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Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. Empirically all matter in isolation exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave. The vast number of interactions in the solids, liquids, and gases we interact with daily suppress this quantum wave behavior (see decoherence).

The concept that matter behaves like a wave was proposed by French physicist Louis de Broglie (/dəˈbrɔɪ/) in 1924.^[1] It is also referred to as the de Broglie hypothesis.^[2] Matter waves are referred to as de Broglie waves.

The de Broglie wavelength, λ, relates to matter's momentum, p, through the Planck constant, h:

${\displaystyle \lambda ={\frac {h}{p))}$.

This wavelength relationship holds for matter and for massless light when we use relativistic momentum for p.

Wave-like behavior of matter was first experimentally demonstrated by George Paget Thomson and Alexander Reid's thin metal diffraction experiment,^[3] and independently in the Davisson–Germer experiment,^[4][5] both using electrons; and it has also been confirmed for other elementary particles, neutral atoms and even molecules.

Observations of diffraction provide direct evidence of wave properties. Before the de Broglie hypothesis in 1924,^[1] small quanta with mass where thought to be "particles" or "corpusles". In 1927 at Bell Labs, Clinton Davisson and Lester Germer fired slow-moving electrons at a crystalline nickel target.^[4][5] The angular dependence of the diffracted electron intensity was measured, and was determined to be similar 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 very thin metal foils to demonstrate the same effect.^[3] When the de Broglie wavelength was inserted into the Bragg condition, structure similar to the experimental diffraction patterns were observed, thereby experimentally supporting the de Broglie hypothesis for electrons.^[10] As also mentioned by Davisson and Germer,^[5] this also supported the quantum mechanical formulation by Erwin Schrödinger^[11]

This was a pivotal result in the development of quantum mechanics. Just as the photoelectric effect demonstrated the particle nature of light, these experiments showed the wave-nature of matter, and completed the theory of wave–particle duality. For physicists this idea was important because it meant that not only could any particle exhibit wave characteristics, but that one could use wave equations to describe phenomena in matter. This was confirmed quantitatively by Hans Bethe who published in 1928^[12] the first formal description of electron diffraction based upon Schrödinger equation,^[11] which is similar to how contemporary electron diffraction is described.^[13][14]

Electrons escape from metals in an electrostatic field at energies less than classical predictions allow. Quantum tunneling explains the result as matter wave penetration of the work function barrier at the surface of the metal.^{[citation needed]}

Numerous different kinds of experiments demonstrate or rely on wave properties of matter:

• Atom interferometer Similar to optical interferometers, atom interferometers measure the difference in phase between atomic matter waves along different paths. Many different types of interferometers have been developed.^{[citation needed]}

• Atomic mirror, a device which quantum reflects neutral atom matter waves in a way similar to the way a conventional mirror reflects visible light.^[15][16][17]

• Atom focusing with a Fresnel diffraction zone plate^[18]

• Atom wave microscope. A beam of He atoms focused by a Fresnel Zone plate scanned across a sample give ${\displaystyle ~2\mu }$ resolution.^[19] " Scanning helium microscopy is able to image such structures completely non-destructively by taking advantage of a neutral helium beam as a chemically, electrically and magnetically inert probe of the sample surface."^[20]

• Optically shaped matter waves^[21]

• Thermal manipulation of matter wave wavelength. 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.^[22]

Recent experiments even 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.^[23] The researchers calculated a De Broglie wavelength of the most probable C₆₀ velocity as 2.5 pm. More recent experiments prove the quantum nature of molecules made of 810 atoms and with a mass of 10,123 u.^[24] As of 2019, this has been pushed to molecules of 25,000 u.^[25]

Empirically, as we see above, we can model the center of mass motion of isolated, bound matter using waves. Collisions with atoms and molecules or the emission or absorption of light degrade the observations^[26], consistent with a transition from wave to particle models. The details of this transition process –-- quantum decoherence – remain unclear; this is an active area of research.^[27]

To avoid decoherence, practical applications of matter wave isolate beams of matter within high vacuum chambers. Design of apparatus for matter waves however can proceed based on direct analogy to the physical optics of light waves. Up until the wave approaches some part of the apparatus it propagates as a free wave; massive particles, charged or not, obey the same equation in free space – the Helmholtz equation – as light. Differences only appear if we consider experimental scenarios where propagation speed matters, for example, the coincident measurement of two particles^[28].

