Matter waves are a central part of the theory ofquantum mechanics, being half ofwave–particle duality. At all scales where measurements have been practical,matter exhibitswave-like behavior. For example, a beam ofelectrons can bediffracted just like a beam of light or a water wave.
The concept that matter behaves like a wave was proposed by French physicistLouis de Broglie (/dəˈbrɔɪ/) in 1924, and so matter waves are also known asde Broglie waves.
Wave-like behavior of matter has been experimentally demonstrated, first for electrons in 1927 and for otherelementary particles, neutralatoms andmolecules in the years since.
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.
At the end of the 19th century, light was thought to consist of waves of electromagnetic fields which propagated according toMaxwell's equations, while matter was thought to consist of localized particles (seehistory of wave and particle duality). In 1900, this division was questioned when, investigating the theory ofblack-body radiation,Max Planck proposed that the thermal energy of oscillating atoms is divided into discrete portions, or quanta.[1] Extending Planck's investigation in several ways, including its connection with thephotoelectric effect,Albert Einstein proposed in 1905 that light is also propagated and absorbed in quanta,[2]: 87 now calledphotons. These quanta would have an energy given by thePlanck–Einstein relation:and a momentum vectorwhereν (lowercaseGreek letter nu) andλ (lowercaseGreek letter lambda) denote thefrequency andwavelength of the light,c the speed of light, andh thePlanck constant.[3] In the modern convention, frequency is symbolized byf as is done in the rest of this article. Einstein's postulate was verified experimentally[2]: 89 byK. T. Compton andO. W. Richardson[4] and by A. L. Hughes[5] in 1912 then more carefully including a measurement of thePlanck constant in 1916 byRobert Millikan.[6]
Propagation ofde Broglie waves in one dimension – real part of thecomplex amplitude is blue, imaginary part is green. The probability (shown as the coloropacity) of finding the particle at a given pointx is spread out like a waveform; there is no definite position of the particle. As the amplitude increases above zero theslope decreases, so the amplitude diminishes again, and vice versa. The result is an alternating amplitude: a wave. Top:plane wave. Bottom:wave packet.
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, in his 1924 PhD thesis,[8] 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 aproper massm0 one may associate a periodic phenomenon of the frequencyν0, such that one finds:hν0 =m0c2. The frequencyν0 is to be measured, of course, in the rest frame of the energy packet. This hypothesis is the basis of our theory."[9][8]: 8 [10][11][12][13] (This frequency is also known asCompton frequency.)
(Modern physics no longer uses this form of the total energy; theenergy–momentum relation has proven more useful.) De Broglie identified the velocity of the particle,v, with the wavegroup velocity in free space:
(The modern definition of group velocity uses angular frequencyω and wave numberk). By applying the differentials to the energy equation and identifying therelativistic momentum:
then integrating, de Broglie arrived at his formula for the relationship between thewavelength,λ, associated with an electron and the modulus of itsmomentum,p, through thePlanck constant,h:[14]
Following up on de Broglie's ideas, physicistPeter Debye made an offhand comment that if particles behaved as waves, they should satisfy some sort of wave equation. Inspired by Debye's remark,Erwin Schrödinger decided to find a proper three-dimensional wave equation for the electron. He was guided byWilliam Rowan Hamilton's analogy between mechanics and optics (seeHamilton's optico-mechanical analogy), encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system – the trajectories oflight rays become sharp tracks that obeyFermat's principle, an analog of theprinciple of least action.[15]
In 1926, Schrödinger published thewave equation that now bears his name[16] – the matter wave analogue ofMaxwell's equations – and used it to derive theenergy spectrum ofhydrogen. Frequencies of solutions of the non-relativistic Schrödinger equation differ from de Broglie waves by theCompton frequency since the energy corresponding to therest mass of a particle is not part of the non-relativistic Schrödinger equation. The Schrödinger equation describes the time evolution of awavefunction, a function that assigns acomplex number to each point in space. Schrödinger tried to interpret themodulus squared of the wavefunction as a charge density. This approach was, however, unsuccessful.[17][18][19]Max Born proposed that the modulus squared of the wavefunction is instead aprobability density, a successful proposal now known as theBorn rule.[17]
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 thefamous biologist) exploredSchrödinger's equation in several idealized scenarios.[20] For an unbound electron in free space he worked out the propagation of the wave, assuming an initialGaussian wave packet. Darwin showed that at time later the position of the packet traveling at velocity would bewhere is the uncertainty in the initial position. This position uncertainty creates uncertainty in velocity (the extra second term in the square root) consistent withHeisenberg'suncertainty relation The wave packet spreads out as show in the figure.
