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. 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.
- 1 Historical context
- 2 The de Broglie hypothesis
- 3 Experimental confirmation
- 4 de Broglie relations
- 5 Interpretations
- 6 See also
- 7 References
- 8 Further reading
- 9 External links
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). In 1900, this division was exposed to doubt, when, investigating the theory of black body thermal radiation, Max Planck proposed that light is emitted in discrete quanta of energy. It was thoroughly challenged in 1905. Extending Planck's investigation in several ways, including its connection with the photoelectric effect, Albert Einstein proposed that light is also propagated and absorbed in quanta. Light quanta are now called photons. These quanta would have an energy
where (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.
The de Broglie hypothesis
De Broglie, in his 1924 PhD thesis, proposed that just as light has both wave-like and particle-like properties, electrons also have wave-like properties. The wavelength, λ, associated with an electron is related to its momentum, p through the Planck constant h:
The relationship is now known to hold for all types of matter: all matter exhibits properties of both particles and waves.
|“||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.||”|
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.
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.
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.
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.
Experiments with Fresnel diffraction and an atomic mirror for specular reflection 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. 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.
This effect has been used to demonstrate atomic holography, and it may allow the construction of an atom probe imaging system with nanometer resolution. The description of these phenomena is based on the wave properties of neutral atoms, confirming the de Broglie hypothesis.
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. 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.
de Broglie relations
where h is Planck's constant. The equations can also be written as
allows the equations to be written as
where denotes the particle's rest mass, its velocity, the Lorentz factor, and 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.
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. The magnitude of the group velocity is equal to the particle's speed.
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.
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
where m is the mass of the particle and v its velocity.
Also in special relativity we find that
where v is the velocity of the particle regardless of wave behavior.
By the de Broglie hypothesis, we see that
Using relativistic relations for energy and momentum, we have
where E is the total energy of the particle (i.e. rest energy plus kinetic energy in kinematic sense), p the momentum, 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 for any particle that has mass (according to special relativity), the phase velocity of matter waves always exceeds c, i.e.
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.
which is frame-independent.
The physical reality underlying de Broglie waves is a subject of ongoing debate. Some theories treat either the particle or the wave aspect as its fundamental nature, seeking to explain the other as an emergent property. Some, such as the hidden variable theory, treat the wave and the particle as distinct entities. 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 Copenhagen interpretation states that the nature of the underlying reality is unknowable and beyond the bounds of scientific enquiry.
Schrödinger's quantum mechanical waves are conceptually different from ordinary physical waves such as of light or sound. Ordinary physical waves are characterized by undulating real-number 'displacements' of dimensioned physical variables at each point of ordinary physical space at each instant of time. Schrödinger's waves are characterized by the undulating value of a dimensionless complex number at each point of an abstract multi-dimensional space, for example of configuration space.
|“||If one wishes to calculate the probabilities of excitation and ionization of atoms [M. Born, Zur Quantenmechanik der Stossvorgange, Z. f. Phys., 37 (1926), 863; [Quantenmechanik der Stossvorgange], ibid., 38 (1926), 803] then one must introduce the coordinates of the atomic electrons as variables on an equal footing with those of the colliding electron. The waves then propagate no longer in three-dimensional space but in multi-dimensional configuration space. From this one sees that the quantum mechanical waves are indeed something quite different from the light waves of the classical theory.||”|
At the same conference, Erwin Schrödinger reported likewise.
|“||Under [the name 'wave mechanics',] at present two theories are being carried on, which are indeed closely related but not identical. The first, which follows on directly from the famous doctoral thesis by L. de Broglie, concerns waves in three-dimensional space. Because of the strictly relativistic treatment that is adopted in this version from the outset, we shall refer to it as the four-dimensional wave mechanics. The other theory is more remote from Mr de Broglie's original ideas, insofar as it is based on a wave-like process in the space of position coordinates (q-space) of an arbitrary mechanical system.[Long footnote about manuscript not copied here.] We shall therefore call it the multi-dimensional wave mechanics. Of course this use of the q-space is to be seen only as a mathematical tool, as it is often applied also in the old mechanics; ultimately, in this version also, the process to be described is one in space and time. In truth, however, a complete unification of the two conceptions has not yet been achieved. Anything over and above the motion of a single electron could be treated so far only in the multi-dimensional version; also, this is the one that provides the mathematical solution to the problems posed by the Heisenberg-Born matrix mechanics.||”|
In 1955, Heisenberg reiterated this.
