# Matter wave

(Redirected from Matter waves)
The de Broglie relations redirect here.

In quantum mechanics, the concept of matter waves or de Broglie waves reflects the wave–particle duality of matter. The theory was proposed by Louis de Broglie in 1924 in his PhD thesis.[1] The de Broglie relations show that the wavelength is inversely proportional to the momentum of a particle and it is therefore called the 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.[2]

## Historical context

Propagation of de Broglie waves in 1d – real part of the complex amplitude is blue, imaginary part is green. The probability (shown as the colour 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.

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 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" (light quanta are now called photons). These quanta would have an energy

$E=h\nu$

where $\scriptstyle \nu$ (lowercase Greek letter nu) 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.

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:

In 1926, Erwin Schrödinger published an equation describing how a matter wave should evolve—the matter wave analogue of Maxwell’s equations—and used it to derive the energy spectrum of hydrogen. Schrödinger's equation describes a wave that moves with respect to an abstract mathematical "space", called phase space, or, more precisely the configuration space, not the same thing as ordinary physical space.[4] That same year Max Born published his finding that the square of the amplitude of such a quantum wave gives the probability of detecting the corresponding particle at any given point in ordinary physical space. He wrote "If one translates this result into terms of particles, only one interpretation is possible. $\Phi_{n,m}(\alpha,\beta,\gamma)$ gives the probability [Footnote addition in proof: More careful consideration shows that the probability is proportional to the square of the quantity $\Phi_{n,m}$.] for the electron, arriving from the z-direction, to be thrown out into the direction designated by the angles $\alpha,\,\beta,\,\gamma$."[5] In 1955, one of the main contributors to the 'Copenhagen interpretation', Werner Heisenberg, wrote a considered opinion: "An important step forward was made by the work of Born [Z. Phys., 37: 863, 1926 and 38: 803, 1926] in the summer of 1926. In this work, the wave in configuration space was interpreted as a probability wave, in order to explain collision processes on Schrödinger's theory. This hypothesis contained two important new features in comparison with that of Bohr, Kramers and Slater."[6] Because of this domain of the wave function in configuration space, discussing the meaning of the quantum mechanical formalism, Niels Bohr, the other main contributor to the Copenhagen interpretation, in 1927 wrote "This entails, however, that in the interpretation of observations a fundamental renunciation regarding the space-time description is unavoidable."[7]

## Experimental confirmation

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.

### Electrons

In 1927 at Bell Labs, Clinton Davisson and Lester Germer 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 pattern as those predicted by Bragg for 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 condition, the observed diffraction pattern was predicted, thereby experimentally confirming the de Broglie hypothesis for electrons.[8]

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 equations to describe phenomena in matter if one uses the de Broglie wavelength.

### Neutral atoms

Further information: Atom optics

Experiments with Fresnel diffraction[9] and an atomic mirror for specular reflection[10][11] of neutral atoms confirm the application of the de Broglie hypothesis to atoms, i.e. the existence of atomic waves which undergo diffraction, interference and allow quantum reflection by the tails of the attractive potential.[12] 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.[13]

This effect has been used to demonstrate atomic holography, and it may allow the construction of an atom probe imaging system with nanometer resolution.[14][15] The description of these phenomena is based on the wave properties of neutral atoms, confirming the de Broglie hypothesis.

The effect has also been used to explain the spatial version of the quantum Zeno effect, in which an otherwise unstable object may be stabilised by rapidly-repeated observations.[11]

### Molecules

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.[16] 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 with a mass up to 6910 amu.[17]

## de Broglie relations

The de Broglie equations relate the wavelength λ to the momentum p, and frequency f to the total energy E of a particle:[2]

 \begin{align} & \lambda = h/p\\ & f = E/h \end{align}

where h is Planck's constant. The equation can also be written as

 \begin{align} & \mathbf p = \hbar \mathbf k\\ & E = \hbar \omega\\ \end{align}

where $\hbar$ is the reduced Planck's constant, $\mathbf k$ is the wave vector, and $\omega$ is the angular frequency.

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

### Special relativity

Using two formulas from special relativity, one for the relativistic momentum and one for the energy

$p = \gamma m_0v$
$E = m c^2 = \gamma m_0 c^2$

allows the equations to be written as[1][18][19][20]

\begin{align}&\lambda =\,\, \frac {h}{\gamma m_0v}\, =\, \frac {h}{m_0v}\,\,\,\, \sqrt{1 - \frac{v^2}{c^2}}\\ & f = \frac{\gamma\,m_0c^2}{h} = \frac {m_0c^2}{h} \bigg/ \sqrt{1 - \frac{v^2}{c^2}} \end{align}

where $m_0$ denotes the particle's rest mass, $v$ its velocity, $\gamma$ the Lorentz factor, and $c$ 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-dispersive medium, they happen to be equal, but otherwise they are not.

#### Group velocity

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

$v_g = \frac{\partial \omega}{\partial k} = \frac{\partial (E/\hbar)}{\partial (p/\hbar)} = \frac{\partial E}{\partial p}$

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

\begin{align} 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{align}

where m is the mass of the particle and v its velocity.

Also in special relativity we find that

\begin{align} v_g &= \frac{\partial E}{\partial p} = \frac{\partial}{\partial p} \left( \sqrt{p^2c^2+m_0^2c^4} \right)\\ &= \frac{pc^2}{\sqrt{p^2c^2 + m_0^2c^4}}\\ &= \frac{pc^2}{E} \end{align}

where $m_0$ is the rest mass of the particle and $c$ is the speed of light in a vacuum. But (see below), using that the phase velocity is $v_p=E/p=c^2/v$, therefore

\begin{align} v_g &= \frac{pc^2}{E}\\ &= \frac{c^2}{v_p}\\ &= v \end{align}

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

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).

