Wave–particle duality

Wave–particle duality is a theory that proposes thatevery elementary particle exhibits the properties of notonly particles, but also waves. A central concept ofquantum mechanics, this duality addresses the inabilityof the classical concepts “particle” or “wave” to fully de-scribe the behavior of quantum-scale objects. As Ein-stein described: "It seems as though we must use some-times the one theory and sometimes the other, while attimes we may use either. We are faced with a new kind ofdifficulty. We have two contradictory pictures of reality;separately neither of them fully explains the phenomena oflight, but together they do".[1]

But Einstein’s 1930s vision and the mainstream of 20thcentury disliked the alternative picture, of the pilot wave,where no duality paradox is found. Developed later as thede Broglie-Bohm theory, this “particle and wave” model,today, supports all the current experimental evidence, tothe same extent as the other interpretations.The standard interpretations of quantum mechanics, nev-ertheless, explain the “duality paradox" as a fundamentalproperty of the universe; and some other alternative in-terpretations explain the duality as an emergent, second-order consequence of various limitations of the observer.The standard treatment focuses on explaining thebehaviour from the perspective of the widely usedCopenhagen interpretation, in which wave-particleduality serves as one aspect of the concept ofcomplementarity.[2]:242, 375–376

1 Origin of theory

The idea of duality originated in a debate over the na-ture of light and matter that dates back to the 17th cen-tury, when Christiaan Huygens and Isaac Newton pro-posed competing theories of light: light was thought ei-ther to consist of waves (Huygens) or of particles (New-ton). Through the work of Max Planck, Albert Einstein,Louis de Broglie, Arthur Compton, Niels Bohr, andmanyothers, current scientific theory holds that all particlesalso have a wave nature (and vice versa).[3] This phe-nomenon has been verified not only for elementary parti-cles, but also for compound particles like atoms and evenmolecules. For macroscopic particles, because of theirextremely short wavelengths, wave properties usually can-not be detected.[4]

2 Brief history of wave and particleviewpoints

Aristotle was one of the first to publicly hypothesizeabout the nature of light, proposing that light is a distur-bance in the element air (that is, it is a wave-like phe-nomenon). On the other hand, Democritus—the originalatomist—argued that all things in the universe, includ-ing light, are composed of indivisible sub-components(light being some form of solar atom).[5] At the begin-ning of the 11th Century, the Arabic scientist Alhazenwrote the first comprehensive treatise on optics; describ-ing refraction, reflection, and the operation of a pinholelens via rays of light traveling from the point of emis-sion to the eye. He asserted that these rays were com-posed of particles of light. In 1630, René Descartes pop-ularized and accredited the opposing wave description inhis treatise on light, showing that the behavior of lightcould be re-created by modeling wave-like disturbancesin a universal medium (“plenum”). Beginning in 1670and progressing over three decades, Isaac Newton devel-oped and championed his corpuscular hypothesis, argu-ing that the perfectly straight lines of reflection demon-strated light’s particle nature; only particles could travelin such straight lines. He explained refraction by posit-ing that particles of light accelerated laterally upon en-tering a denser medium. Around the same time, New-ton’s contemporaries Robert Hooke and Christiaan Huy-gens—and later Augustin-Jean Fresnel—mathematicallyrefined the wave viewpoint, showing that if light traveledat different speeds in different media (such as water andair), refraction could be easily explained as the medium-dependent propagation of light waves. The resultingHuygens–Fresnel principle was extremely successful atreproducing light’s behavior and, subsequently supportedby Thomas Young's 1803 discovery of double-slit inter-ference, was the beginning of the end for the particle lightcamp.[6][7]

The final blow against corpuscular theory came whenJames Clerk Maxwell discovered that he could combinefour simple equations, which had been previously discov-ered, along with a slight modification to describe self-propagating waves of oscillating electric and magneticfields. When the propagation speed of these electromag-netic waves was calculated, the speed of light fell out.It quickly became apparent that visible light, ultravioletlight, and infrared light (phenomena thought previouslyto be unrelated) were all electromagnetic waves of dif-fering frequency. The wave theory had prevailed—or at