|“||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. The first of these was the assertion that, in considering "probability waves", we are concerned with processes not in ordinary three-dimensional space, but in an abstract configuration space (a fact which is, unfortunately, sometimes overlooked even today); the second was the recognition that the probability wave is related to an individual process.||”|
This conceptual difference entails that, in contrast to de Broglie's pre-quantum mechanical wave description, which is 'separable', the quantum mechanical description is 'non-separable'. A non-separable description does not express causality, the principle that causal efficacy propagates no faster than light. The physical reasoning for this was first recognized by Einstein.
- Bohr model
- 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.
- 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.
- J. P. McEvoy and Oscar Zarate (2004). Introducing Quantum Theory. Totem Books. pp. 110–114. ISBN 1-84046-577-8.
- Louis de Broglie "The Reinterpretation of Wave Mechanics" Foundations of Physics, Vol. 1 No. 1 (1970)
- Mauro Dardo, Nobel Laureates and Twentieth-Century Physics, Cambridge University Press 2004, pp. 156–157
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- Holden, Alan (1971). Stationary states. New York: Oxford University Press. ISBN 0-19-501497-9.
- Williams, W.S.C. (2002). Introducing Special Relativity, Taylor & Francis, London, ISBN 0-415-27761-2, p. 192.
- de Broglie, L. (1970). The reinterpretation of wave mechanics, Foundations of Physics 1(1): 5–15, p. 9.
- Born, M., Heisenberg, W. (1928). Quantum mechanics, pp. 143–181 of Électrons et Photons: Rapports et Discussions du Cinquième Conseil de Physique, tenu à Bruxelles du 24 au 29 Octobre 1927, sous les Auspices de l'Institut International de Physique Solvay, Gauthier-Villars, Paris, p. 166; this translation at p. 425 of Bacciagaluppi, G., Valentini, A. (2009), Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference, Cambridge University Press, Cambridge UK, ISBN 978-0-521-81421-8.
- Schrödinger, E. (1928). Wave mechanics, pp. 185–206 of Électrons et Photons: Rapports et Discussions du Cinquième Conseil de Physique, tenu à Bruxelles du 24 au 29 Octobre 1927, sous les Auspices de l'Institut International de Physique Solvay, Gauthier-Villars, Paris, pp. 185–186; this translation at p. 447 of Bacciagaluppi, G., Valentini, A. (2009), Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference, Cambridge University Press, Cambridge UK, ISBN 978-0-521-81421-8.
- 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.
- Fine, A. (1986). The Shaky Game: Einstein Realism and the Quantum Theory, University of Chicago, Chicago, ISBN 0-226-24946-8
- Howard, D. (1990). "Nicht sein kann was nicht sein darf", or the prehistory of the EPR, 1909–1935; Einstein's early worries about the quantum mechanics of composite systems, pp. 61–112 in Sixty-two Years of Uncertainty: Historical Philosophical and Physical Inquiries into the Foundations of Quantum Mechanics, edited by A.I. Miller, Plenum Press, New York, ISBN 978-1-4684-8773-2.
- 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.
- Web version of Thesis, translated by Kracklauer (English)
- Zumdahl, Steven S. (2005). Chemical Principles (5th ed.). Boston: Houghton Mifflin. ISBN 0-618-37206-7.
- An extensive review article "Optics and interferometry with atoms and molecules" appeared in July 2009: http://www.atomwave.org/rmparticle/RMPLAO.pdf.
- 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):