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]

#### Phase velocity

In quantum mechanics, particles also behave as waves with complex phases. By the de Broglie hypothesis, we see that

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

Using relativistic relations for energy and momentum, we have

$v_\mathrm{p} = \frac{E}{p} = \frac{\gamma m_0 c^2}{\gamma m_0 v} = \frac{c^2}{v} = \frac{c}{\beta}$

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

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

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.

### Four-vectors

Main article: Four-vector

Using the four-momentum P = (E/c, p) and the four-wavevector K = (ω/c, k), the De Broglie relations form a single equation:

 $\mathbf{P}= \hbar\mathbf{K}$

which is frame-independent.

## Interpretations

The physical reality underlying de Broglie waves is a subject of ongoing debate. Some theories, such as the hidden variable theory, treat the wave as a distinct entity from the massive particle, while others treat the particle aspect as a manifestation of its fundamental wave nature and yet others propose some intermediate entity that is neither quite wave nor quite particle but only appears as such when we measure one or the other property. The recent development of technologies such as the matter wave clock has brought the debate back into focus.[citation needed]

## References

1. ^ a b 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.
2. ^ a b Resnick, R.; Eisberg, R. (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd ed.). New York: John Wiley & Sons. ISBN 0-471-87373-X.
3. ^
4. ^ Pais, A. (1982). Max Born's statistical interpretation of quantum mechanics, Science 218: 1193–1198, p. 1197.
5. ^ Born, M. (1926). On the quantum mechanics of collisions, Z. Phys. 37: 863–867, translated by J.A. Wheeler and W.H. Zurek, pp. 52–55 in Quantum Theory and Measurement, edited by those translators, Princeton University Press, Princeton NJ, 1983, p. 54.
6. ^ Heisenberg, W. (1955). The development of the interpretation of the quantum theory, pp. 12–29, in Niels Bohr and the Development of Physics: Essays dedicated to Niels Bohr on the occasion of his seventieth birthday, edited by W. Pauli, with the assistance of L. Rosenfeld and V. Weisskopf, Pergamon Press, London, p. 13.
7. ^ Bohr, N. (1927/1928). The quantum postulate and the recent development of atomic theory, Nature Supplement April 14 1928, 121: 580–590, p. 587.
8. ^ Mauro Dardo, Nobel Laureates and Twentieth-Century Physics, Cambridge University Press 2004, pp. 156–157
9. ^ R.B.Doak; R.E.Grisenti, S.Rehbein, G.Schmahl, J.P.Toennies2, and Ch. Wöll (1999). "Towards Realization of an Atomic de Broglie Microscope: Helium Atom Focusing Using Fresnel Zone Plates". Physical Review Letters 83 (21): 4229–4232. Bibcode:1999PhRvL..83.4229D. doi:10.1103/PhysRevLett.83.4229.
10. ^ F. Shimizu (2000). "Specular Reflection of Very Slow Metastable Neon Atoms from a Solid Surface". Physical Review Letters 86 (6): 987–990. Bibcode:2001PhRvL..86..987S. doi:10.1103/PhysRevLett.86.987. PMID 11177991.
11. ^ a b D. Kouznetsov; H. Oberst (2005). "Reflection of Waves from a Ridged Surface and the Zeno Effect". Optical Review 12 (5): 1605–1623. Bibcode:2005OptRv..12..363K. doi:10.1007/s10043-005-0363-9.
12. ^ H.Friedrich; G.Jacoby, C.G.Meister (2002). "quantum reflection by Casimir–van der Waals potential tails". Physical Review A 65 (3): 032902. Bibcode:2002PhRvA..65c2902F. doi:10.1103/PhysRevA.65.032902.
13. ^ Pierre Cladé; Changhyun Ryu, Anand Ramanathan, Kristian Helmerson, William D. Phillips (2008). "Observation of a 2D Bose Gas: From thermal to quasi-condensate to superfluid". arXiv:0805.3519.
14. ^ Shimizu; J.Fujita (2002). "Reflection-Type Hologram for Atoms". Physical Review Letters 88 (12): 123201. Bibcode:2002PhRvL..88l3201S. doi:10.1103/PhysRevLett.88.123201. PMID 11909457.
15. ^ D. Kouznetsov; H. Oberst, K. Shimizu, A. Neumann, Y. Kuznetsova, J.-F. Bisson, K. Ueda, S. R. J. Brueck (2006). "Ridged atomic mirrors and atomic nanoscope". Journal of Physics B 39 (7): 1605–1623. Bibcode:2006JPhB...39.1605K. doi:10.1088/0953-4075/39/7/005.
16. ^ Arndt, M.; O. Nairz, J. Voss-Andreae, C. Keller, G. van der Zouw, A. Zeilinger (14 October 1999). "Wave-particle duality of C60". Nature 401 (6754): 680–682. Bibcode:1999Natur.401..680A. doi:10.1038/44348. PMID 18494170.
17. ^ Gerlich, S.; S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tüxen, M. Mayor & M. Arndt (5 April 2011). "Quantum interference of large organic molecules". Nature Communications 2 (263): 263–. Bibcode:2011NatCo...2E.263G. doi:10.1038/ncomms1263. PMC 3104521. PMID 21468015.
18. ^ Holden, Alan (1971). Stationary states. New York: Oxford University Press. ISBN 0-19-501497-9.
19. ^ Williams, W.S.C. (2002). Introducing Special Relativity, Taylor & Francis, London, ISBN 0-415-27761-2, p. 192.
20. ^ de Broglie, L. (1970). The reinterpretation of wave mechanics, Foundations of Physics 1(1): 5–15, p. 9.