Original electron diffraction camera made and used by Nobel laureate G P Thomson and his student Alexander Reid in 1925
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 have been shown to be wave-like.[24]
The first electron wave interference patterns directly demonstratingwave–particle duality used electron biprisms[25][26] (essentially a wire placed in an electron microscope) and measured single electrons building up the diffraction pattern.Recently, a close copy of the famousdouble-slit experiment[27]: 260 using electrons through physical apertures gave the movie shown.[28]
Matter wavedouble 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.[28]
In 1927 at Bell Labs,Clinton Davisson andLester Germerfired slow-movingelectrons at acrystallinenickel target.[22][23] The diffracted electron intensity was measured, and was determined to have a similar angular dependence todiffraction patterns predicted byBragg forx-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.[21] (Alexander Reid, who was Thomson's graduate student, performed the first experiments but he died soon after in a motorcycle accident[29] 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 anydiffraction effects by matter demonstrated the wave-like nature of matter.[30] The matter wave interpretation was placed onto a solid foundation in 1928 byHans Bethe,[31] who solved theSchrödinger equation,[16] showing how this could explain the experimental results. His approach is similar to what is used in modernelectron diffraction approaches.[32][33]
This was a pivotal result in the development ofquantum mechanics. Just as thephotoelectric effect demonstrated the particle nature of light, these experiments showed the wave nature of matter.
Neutrons, produced innuclear reactors with kinetic energy of around1 MeV,thermalize to around0.025 eV as they scatter from light atoms. The resulting de Broglie wavelength (around180 pm) matches interatomic spacing and neutrons scatter strongly from hydrogen atoms. Consequently, neutron matter waves are used incrystallography, especially for biological materials.[34] Neutrons were discovered in the early 1930s, and their diffraction was observed in 1936.[35] In 1944,Ernest O. Wollan, with a background in X-ray scattering from his PhD work[36] underArthur Compton, recognized the potential for applying thermal neutrons from the newly operationalX-10 nuclear reactor tocrystallography. Joined byClifford G. Shull, they developed[37]neutron diffraction throughout the 1940s. In the 1970s, aneutron interferometer demonstrated the action ofgravity in relation to wave–particle duality.[38] The double-slit experiment was performed using neutrons in 1988.[39]
Interference of atom matter waves was first observed byImmanuel Estermann andOtto Stern in 1930, when a Na beam was diffracted off a surface of NaCl.[40] The short de Broglie wavelength of atoms prevented progress for many years until two technological breakthroughs revived interest:microlithography allowing precise small devices andlaser cooling allowing atoms to be slowed, increasing their de Broglie wavelength.[41] The double-slit experiment on atoms was performed in 1991.[42]
Advances inlaser cooling allowed cooling of neutral atoms down to nanokelvin temperatures. At these temperatures, the de Broglie wavelengths come into the micrometre range. UsingBragg diffraction of atoms and a Ramsey interferometry technique, the de Broglie wavelength of coldsodium atoms was explicitly measured and found to be consistent with the temperature measured by a different method.[43]
Recent experiments confirm the relations for molecules and evenmacromolecules that otherwise might be supposed too large to undergo quantum mechanical effects. In 1999, a research team inVienna demonstrated diffraction for molecules as large asfullerenes.[44] The researchers calculated a de Broglie wavelength of the most probable C60 velocity as2.5 pm.More recent experiments prove the quantum nature of molecules made of 810 atoms and with a mass of10123Da.[45] As of 2019, this has been pushed to molecules of25000 Da.[46]
In these experiments the build-up of such interference patterns could be recorded in real time and with single molecule sensitivity.[47]Large molecules are already so complex that they give experimental access to some aspects of the quantum-classical interface, i.e., to certaindecoherence mechanisms.[48][49]
Waves have more complicated concepts forvelocity than solid objects.The simplest approach is to focus on the description in terms of plane matter waves for afree particle, that is a wave function described bywhere is a position in real space, is thewave vector in units of inverse meters,ω is theangular frequency with units of inverse time and is time. (Here the physics definition for the wave vector is used, which is times the wave vector used incrystallography, seewavevector.) The de Broglie equations relate thewavelengthλ to the modulus of themomentum, andfrequencyf to the total energyE of afree particle as written above:[54]whereh is thePlanck constant. The equations can also be written asHere,ħ =h/2π is the reduced Planck constant. The second equation is also referred to as thePlanck–Einstein relation.
In the de Broglie hypothesis, the velocity of a particle equals thegroup velocity of the matter wave.[2]: 214 In isotropic media or a vacuum thegroup velocity of a wave is defined by:The relationship between the angular frequency and wavevector is called thedispersion relationship. For the non-relativistic case this is:where is the rest mass. Applying the derivative gives the (non-relativistic)matter wave group velocity:For comparison, the group velocity of light, with adispersion, is thespeed of light.
As an alternative, using the relativisticdispersion relationship for matter wavesthenThis relativistic form relates to the phase velocity as discussed below.
