' the [[momentum]], <math>\gamma</math> the [[Lorentz factor]], ''c'' the [[speed of light]], and β the speed as a fraction of ''c''. The variable ''v'' can either be taken to be the speed of the particle or the group velocity of the corresponding matter wave. Since the particle speed <math>v < c </math> for any particle that has mass (according to [[special relativity]]), the phase velocity of matter waves always exceeds ''c'', i.e.
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| :<math>v_\mathrm{p} > c, \,</math>
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| and as we can see, it approaches ''c'' when the particle speed is in the relativistic range. The [[superluminal]] phase velocity does not violate special relativity, as it carries no information. See the article on ''[[signal velocity]]'' for details.
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| ===Four-vectors===
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| {{Main|Four-vector}}
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| Using the [[four-momentum]] '''P''' = (''E/c'', '''p''') and the [[Four-vector#Four-wavevector|four-wavevector]] '''K''' = (''ω/c'', '''k'''), the De Broglie relations form a single equation:
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| {{Equation box 1
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| |indent=:
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| |equation = <math>\mathbf{P}= \hbar\mathbf{K}</math>
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| |cellpadding
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| |border
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| |border colour = #50C878
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| |background colour = #ECFCF4}}
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| which is [[Inertial frame of reference|frame]]-independent.
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| ==Experimental confirmation==
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| Matter waves were first experimentally confirmed to occur in the Davisson-Germer experiment for electrons, and the de Broglie hypothesis has been confirmed for other [[elementary particles]]. Furthermore, neutral atoms and even molecules have been shown to be wave-like.
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| ===Electrons===
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| {{further|Davisson–Germer experiment}}
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| In 1927 at Bell Labs, [[Clinton Davisson]] and [[Lester Germer]] [[Davisson–Germer experiment|fired]] slow-moving [[electrons]] at a [[crystalline]] [[nickel]] target. The angular dependence of the reflected electron intensity was measured, and was determined to have the same [[diffraction|diffraction pattern]] as those predicted by [[William Lawrence Bragg|Bragg]] for [[X rays|x-rays]]. Before the acceptance of the de Broglie hypothesis, diffraction was a property that was thought to be only exhibited by waves. Therefore, the presence of any [[diffraction]] effects by matter demonstrated the wave-like nature of matter. When the de Broglie wavelength was inserted into the [[Bragg's law|Bragg condition]], the observed diffraction pattern was predicted, thereby experimentally confirming the de Broglie hypothesis for electrons.<ref>Mauro Dardo, ''Nobel Laureates and Twentieth-Century Physics'', Cambridge University Press 2004, pp. 156–157</ref>
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| This was a pivotal result in the development of [[quantum mechanics]]. Just as the [[photoelectric effect]] demonstrated the particle nature of light, the [[Davisson–Germer experiment]] showed the wave-nature of matter, and completed the theory of [[wave-particle duality]]. For [[physicists]] this idea was important because it means that not only can any particle exhibit wave characteristics, but that one can use [[wave equation]]s to describe phenomena in matter if one uses the de Broglie wavelength.
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| ===Neutral atoms===
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| Experiments with [[Fresnel diffraction]]<ref name="doak">
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| {{cite journal
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| | author=R.B.Doak
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| | year=1999
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| | coauthors=R.E.Grisenti, S.Rehbein, G.Schmahl, J.P.Toennies2, and Ch. Wöll
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| | title=Towards Realization of an Atomic de Broglie Microscope: Helium Atom Focusing Using Fresnel Zone Plates
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| | journal=[[Physical Review Letters]]
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| | volume=83 | pages=4229–4232
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| | doi=10.1103/PhysRevLett.83.4229
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| | bibcode=1999PhRvL..83.4229D
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| | issue=21
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| }}</ref> and [[specular reflection]]<ref name="sh">
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| {{cite journal
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| | author= F. Shimizu
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| | year=2000
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| | title=Specular Reflection of Very Slow Metastable Neon Atoms from a Solid Surface
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| | journal=[[Physical Review Letters]]
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| | volume=86| pages=987–990
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| | doi=10.1103/PhysRevLett.86.987
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| | bibcode=2001PhRvL..86..987S
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| | pmid=11177991
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| | issue= 6
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| }}</ref><ref name="zeno">
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| {{cite journal
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| | author= D. Kouznetsov
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| | coauthors= H. Oberst
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| | year=2005
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| | title=Reflection of Waves from a Ridged Surface and the Zeno Effect
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| | journal=[[Optical Review]]
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| | volume=12 | pages=1605–1623
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| | doi=10.1007/s10043-005-0363-9
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| | bibcode=2005OptRv..12..363K
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| | issue= 5
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| }}</ref> of neutral atoms confirm the application of the de Broglie hypothesis to atoms, i.e. the existence of atomic waves which undergo
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| [[diffraction]], [[Interference (wave propagation)|interference]] and allow [[quantum reflection]] by the tails of the attractive potential.<ref name="Fri">
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| {{cite journal
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| |author=H.Friedrich
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| |coauthors=G.Jacoby, C.G.Meister
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| |year=2002
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| |journal=[[Physical Review A]]
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| |title=quantum reflection by Casimir–van der Waals potential tails
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| |volume=65 |pages=032902
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| |doi=10.1103/PhysRevA.65.032902
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| |bibcode = 2002PhRvA..65c2902F
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| |issue=3 }}</ref> Advances in [[laser cooling]] have allowed cooling of neutral atoms down to nanokelvin temperatures. At these temperatures, the thermal 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">
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| {{cite arXiv
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| |author=Pierre Cladé
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| |coauthors=Changhyun Ryu, Anand Ramanathan, Kristian Helmerson, William D. Phillips
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| |title=Observation of a 2D Bose Gas: From thermal to quasi-condensate to superfluid
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| |year=2008
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| |arxiv=0805.3519
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| }}</ref>
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| This effect has been used to demonstrate atomic [[holography]], and it may allow the construction of an atom probe imaging system with nanometer resolution.<ref name="holo">{{cite journal |title=Reflection-Type Hologram for Atoms |author=Shimizu |coauthors=J.Fujita |journal=[[Physical Review Letters]] |volume=88 |issue=12 |pages=123201 |year=2002 |doi=10.1103/PhysRevLett.88.123201 |pmid=11909457 |bibcode=2002PhRvL..88l3201S}}</ref><ref name="nanoscope">{{cite journal |author=D. Kouznetsov |coauthors=H. Oberst, K. Shimizu, A. Neumann, Y. Kuznetsova, J.-F. Bisson, K. Ueda, S. R. J. Brueck |title=Ridged atomic mirrors and atomic nanoscope |journal=[[Journal of Physics B]] |volume=39 |pages=1605–1623 |year=2006 |doi=10.1088/0953-4075/39/7/005|bibcode = 2006JPhB...39.1605K |issue=7 }}</ref> The description of these phenomena is based on the wave properties of neutral atoms, confirming the de Broglie hypothesis.