1

2 3 TURN OF THE 20TH CENTURY AND THE PARADIGM SHIFT

Thomas Young’s sketch of two-slit diffraction of waves, 1803

least it seemed to.While the 19th century had seen the success of the wavetheory at describing light, it had also witnessed the rise ofthe atomic theory at describing matter. In 1789, AntoineLavoisier securely differentiated chemistry from alchemyby introducing rigor and precision into his laboratorytechniques; allowing him to deduce the conservation ofmass and categorize many new chemical elements andcompounds. However, the nature of these essentialchemical elements remained unknown. In 1799, JosephLouis Proust advanced chemistry towards the atom byshowing that elements combined in definite proportions.This led John Dalton to resurrect Democritus’ atom in1803, when he proposed that elements were invisiblesub components; which explained why the varying oxidesof metals (e.g. stannous oxide and cassiterite, SnO andSnO2 respectively) possess a 1:2 ratio of oxygen to oneanother. But Dalton and other chemists of the time hadnot considered that some elements occur in monatomicform (like Helium) and others in diatomic form (like Hy-drogen), or that water was H2O, not the simpler and moreintuitive HO—thus the atomic weights presented at thetime were varied and often incorrect. Additionally, theformation of H2O by two parts of hydrogen gas and onepart of oxygen gas would require an atom of oxygen tosplit in half (or two half-atoms of hydrogen to come to-gether). This problem was solved by Amedeo Avogadro,who studied the reacting volumes of gases as they formedliquids and solids. By postulating that equal volumes ofelemental gas contain an equal number of atoms, he wasable to show that H2Owas formed from two parts H2 andone part O2. By discovering diatomic gases, Avogadrocompleted the basic atomic theory, allowing the correctmolecular formulae of most known compounds—as wellas the correct weights of atoms—to be deduced and cate-gorized in a consistent manner. The final stroke in classi-cal atomic theory came when Dimitri Mendeleev saw anorder in recurring chemical properties, and created a tablepresenting the elements in unprecedented order and sym-metry. But there were holes in Mendeleev’s table, withno element to fill them in. His critics initially cited this asa fatal flaw, but were silenced when new elements werediscovered that perfectly fit into these holes. The successof the periodic table effectively converted any remainingopposition to atomic theory; even though no single atom

had ever been observed in the laboratory, chemistry wasnow an atomic science.

Animation showing the wave-particle duality with a double slitexperiment and effect of an observer. Increase size to see expla-nations in the video itself. See also quiz based on this animation.

Particle impacts make visible the interference pattern ofwaves.

A quantum particle is represented by a wave packet.Interference of a quantum particle with itself.Click images for animations.

3 Turn of the 20th century and theparadigm shift

3.1 Particles of electricity

At the close of the 19th century, the reductionism ofatomic theory began to advance into the atom itself; de-termining, through physics, the nature of the atom and theoperation of chemical reactions. Electricity, first thoughtto be a fluid, was now understood to consist of parti-cles called electrons. This was first demonstrated by J.J. Thomson in 1897 when, using a cathode ray tube, hefound that an electrical charge would travel across a vac-uum (which would possess infinite resistance in classicaltheory). Since the vacuum offered no medium for anelectric fluid to travel, this discovery could only be ex-plained via a particle carrying a negative charge and mov-ing through the vacuum. This electron flew in the face of

3.2 Radiation quantization 3

classical electrodynamics, which had successfully treatedelectricity as a fluid for many years (leading to the in-vention of batteries, electric motors, dynamos, and arclamps). More importantly, the intimate relation betweenelectric charge and electromagnetism had been well doc-umented following the discoveries of Michael Faradayand James Clerk Maxwell. Since electromagnetism wasknown to be a wave generated by a changing electric ormagnetic field (a continuous, wave-like entity itself) anatomic/particle description of electricity and charge wasa non sequitur. Furthermore, classical electrodynamicswas not the only classical theory rendered incomplete.

3.2 Radiation quantization

Black-body radiation, the emission of electromagneticenergy due to an object’s heat, could not be explainedfrom classical arguments alone. The equipartition theo-rem of classical mechanics, the basis of all classical ther-modynamic theories, stated that an object’s energy is par-titioned equally among the object’s vibrational modes.This worked well when describing thermal objects, whosevibrational modes were defined as the speeds of their con-stituent atoms, and the speed distribution derived fromegalitarian partitioning of these vibrational modes closelymatched experimental results. Speeds much higher thanthe average speed were suppressed by the fact that kineticenergy is quadratic—doubling the speed requires fourtimes the energy—thus the number of atoms occupyinghigh energy modes (high speeds) quickly drops off be-cause the constant, equal partition can excite successivelyfewer atoms. Low speed modes would ostensibly dom-inate the distribution, since low speed modes would re-quire ever less energy, and prima facie a zero-speed modewould require zero energy and its energy partition wouldcontain an infinite number of atoms. But this would onlyoccur in the absence of atomic interaction; when colli-sions are allowed, the low speed modes are immediatelysuppressed by jostling from the higher energy atoms, ex-citing them to higher energy modes. An equilibrium isswiftly reached where most atoms occupy a speed pro-portional to the temperature of the object (thus definingtemperature as the average kinetic energy of the object).But applying the same reasoning to the electromagneticemission of such a thermal object was not so successful. Ithad been long known that thermal objects emit light. Hotmetal glows red, and upon further heating, white (this isthe underlying principle of the incandescent bulb). Sincelight was known to be waves of electromagnetism, physi-cists hoped to describe this emission via classical laws.This became known as the black body problem. Since theequipartition theoremworked so well in describing the vi-brational modes of the thermal object itself, it was trivialto assume that it would perform equally well in describ-ing the radiative emission of such objects. But a problemquickly arose when determining the vibrational modes oflight. To simplify the problem (by limiting the vibrational

modes) a longest allowable wavelength was defined byplacing the thermal object in a cavity. Any electromag-netic mode at equilibrium (i.e. any standing wave) couldonly exist if it used the walls of the cavities as nodes. Thusthere were no waves/modes with a wavelength larger thantwice the length (L) of the cavity.