For non-isotropic media we use theEnergy–momentum form instead:
But (see below), since the phase velocity is, thenwhere is the velocity of the center of mass of the particle, identical to the group velocity.
Thephase velocity in isotropic media is defined as:Using the relativistic group velocity above:[2]: 215 This shows that as reported by R.W. Ditchburn in 1948 and J. L. Synge in 1952. Electromagnetic waves also obey, as both and. Since for matter waves,, it follows that, but only the group velocity carries information. Thesuperluminal phase velocity therefore does not violate special relativity, as it does not carry information.
For non-isotropic media, then
Using therelativistic relations for energy and momentum yieldsThe variable 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 for any particle that has nonzero mass (according tospecial relativity), the phase velocity of matter waves always exceedsc, i.e.,which approachesc when the particle speed is relativistic. Thesuperluminal phase velocity does not violate special relativity, similar to the case above for non-isotropic media. See the article onDispersion (optics) for further details.
Using two formulas fromspecial relativity, one for the relativistic mass energy and one for therelativistic momentumallows the equations for de Broglie wavelength and frequency to be written aswhere is thevelocity, theLorentz factor, and thespeed of light in vacuum.[55][56] This shows that as the velocity of a particle approaches zero (rest) the de Broglie wavelength approaches infinity.
Using four-vectors, the de Broglie relations form a single equation:which isframe-independent.Likewise, the relation between group/particle velocity and phase velocity is given in frame-independent form by:where
The preceding sections refer specifically tofree 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.
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 towhere now there is an additional spatial term 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 aneffective mass which in general is a tensor given byso that in the simple case where all directions are the same the form is similar to that of a free wave above.In general the group velocity would be replaced by theprobability current[57]where is thedel orgradientoperator. The momentum would then be described using thekinetic momentum operator,[57]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:
Evanescent waves, where the component of the wavevector in one direction is complex. These are common when matter waves are being reflected, particularly forgrazing-incidence diffraction.
Other classes of matter waves involve more than one particle, so are called collective waves and are oftenquasiparticles. Many of these occur in solids – seeAshcroft and Mermin. Examples include:
In solids, anelectron quasiparticle is anelectron where interactions with other electrons in the solid have been included. An electron quasiparticle has the samecharge andspin as a "normal" (elementary particle) electron and, like a normal electron, it is afermion. However, itseffective mass can differ substantially from that of a normal electron.[60] Its electric field is also modified, as a result ofelectric field screening.
Ahole is a quasiparticle which can be thought of as avacancy of an electron in a state; it is most commonly used in the context of empty states in thevalence band of asemiconductor.[60] A hole has the opposite charge of an electron.
Apolaron is a quasiparticle where an electron interacts with thepolarization of nearby atoms.
Anexciton is an electron and hole pair which are bound together.
ACooper pair is two electrons bound together so they behave as a single matter wave.
Some trajectories of a particle in a box according toNewton's laws ofclassical 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 thewavefunction. The states (B,C,D) areenergy 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 zerogroup velocity orprobability flux. The simplest of these, similar to the notation above would beThese occur as part of theparticle in a box, and other cases such as in aring. 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 theBohr–Sommerfeld condition in the early approaches to quantum mechanics.[61] In that senseatomic orbitals around atoms, and alsomolecular orbitals are electron matter waves.[62][63][64]
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. Obviouslyvisible 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.[67] With special apparatus, high velocity electrons can be used to studyliquids andgases. Neutrons, an important exception, interact primarily by collisions with nuclei, and thus travel several hundred feet in air.[68]
Dispersion. Light waves of all frequencies travel at the samespeed 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 adispersion relation. Light waves in a vacuum have linear dispersion relation between frequency:. For matter waves the relation is non-linear:This non-relativisticmatter wave dispersion relation says the frequency in vacuum varies with wavenumber () in two parts: a constant part due to the de Broglie frequency of the rest mass () and a quadratic part due to kinetic energy. The quadratic term causes rapid spreading ofwave packets of matter waves.
Coherence The visibility of diffraction features using an optical theory approach depends on the beamcoherence,[27] which at the quantum level is equivalent to adensity matrix approach.[69][70] As with light, transverse coherence (across the direction of propagation) can be increased bycollimation. 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.[71] 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.[66]: 154
Optically shaped matter wavesOptical 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."[72] Laser light momentum transfer cancool matter particles and alter the internal excitation state of atoms.[73]
Multi-particle experimentsWhile single-particle free-space optical and matter wave equations are identical, multiparticle systems likecoincidence experiments are not.[74]
The following subsections provide links to pages describing applications of matter waves as probes of materials or of fundamentalquantum properties. In most cases these involve some method of producing travelling matter waves which initially have the simple form, then using these to probe materials.