The de Broglie equations relate the wavelength λ to the momentum p, and frequency f to the total energy E of a free particle:^[29]

$${\displaystyle {\begin{aligned}&\lambda ={\frac {h}{p))\\&f={\frac {E}{h))\end{aligned))}$$

where h is the Planck constant. The equations can also be written as

$${\displaystyle {\begin{aligned}&\mathbf {p} =\hbar \mathbf {k} \\&E=\hbar \omega \\\end{aligned))}$$

or ^[30]

$${\displaystyle {\begin{aligned}&\mathbf {p} =\hbar {\boldsymbol {\beta ))\\&E=\hbar \omega \\\end{aligned))}$$

where ħ = h/2π is the reduced Planck constant, k is the wave vector, β is the phase constant, and ω is the angular frequency.

In each pair, the second equation is also referred to as the Planck–Einstein relation, since it was also proposed by Planck and Einstein.

Using two formulas from special relativity, one for the relativistic mass energy and one for the relativistic momentum^{[citation needed]}

${\displaystyle E=mc^{2}=\gamma m_{0}c^{2))$

${\displaystyle \mathbf {p} =m\mathbf {v} =\gamma m_{0}\mathbf {v} }$

allows the de Broglie equations to be written as

${\displaystyle {\begin{aligned}&\lambda =\,\,{\frac {h}{\gamma m_{0}v))\,=\,{\frac {h}{m_{0}v))\,\,\,\,{\sqrt {1-{\frac {v^{2)){c^{2))))}\\&f={\frac {\gamma \,m_{0}c^{2)){h))={\frac {m_{0}c^{2)){h{\sqrt {1-{\frac {v^{2)){c^{2))))))}\end{aligned))}$

where ${\displaystyle m_{0))$ denotes the particle's rest mass, ${\displaystyle v}$ its velocity, ${\displaystyle \gamma }$ the Lorentz factor, and ${\displaystyle c}$ the speed of light in vacuum.^[31][32][33] See below for details of the derivation of the de Broglie relations.

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, should always equal the group velocity of the corresponding wave and not the phase velocity. The magnitude of the group velocity is equal to the particle's speed; the phase velocity equals the product of the particle's frequency and its wavelength.

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.^[23]

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

${\displaystyle v_{g}={\frac {\partial \omega }{\partial k))={\frac {\partial (E/\hbar )}{\partial (p/\hbar )))={\frac {\partial E}{\partial p))}$

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

${\displaystyle {\begin{aligned}v_{g}&={\frac {\partial E}{\partial p))={\frac {\partial }{\partial p))\left({\frac {1}{2)){\frac {p^{2)){m))\right)\\&={\frac {p}{m))\\&=v\end{aligned))}$

where m is the mass of the particle and v its velocity.^{[clarification needed]}

Also in special relativity we find that

${\displaystyle {\begin{aligned}v_{g}&={\frac {\partial E}{\partial p))={\frac {\partial }{\partial p))\left({\sqrt {p^{2}c^{2}+m_{0}^{2}c^{4))}\right)\\&={\frac {pc^{2)){\sqrt {p^{2}c^{2}+m_{0}^{2}c^{4))))\\&={\frac {pc^{2)){E))\end{aligned))}$

where m₀ is the rest mass of the particle and c is the speed of light in vacuum. But (see below), using that the phase velocity is v_p = E/p = c²/v, therefore

${\displaystyle {\begin{aligned}v_{g}&={\frac {pc^{2)){E))\\&={\frac {c^{2)){v_{p))}\\&=v\end{aligned))}$

where v is the velocity of the particle regardless of wave behavior.