Standing waves in a cavity

The first few allowable modes would therefore have wave-lengths of : 2L, L, 2L/3, L/2, etc. (each successive wave-length adding one node to the wave). However, whilethe wavelength could never exceed 2L, there was no suchlimit on decreasing the wavelength, and adding nodes toreduce the wavelength could proceed ad infinitum. Sud-denly it became apparent that the short wavelength modescompletely dominated the distribution, since ever shorterwavelength modes could be crammed into the cavity. Ifeach mode received an equal partition of energy, theshort wavelength modes would consume all the energy.This became clear when plotting the Rayleigh–Jeans lawwhich, while correctly predicting the intensity of longwavelength emissions, predicted infinite total energy asthe intensity diverges to infinity for short wavelengths.This became known as the ultraviolet catastrophe.The solution arrived in 1900 when Max Planck hypothe-sized that the frequency of light emitted by the black bodydepended on the frequency of the oscillator that emittedit, and the energy of these oscillators increased linearlywith frequency (according to his constant h, where E =hν). This was not an unsound proposal considering thatmacroscopic oscillators operate similarly: when study-ing five simple harmonic oscillators of equal amplitudebut different frequency, the oscillator with the highestfrequency possesses the highest energy (though this re-lationship is not linear like Planck’s). By demanding thathigh-frequency light must be emitted by an oscillator ofequal frequency, and further requiring that this oscilla-tor occupy higher energy than one of a lesser frequency,Planck avoided any catastrophe; giving an equal partitionto high-frequency oscillators produced successively fewer

4 4 DEVELOPMENTAL MILESTONES

oscillators and less emitted light. And as in the Maxwell–Boltzmann distribution, the low-frequency, low-energyoscillators were suppressed by the onslaught of thermaljiggling from higher energy oscillators, which necessarilyincreased their energy and frequency.The most revolutionary aspect of Planck’s treatment ofthe black body is that it inherently relies on an integernumber of oscillators in thermal equilibrium with theelectromagnetic field. These oscillators give their entireenergy to the electromagnetic field, creating a quantum oflight, as often as they are excited by the electromagneticfield, absorbing a quantum of light and beginning to os-cillate at the corresponding frequency. Planck had inten-tionally created an atomic theory of the black body, buthad unintentionally generated an atomic theory of light,where the black body never generates quanta of light ata given frequency with an energy less than hν. However,once realizing that he had quantized the electromagneticfield, he denounced particles of light as a limitation of hisapproximation, not a property of reality.

3.3 Photoelectric effect illuminated

Yet while Planck had solved the ultraviolet catastropheby using atoms and a quantized electromagnetic field,most physicists immediately agreed that Planck’s “lightquanta” were unavoidable flaws in his model. A morecomplete derivation of black body radiation would pro-duce a fully continuous, fully wave-like electromagneticfield with no quantization. However, in 1905 Albert Ein-stein took Planck’s black body model in itself and sawa wonderful solution to another outstanding problem ofthe day: the photoelectric effect, the phenomenon whereelectrons are emitted from atoms when they absorb en-ergy from light. Ever since the discovery of electronseight years previously, electrons had been the thing tostudy in physics laboratories worldwide.In 1902 Philipp Lenard discovered that (within the rangeof the experimental parameters he was using) the energyof these ejected electrons did not depend on the inten-sity of the incoming light, but on its frequency. So if oneshines a little low-frequency light upon a metal, a few lowenergy electrons are ejected. If one now shines a very in-tense beam of low-frequency light upon the same metal,a whole slew of electrons are ejected; however they pos-sess the same low energy, there are merely more of them.In order to get high energy electrons, one must illumi-nate the metal with high-frequency light. The more lightthere is, the more electrons are ejected. Like blackbodyradiation, this was at odds with a theory invoking con-tinuous transfer of energy between radiation and matter.However, it can still be explained using a fully classicaldescription of light, as long as matter is quantummechan-ical in nature.[8]