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| ===Molecules===
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| Recent experiments even confirm the relations for molecules and even [[macromolecules]], which are normally considered too large to undergo quantum mechanical effects. In 1999, a research team in [[Vienna]] demonstrated diffraction for molecules as large as [[fullerenes]].<ref>{{cite journal| title=Wave-particle duality of C60| first=M.| last=Arndt| coauthors=O. Nairz, [[Julian Voss-Andreae|J. Voss-Andreae]], C. Keller, G. van der Zouw, [[Anton Zeilinger|A. Zeilinger]]| journal=Nature| volume=401| issue=6754| pages=680–682| month=14 October| pmid=18494170| year=1999| doi=10.1038/44348| bibcode=1999Natur.401..680A}}</ref> The researchers calculated a De Broglie wavelength of the most probable C<sub>60</sub> velocity as 2.5 [[Pico-|pm]].
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| More recent experiments prove the quantum nature of molecules with a mass up to 6910 [[Atomic mass unit|amu]].<ref>{{cite journal| title=Quantum interference of large organic molecules| first=S.| last=Gerlich| coauthors=S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüxen, M. Mayor & [[Markus Arndt|M. Arndt]]| journal=Nature Communications| volume=2| issue=263|date=5 April 2011| doi=10.1038/ncomms1263|bibcode = 2011NatCo...2E.263G| pmid=21468015| pmc=3104521| pages=263–}}</ref>
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| In general, the De Broglie hypothesis is expected to apply to any well isolated object.
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| ==Spatial Zeno effect==
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| {{confusing section|date=December 2013}}
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| The matter wave leads to the spatial version of the [[quantum Zeno effect]]. If an object (particle) is observed with frequency Ω >> ω in a half-space (say, ''y'' < 0), then this observation prevents the particle, which stays in the half-space ''y'' > 0 from entry into this half-space ''y'' < 0. Such an "observation" can be realized with a set of rapidly moving absorbing ridges, filling one half-space. In the system of coordinates related to the ridges, this phenomenon appears as a [[specular reflection]] of a particle from a [[ridged mirror]], assuming the grazing incidence (small values of the [[grazing angle]]). Such a ridged mirror is universal; while we consider the idealised "absorption" of the de Broglie wave at the ridges, the reflectivity is determined by wavenumber ''k'' and does not depend on other properties of a particle.<ref name="zeno"/>
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| ==See also==
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| * [[Atom optics]]
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| * [[Atomic de Broglie microscope]]
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| * [[Atomic mirror]]
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| * [[Bohr model]]
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| * [[Electron diffraction]]
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| * [[Faraday wave]]
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| * [[Schrödinger equation]]
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| * [[Theoretical and experimental justification for the Schrödinger equation]]
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| * [[Thermal de Broglie wavelength]]
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| * [http://www.nobelprize.org/nobel_prizes/physics/laureates/1929/broglie-lecture.pdf Broglie, Louis de, ''The wave nature of the electron'' Nobel Lecture, 12, 1929]
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| * 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. 203–4, 222–3, 236.
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| * [http://ebookbrowse.com/de-broglie-kracklauer-pdf-d90199080 Web version of Thesis, translated by Kracklauer (English)]
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| * {{cite book |first=Steven S. |last=Zumdahl |title=Chemical Principles |edition=5th |year=2005 |location=Boston |publisher=Houghton Mifflin |isbn=0-618-37206-7 }}
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| * An extensive review article "Optics and interferometry with atoms and molecules" appeared in July 2009: http://www.atomwave.org/rmparticle/RMPLAO.pdf.
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| * [http://arxiv.org/abs/1005.4534 This paper appeared in a collection of papers titled "Scientific Papers Presented to Max Born on his retirement from the Tait Chair of Natural Philosophy in the University of Edinburgh", published in 1953 (Oliver and Boyd):]
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| ==External links==
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| * {{cite web|last=Bowley|first=Roger|title=de Broglie Waves|url=http://www.sixtysymbols.com/videos/debroglie.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]}}
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| {{Use dmy dates|date=September 2010}}
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| {{DEFAULTSORT:Matter Wave}}
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| [[Category:Waves]]
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| [[Category:Matter]]
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| [[Category:Foundational quantum physics]]
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