As shown in the table below, matter wavemass ranges over 6orders of magnitude andenergy over 9 orders but the wavelengths are all in thepicometre range, comparable to atomic spacings. (Atomic diameters range from 62 to 520 pm, and the typical length of acarbon–carbon single bond is 154 pm.) Reaching longer wavelengths requires special techniques likelaser cooling to reach lower energies; shorter wavelengths make diffraction effects more difficult to discern.[41] Therefore, many applications focus onmaterial structures, in parallel with applications of electromagnetic waves, especiallyX-rays. Unlike light, matter wave particles may havemass,electric charge,magnetic moments, and internal structure, presenting new challenges and opportunities.
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.
The measurements of the energy they lose inelectron energy loss spectroscopy provides information about the chemistry and electronic structure of materials. Beams of electrons also lead to characteristic X-rays inenergy 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.
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.
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 or properties of individualized complex molecules.[85]
^abcdeWhittaker, Sir Edmund (1 January 1989).A History of the Theories of Aether and Electricity. Vol. 2. Courier Dover Publications.ISBN0-486-26126-3.
^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.
^abde Broglie, Louis Victor."On the Theory of Quanta"(PDF).Foundation of Louis de Broglie (English translation by A.F. Kracklauer, 2004. ed.). Retrieved25 February 2023.
^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.
^Merli, P. G., G. F. Missiroli, and G. Pozzi. "On the statistical aspect of electron interference phenomena." American Journal of Physics 44 (1976): 306
^Tonomura, A.; Endo, J.; Matsuda, T.; Kawasaki, T.; Ezawa, H. (1989). "Demonstration of single-electron buildup of an interference pattern".American Journal of Physics.57 (2). American Association of Physics Teachers (AAPT):117–120.Bibcode:1989AmJPh..57..117T.doi:10.1119/1.16104.ISSN0002-9505.
^abPeng, L.-M.; Dudarev, S. L.; Whelan, M. J. (2011).High energy electron diffraction and microscopy. Oxford: Oxford University Press.ISBN978-0-19-960224-7.OCLC656767858.
^Holden, Alan (1971).Stationary states. New York: Oxford University Press.ISBN978-0-19-501497-6.
^Williams, W.S.C. (2002).Introducing Special Relativity, Taylor & Francis, London,ISBN0-415-27761-2, p. 192.
^abSchiff, Leonard I. (1987).Quantum mechanics. International series in pure and applied physics (3. ed., 24. print ed.). New York: McGraw-Hill.ISBN978-0-07-085643-1.
^Metherell, A. J. (1972).Electron Microscopy in Materials Science. Commission of the European Communities. pp. 397–552.
^Jammer, Max (1989).The conceptual development of quantum mechanics. The history of modern physics (2nd ed.). Los Angeles (Calif.): Thomas publishers.ISBN978-0-88318-617-6.
^Griffiths, David J. (1995).Introduction to quantum mechanics. Englewood Cliffs, NJ: Prentice Hall.ISBN978-0-13-124405-4.
^Levine, Ira N. (2000).Quantum chemistry (5th ed.). Upper Saddle River, NJ: Prentice Hall.ISBN978-0-13-685512-5.
^Schrödinger, Erwin (2001).Collected papers on wave mechanics: together with his Four lectures on wave mechanics. Translated by Shearer, J. F.; Deans, Winifred Margaret (Third (augmented) edition, New York 1982 ed.). Providence, Rhode Island: AMS Chelsea Publishing, American Mathematical Society.ISBN978-0-8218-3524-1.
^Grisenti, R. E.; W. Schöllkopf; J. P. Toennies; J. R. Manson; T. A. Savas; Henry I. Smith (2000). "He-atom diffraction from nanostructure transmission gratings: The role of imperfections".Physical Review A.61 (3): 033608.Bibcode:2000PhRvA..61c3608G.doi:10.1103/PhysRevA.61.033608.
^Chapman, Michael S.; Christopher R. Ekstrom; Troy D. Hammond; Richard A. Rubenstein; Jörg Schmiedmayer; Stefan Wehinger; David E. Pritchard (1995). "Optics and interferometry with Na2 molecules".Physical Review Letters.74 (24):4783–4786.Bibcode:1995PhRvL..74.4783C.doi:10.1103/PhysRevLett.74.4783.PMID10058598.
^Gerlich, Stefan; Fein, Yaakov Y.; Shayeghi, Armin; Köhler, Valentin; Mayor, Marcel; Arndt, Markus (2021), Friedrich, Bretislav; Schmidt-Böcking, Horst (eds.), "Otto Stern's Legacy in Quantum Optics: Matter Waves and Deflectometry",Molecular Beams in Physics and Chemistry: From Otto Stern's Pioneering Exploits to Present-Day Feats, Cham: Springer International Publishing, pp. 547–573,Bibcode:2021mbpc.book..547G,doi:10.1007/978-3-030-63963-1_24,ISBN978-3-030-63963-1
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.