If one used Planck’s energy quanta, and demanded thatelectromagnetic radiation at a given frequency could only

transfer energy to matter in integer multiples of an en-ergy quantum hν, then the photoelectric effect could beexplained very simply. Low-frequency light only ejectslow-energy electrons because each electron is excited bythe absorption of a single photon. Increasing the intensityof the low-frequency light (increasing the number of pho-tons) only increases the number of excited electrons, nottheir energy, because the energy of each photon remainslow. Only by increasing the frequency of the light, andthus increasing the energy of the photons, can one ejectelectrons with higher energy. Thus, using Planck’s con-stant h to determine the energy of the photons based upontheir frequency, the energy of ejected electrons shouldalso increase linearly with frequency; the gradient of theline being Planck’s constant. These results were not con-firmed until 1915, when Robert Andrews Millikan, whohad previously determined the charge of the electron,produced experimental results in perfect accord with Ein-stein’s predictions. While the energy of ejected elec-trons reflected Planck’s constant, the existence of pho-tons was not explicitly proven until the discovery of thephoton antibunching effect, of which a modern exper-iment can be performed in undergraduate-level labs.[9]This phenomenon could only be explained via photons,and not through any semi-classical theory (which couldalternatively explain the photoelectric effect). When Ein-stein received his Nobel Prize in 1921, it was not for hismore difficult and mathematically laborious special andgeneral relativity, but for the simple, yet totally revolu-tionary, suggestion of quantized light. Einstein’s “lightquanta” would not be called photons until 1925, but evenin 1905 they represented the quintessential example ofwave-particle duality. Electromagnetic radiation prop-agates following linear wave equations, but can only beemitted or absorbed as discrete elements, thus acting as awave and a particle simultaneously.

4 Developmental milestones

4.1 Huygens and Newton

The earliest comprehensive theory of light was advancedby Christiaan Huygens, who proposed a wave theory oflight, and in particular demonstrated how waves might in-terfere to form a wavefront, propagating in a straight line.However, the theory had difficulties in other matters, andwas soon overshadowed by Isaac Newton’s corpusculartheory of light. That is, Newton proposed that light con-sisted of small particles, with which he could easily ex-plain the phenomenon of reflection. With considerablymore difficulty, he could also explain refraction througha lens, and the splitting of sunlight into a rainbow by aprism. Newton’s particle viewpoint went essentially un-challenged for over a century.[10]

4.5 De Broglie’s wavelength 5

4.2 Young, Fresnel, and Maxwell

In the early 19th century, the double-slit experiments byYoung and Fresnel provided evidence for Huygens’ wavetheories. The double-slit experiments showed that whenlight is sent through a grid, a characteristic interferencepattern is observed, very similar to the pattern resultingfrom the interference of water waves; the wavelength oflight can be computed from such patterns. The wave viewdid not immediately displace the ray and particle view,but began to dominate scientific thinking about light inthe mid 19th century, since it could explain polarizationphenomena that the alternatives could not.[11]

In the late 19th century, James Clerk Maxwell explainedlight as the propagation of electromagnetic waves accord-ing to the Maxwell equations. These equations were ver-ified by experiment by Heinrich Hertz in 1887, and thewave theory became widely accepted.

4.3 Planck’s formula for black-body radi-ation

Main article: Planck’s law

In 1901, Max Planck published an analysis that suc-ceeded in reproducing the observed spectrum of lightemitted by a glowing object. To accomplish this, Planckhad to make an ad hoc mathematical assumption of quan-tized energy of the oscillators (atoms of the black body)that emit radiation. It was Einstein who later proposedthat it is the electromagnetic radiation itself that is quan-tized, and not the energy of radiating atoms.

4.4 Einstein’s explanation of the photo-electric effect

Main article: Photoelectric effectIn 1905, Albert Einstein provided an explanation of the

The photoelectric effect. Incoming photons on the left strike ametal plate (bottom), and eject electrons, depicted as flying off tothe right.

photoelectric effect, a hitherto troubling experiment thatthe wave theory of light seemed incapable of explaining.He did so by postulating the existence of photons, quantaof light energy with particulate qualities.In the photoelectric effect, it was observed that shining alight on certain metals would lead to an electric currentin a circuit. Presumably, the light was knocking electronsout of the metal, causing current to flow. However, us-ing the case of potassium as an example, it was also ob-served that while a dim blue light was enough to causea current, even the strongest, brightest red light availablewith the technology of the time caused no current at all.According to the classical theory of light and matter, thestrength or amplitude of a light wave was in proportionto its brightness: a bright light should have been easilystrong enough to create a large current. Yet, oddly, thiswas not so.Einstein explained this conundrum by postulating thatthe electrons can receive energy from electromagneticfield only in discrete portions (quanta that were calledphotons): an amount of energy E that was related to thefrequency f of the light by

E = hf

where h is Planck’s constant (6.626 × 10−34 J seconds).Only photons of a high enough frequency (above a cer-tain threshold value) could knock an electron free. Forexample, photons of blue light had sufficient energy tofree an electron from the metal, but photons of red lightdid not. More intense light above the threshold frequencycould release more electrons, but no amount of light (us-ing technology available at the time) below the thresholdfrequency could release an electron. To “violate” this lawwould require extremely high intensity lasers which hadnot yet been invented. Intensity-dependent phenomenahave now been studied in detail with such lasers.[12]

Einstein was awarded the Nobel Prize in Physics in 1921for his discovery of the law of the photoelectric effect.