In quantum mechanics, particles also behave as waves with complex phases. The phase velocity is equal to the product of the frequency multiplied by the wavelength. In the case of a non-dispersive medium, the phase velocity equals the group velocity; otherwise they are not equal.

By the de Broglie hypothesis, we see that

${\displaystyle v_{\mathrm {p} }={\frac {\omega }{k))={\frac {E/\hbar }{p/\hbar ))={\frac {E}{p)).}$

Using relativistic relations for energy and momentum, we have

${\displaystyle v_{\mathrm {p} }={\frac {E}{p))={\frac {mc^{2)){mv))={\frac {\gamma m_{0}c^{2)){\gamma m_{0}v))={\frac {c^{2)){v))={\frac {c}{\beta ))}$

where E is the total energy of the particle (i.e., rest energy plus kinetic energy in the kinematic sense), p the momentum, ${\displaystyle \gamma }$ 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 ${\displaystyle v<c}$ for any particle that has mass (according to special relativity), the phase velocity of matter waves always exceeds c, i.e.,

${\displaystyle v_{\mathrm {p} }>c,\,}$

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, because phase propagation carries no energy. See the article on Dispersion (optics) for details.

Using four-vectors, the De Broglie relations form a single equation: $${\displaystyle \mathbf {P} =\hbar \mathbf {K} }$$ which is frame-independent.

Likewise, the relation between group/particle velocity and phase velocity is given in frame-independent form by: $${\displaystyle \mathbf {K} =\left({\frac {\omega _{o)){c^{2))}\right)\mathbf {U} }$$ where

• Four-momentum ${\displaystyle \mathbf {P} =\left({\frac {E}{c)),{\vec {\mathbf {p} ))\right)}$
• Four-wavevector ${\displaystyle \mathbf {K} =\left({\frac {\omega }{c)),{\vec {\mathbf {k} ))\right)=\left({\frac {\omega }{c)),{\frac {\omega }{v_{p))}\mathbf {\hat {n)) \right)}$
• Four-velocity ${\displaystyle \mathbf {U} =\gamma (c,{\vec {\mathbf {u} )))=\gamma (c,v_{g}{\hat {\mathbf {n} )))}$

The concept that matter behaves like a wave was initially proposed by French physicist Louis de Broglie in his 1924 PhD thesis.^[1] The De Broglie hypothesis had widespread impact on early quantum theory.

Erwin Schrödinger picking up de Broglie's hypothesis,^[11] applied Hamilton's optico-mechanical analogy to derive his eponymous wave equation. That is, he reasoned that "the failure of classical physics to account for quantum phenomena is analogous to the failure of geometrical optics to account for interference and diffraction: and he proposed to create in connection with de Broglie waves a theory analogous to Physical Optics".^[34] The many optical applications of matter waves bears out this historical connection.

Matter waves were first experimentally confirmed to occur in George Paget Thomson and Alexander Reid's electron diffraction experiment^[3] and the Davisson-Germer experiment for electrons.^[4][5] The de Broglie hypothesis has been confirmed for other elementary particles. Furthermore, neutral atoms and even molecules have been shown to be wave-like.

• Bohr model
• Compton wavelength
• Faraday wave
• Kapitsa–Dirac effect
• Matter wave clock
• Schrödinger equation
• Theoretical and experimental justification for the Schrödinger equation
• Thermal de Broglie wavelength

• 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). English translation by A.F. Kracklauer.
• 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. 203–4, 222–3, 236.
• Zumdahl, Steven S. (2005). Chemical Principles (5th ed.). Boston: Houghton Mifflin. ISBN 978-0-618-37206-5.
• An extensive review article "Optics and interferometry with atoms and molecules" appeared in July 2009: https://web.archive.org/web/20110719220930/http://www.atomwave.org/rmparticle/RMPLAO.pdf.
• "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)

• Bowley, Roger. "de Broglie Waves". Sixty Symbols. Brady Haran for the University of Nottingham.