4.5 De Broglie’s wavelength

Main article: Matter waveIn 1924, Louis-Victor de Broglie formulated the deBroglie hypothesis, claiming that all matter,[13][14] notjust light, has a wave-like nature; he related wavelength(denoted as λ), and momentum (denoted as p):

λ =h

p

This is a generalization of Einstein’s equation above, sincethe momentum of a photon is given by p = E

c and thewavelength (in a vacuum) by λ = c

f , where c is the speedof light in vacuum.

6 4 DEVELOPMENTAL MILESTONES

i(p x t) A e n n n = n

x

x

i(px t) = Ae

Propagation of de Broglie waves in 1d—real part of the complexamplitude is blue, imaginary part is green. The probability(shown as the colour opacity) of finding the particle at a givenpoint x is spread out like a waveform; there is no definite po-sition of the particle. As the amplitude increases above zerothe curvature decreases, so the amplitude decreases again, andvice versa—the result is an alternating amplitude: a wave. Top:Plane wave. Bottom: Wave packet.

De Broglie’s formula was confirmed three years laterfor electrons (which differ from photons in having arest mass) with the observation of electron diffraction intwo independent experiments. At the University of Ab-erdeen, George Paget Thomson passed a beam of elec-trons through a thin metal film and observed the pre-dicted interference patterns. At Bell Labs Clinton JosephDavisson and Lester Halbert Germer guided their beamthrough a crystalline grid.De Broglie was awarded the Nobel Prize for Physics in1929 for his hypothesis. Thomson and Davisson sharedthe Nobel Prize for Physics in 1937 for their experimentalwork.

4.6 Heisenberg’s uncertainty principle

Main article: Heisenberg uncertainty principle

In his work on formulating quantum mechanics, Werner

Heisenberg postulated his uncertainty principle, whichstates:

∆x∆p ≥ ℏ2

where

∆ here indicates standard deviation, a measureof spread or uncertainty;x and p are a particle’s position and linear mo-mentum respectively.ℏ is the reduced Planck’s constant (Planck’sconstant divided by 2 π ).

Heisenberg originally explained this as a consequence ofthe process of measuring: Measuring position accuratelywould disturb momentum and vice-versa, offering an ex-ample (the “gamma-ray microscope”) that depended cru-cially on the de Broglie hypothesis. It is now thought,however, that this only partly explains the phenomenon,but that the uncertainty also exists in the particle itself,even before the measurement is made.In fact, the modern explanation of the uncertainty prin-ciple, extending the Copenhagen interpretation first putforward by Bohr andHeisenberg, depends evenmore cen-trally on the wave nature of a particle: Just as it is nonsen-sical to discuss the precise location of a wave on a string,particles do not have perfectly precise positions; likewise,just as it is nonsensical to discuss the wavelength of a“pulse” wave traveling down a string, particles do not haveperfectly precise momenta (which corresponds to the in-verse of wavelength). Moreover, when position is rela-tively well defined, the wave is pulse-like and has a veryill-defined wavelength (and thus momentum). And con-versely, when momentum (and thus wavelength) is rela-tively well defined, the wave looks long and sinusoidal,and therefore it has a very ill-defined position.

4.7 de Broglie–Bohm theory

Couder experiments,[15] “materializing” the pilot wave model.

De Broglie himself had proposed a pilot wave constructto explain the observed wave-particle duality. In this

7

view, each particle has a well-defined position and mo-mentum, but is guided by a wave function derived fromSchrödinger’s equation. The pilot wave theory was ini-tially rejected because it generated non-local effects whenapplied to systems involvingmore than one particle. Non-locality, however, soon became established as an integralfeature of quantum theory (see EPR paradox), and DavidBohm extended de Broglie’s model to explicitly include it.In the resulting representation, also called the de Broglie–Bohm theory or Bohmian mechanics,[16] the wave-particle duality vanishes, and explains the wave behaviouras a scattering with wave appearance, because the par-ticle’s motion subject to a guiding equation or quantumpotential. “This idea seems to me so natural and simple,to resolve the wave-particle dilemma in such a clear andordinary way, that it is a great mystery to me that it was sogenerally ignored”,[17] J.S.Bell.The best illustration of the pilot-wave model was givenby Couder’s 2010 “walking droplets” experiments,[18]demonstrating the pilot-wave behaviour in a macroscopicmechanical analog.[15]

5 Wave behavior of large objects

Since the demonstrations of wave-like properties inphotons and electrons, similar experiments have beenconducted with neutrons and protons. Among the mostfamous experiments are those of Estermann and OttoStern in 1929.[19] Authors of similar recent experimentswith atoms and molecules, described below, claim thatthese larger particles also act like waves.A dramatic series of experiments emphasizing the ac-tion of gravity in relation to wave–particle duality wereconducted in the 1970s using the neutron interferome-ter.[20] Neutrons, one of the components of the atomicnucleus, provide much of the mass of a nucleus and thusof ordinary matter. In the neutron interferometer, theyact as quantum-mechanical waves directly subject to theforce of gravity. While the results were not surprisingsince gravity was known to act on everything, includinglight (see tests of general relativity and the Pound–Rebkafalling photon experiment), the self-interference of thequantummechanical wave of a massive fermion in a grav-itational field had never been experimentally confirmedbefore.In 1999, the diffraction of C60 fullerenes by re-searchers from the University of Vienna was reported.[21]Fullerenes are comparatively large and massive objects,having an atomic mass of about 720 u. The de Brogliewavelength is 2.5 pm, whereas the diameter of themolecule is about 1 nm, about 400 times larger. In 2012,these far-field diffraction experiments could be extendedto phthalocyanine molecules and their heavier derivatives,which are composed of 58 and 114 atoms respectively.In these experiments the build-up of such interference

patterns could be recorded in real time and with singlemolecule sensitivity.[22][23]

In 2003, the Vienna group also demonstrated the wavenature of tetraphenylporphyrin[24]—a flat biodye with anextension of about 2 nm and a mass of 614 u. Forthis demonstration they employed a near-field TalbotLau interferometer.[25][26] In the same interferometerthey also found interference fringes for C60F₄₈., a flu-orinated buckyball with a mass of about 1600 u, com-posed of 108 atoms.[24] Large molecules are already socomplex that they give experimental access to some as-pects of the quantum-classical interface, i.e., to certaindecoherence mechanisms.[27][28] In 2011, the interfer-ence of molecules as heavy as 6910 u could be demon-strated in a Kapitza–Dirac–Talbot–Lau interferometer.These are the largest objects that so far showed de Brogliematter-wave interference.[29] In 2013, the interference ofmolecules beyond 10,000 u has been demonstrated.[30]

Whether objects heavier than the Planck mass (about theweight of a large bacterium) have a de Broglie wavelengthis theoretically unclear and experimentally unreachable;above the Planck mass a particle’s Compton wavelengthwould be smaller than the Planck length and its ownSchwarzschild radius, a scale at which current theories ofphysics may break down or need to be replaced by moregeneral ones.[31]

Recently Couder, Fort, et al. showed[32] that we can usemacroscopic oil droplets on a vibrating surface as a modelof wave–particle duality—localized droplet creates pe-riodical waves around and interaction with them leadsto quantum-like phenomena: interference in double-slitexperiment,[33] unpredictable tunneling[34] (depending incomplicated way on practically hidden state of field), or-bit quantization[35] (that particle has to 'find a resonance'with field perturbations it creates—after one orbit, its in-ternal phase has to return to the initial state) and Zeemaneffect.[36]

6 Treatment in modern quantummechanics

Wave–particle duality is deeply embedded into the foun-dations of quantum mechanics. In the formalism of thetheory, all the information about a particle is encodedin its wave function, a complex-valued function roughlyanalogous to the amplitude of a wave at each point inspace. This function evolves according to a differentialequation (generically called the Schrödinger equation).For particles with mass this equation has solutions thatfollow the form of the wave equation. Propagation ofsuch waves leads to wave-like phenomena such as inter-ference and diffraction. Particles without mass, like pho-tons, has no solutions of the Schrödinger equation so haveanother wave.

8 7 ALTERNATIVE VIEWS

The particle-like behavior is most evident due to phenom-ena associated with measurement in quantum mechanics.Upon measuring the location of the particle, the parti-cle will be forced into a more localized state as givenby the uncertainty principle. When viewed through thisformalism, the measurement of the wave function willrandomly "collapse", or rather "decohere", to a sharplypeaked function at some location. For particles withmass the likelihood of detecting the particle at any par-ticular location is equal to the squared amplitude of thewave function there. The measurement will return a well-defined position, (subject to uncertainty), a property tra-ditionally associated with particles. It is important to notethat a measurement is only a particular type of interactionwhere some data is recorded and the measured quantityis forced into a particular eigenstate. The act of measure-ment is therefore not fundamentally different from anyother interaction.Following the development of quantum field theory theambiguity disappeared. The field permits solutions thatfollow the wave equation, which are referred to as thewave functions. The term particle is used to label theirreducible representations of the Lorentz group that arepermitted by the field. An interaction as in a Feynman di-agram is accepted as a calculationally convenient approx-imation where the outgoing legs are known to be simpli-fications of the propagation and the internal lines are forsome order in an expansion of the field interaction. Sincethe field is non-local and quantized, the phenomena whichpreviously were thought of as paradoxes are explained.Within the limits of the wave-particle duality the quan-tum field theory gives the same results.

6.1 Visualization

There are two ways to visualize the wave-particle be-haviour: by the “standard model”, described below; andby the Broglie–Bohm model, where no duality is per-ceived.Below is an illustration of wave–particle duality as it re-lates to De Broglie’s hypothesis and Heisenberg’s un-certainty principle (above), in terms of the position andmomentum space wavefunctions for one spinless particlewith mass in one dimension. These wavefunctions areFourier transforms of each other.The more localized the position-space wavefunction, themore likely the particle is to be found with the posi-tion coordinates in that region, and correspondingly themomentum-space wavefunction is less localized so thepossible momentum components the particle could haveare more widespread.Conversely the more localized the momentum-spacewavefunction, the more likely the particle is to be foundwith those values of momentum components in that re-gion, and correspondingly the less localized the position-space wavefunction, so the position coordinates the par-

ticle could occupy are more widespread.

(p) Re[ ]

p x

p x

(p) 2

x

x

(x) Re[ ] (x) 2

x

x

(x) Re[ ] (x) 2

p x

p x

(p) 2 (p) Re[ ]

= ?

Position x and momentum p wavefunctions corresponding toquantum particles. The colour opacity (%) of the particles cor-responds to the probability density of finding the particle withposition x or momentum component p.Top: If wavelength λ is unknown, so are momentum p, wave-vector k and energy E (de Broglie relations). As the particle ismore localized in position space, Δx is smaller than for Δpₓ.Bottom: If λ is known, so are p, k, and E. As the particle is morelocalized in momentum space, Δp is smaller than for Δx.

7 Alternative views

Wave–particle duality is an ongoing conundrum in mod-ern physics. Most physicists accept wave-particle dual-ity as the best explanation for a broad range of observedphenomena; however, it is not without controversy. Al-ternative views are also presented here. These views arenot generally accepted by mainstream physics, but serveas a basis for valuable discussion within the community.

7.1 Both-particle-and-wave view

The pilot wave model, originally developed by Louis deBroglie and further developed by David Bohm into thehidden variable theory proposes that there is no dual-ity, but rather a system exhibits both particle proper-ties and wave properties simultaneously, and particles areguided, in a deterministic fashion, by the pilot wave (orits "quantum potential") which will direct them to ar-eas of constructive interference in preference to areas ofdestructive interference. This idea is held by a significantminority within the physics community.[37]

At least one physicist considers the “wave-duality” a mis-nomer, as L. Ballentine, Quantum Mechanics, A ModernDevelopment, p. 4, explains:

When first discovered, particle diffractionwas a source of great puzzlement. Are “par-ticles” really “waves?" In the early experi-ments, the diffraction patterns were detectedholistically by means of a photographic plate,which could not detect individual particles.As a result, the notion grew that particle and

7.3 Neither-wave-nor-particle view 9

wave properties were mutually incompatible,or complementary, in the sense that differentmeasurement apparatuses would be required toobserve them. That idea, however, was only anunfortunate generalization from a technologi-cal limitation. Today it is possible to detect thearrival of individual electrons, and to see thediffraction pattern emerge as a statistical pat-tern made up of many small spots (Tonomuraet al., 1989). Evidently, quantum particles areindeed particles, but whose behaviour is verydifferent from classical physics would have usto expect.

The Afshar experiment[38] (2007) has demonstrated thatit is possible to simultaneously observe both wave andparticle properties of photons.

7.2 Wave-only view

At least one scientist proposes that the duality can be re-placed by a “wave-only” view. In his bookCollective Elec-trodynamics: Quantum Foundations of Electromagnetism(2000), Carver Mead purports to analyze the behavior ofelectrons and photons purely in terms of electron wavefunctions, and attributes the apparent particle-like behav-ior to quantization effects and eigenstates. According toreviewer David Haddon:[39]

Mead has cut the Gordian knot of quantumcomplementarity. He claims that atoms, withtheir neutrons, protons, and electrons, are notparticles at all but pure waves of matter. Meadcites as the gross evidence of the exclusivelywave nature of both light andmatter the discov-ery between 1933 and 1996 of ten examplesof pure wave phenomena, including the ubiqui-tous laser of CD players, the self-propagatingelectrical currents of superconductors, and theBose–Einstein condensate of atoms.

Albert Einstein, who, in his search for a Unified FieldTheory, did not accept wave-particle duality, wrote:[40]

This double nature of radiation (and ofmaterial corpuscles)...has been interpreted byquantum-mechanics in an ingenious and amaz-ingly successful fashion. This interpreta-tion...appears to me as only a temporary wayout...

The many-worlds interpretation (MWI) is sometimespresented as a waves-only theory, including by its orig-inator, Hugh Everett who referred to MWI as “the waveinterpretation”.[41]

The Three Wave Hypothesis of R. Horodecki relates theparticle to wave.[42][43] The hypothesis implies that a mas-sive particle is an intrinsically spatially as well as tempo-rally extended wave phenomenon by a nonlinear law.

7.3 Neither-wave-nor-particle view

It has been argued that there are never exact parti-cles or waves, but only some compromise or interme-diate between them. One consideration is that zero-dimensional mathematical points cannot be observed.Another is that the formal representation of such points,the Kronecker delta function is unphysical, because itcannot be normalized. Parallel arguments apply to purewave states. Roger Penrose states:[44]

“Such 'position states’ are idealized wave-functions in the opposite sense from the mo-mentum states. Whereas the momentum statesare infinitely spread out, the position states areinfinitely concentrated. Neither is normaliz-able [...].”

7.4 Relational approach to wave–particleduality

Relational quantum mechanics is developed which re-gards the detection event as establishing a relationship be-tween the quantized field and the detector. The inherentambiguity associated with applying Heisenberg’s uncer-tainty principle and thus wave–particle duality is subse-quently avoided.[45]

8 Applications

Although it is difficult to draw a line separating wave–particle duality from the rest of quantum mechanics, itis nevertheless possible to list some applications of thisbasic idea.

• Wave–particle duality is exploited in electron mi-croscopy, where the small wavelengths associatedwith the electron can be used to view objects muchsmaller than what is visible using visible light.

• Similarly, neutron diffraction uses neutrons with awavelength of about 0.1 nm, the typical spacing ofatoms in a solid, to determine the structure of solids.

9 See also• Arago spot

• Afshar experiment

10 10 NOTES AND REFERENCES

• Basic concepts of quantum mechanics

• Complementarity (physics)

• Kapitsa–Dirac effect

• Electron wave-packet interference

• Faraday wave

• Hanbury Brown and Twiss effect

• Photon polarization

• Scattering theory

• Wavelet

• Wheeler’s delayed choice experiment

10 Notes and references[1] Harrison, David (2002). “Complementarity and the

Copenhagen Interpretation of QuantumMechanics”. UP-SCALE. Dept. of Physics, U. of Toronto. Retrieved 2008-06-21.

[2] Kumar, Manjit (2011). Quantum: Einstein, Bohr, andthe Great Debate about the Nature of Reality (Reprintedition ed.). W. W. Norton & Company. ISBN 978-0393339888.

[3] Walter Greiner (2001). Quantum Mechanics: An Intro-duction. Springer. ISBN 3-540-67458-6.

[4] R. Eisberg and R. Resnick (1985). Quantum Physics ofAtoms, Molecules, Solids, Nuclei, and Particles (2nd ed.).John Wiley & Sons. pp. 59–60. ISBN 047187373X.“For both large and small wavelengths, both matter andradiation have both particle and wave aspects.... But thewave aspects of their motion become more difficult to ob-serve as their wavelengths become shorter.... For ordi-nary macroscopic particles the mass is so large that themomentum is always sufficiently large to make the deBroglie wavelength small enough to be beyond the rangeof experimental detection, and classical mechanics reignssupreme.”

[5] Nathaniel Page Stites, M.A./M.S. “Light I: Parti-cle or Wave?,” Visionlearning Vol. PHY-1 (3),2005. http://www.visionlearning.com/library/module_viewer.php?mid=132

[6] Young, Thomas (1804). “Bakerian Lecture: Ex-periments and calculations relative to physicaloptics”. Philosophical Transactions of the RoyalSociety 94: 1–16. Bibcode:1804RSPT...94....1Y.doi:10.1098/rstl.1804.0001.

[7] Thomas Young: The Double Slit Experiment

[8] Lamb, Willis E.; Scully, Marlan O. (1968). “The photo-electric effect without photons”.

[9] http://dx.doi.org/10.1119/1.1737397

[10] "light", The Columbia Encyclopedia, Sixth Edition.2001–05.

[11] Buchwald, Jed (1989). The Rise of the Wave Theory ofLight: Optical Theory and Experiment in the Early Nine-teenth Century. Chicago: University of Chicago Press.ISBN 0-226-07886-8. OCLC 18069573 59210058.

[12] Zhang, Q (1996). “Intensity dependence of thephotoelectric effect induced by a circularly polar-ized laser beam”. Physics Letters A 216 (1-5):125. Bibcode:1996PhLA..216..125Z. doi:10.1016/0375-9601(96)00259-9.

[13] Donald H Menzel, "Fundamental formulas of Physics",volume 1, page 153; Gives the de Broglie wavelengths forcomposite particles such as protons and neutrons.

[14] Brian Greene, The Elegant Universe, page 104 “all matterhas a wave-like character”

[15] See this Science Channel production (Season II, EpisodeVI “How Does The Universe Work?"), presented byMorgan Freeman, https://www.youtube.com/watch?v=W9yWv5dqSKk

[16] Bohmian Mechanics, Stanford Encyclopedia of Philoso-phy.

[17] Bell, J. S., “Speakable and Unspeakable in Quantum Me-chanics”, Cambridge: Cambridge University Press, 1987.

[18] Y. Couder, A. Boudaoud, S. Protière, Julien Moukhtar,E. Fort: Walking droplets: a form of wave-particle dualityat macroscopic level? , doi:10.1051/epn/2010101, (PDF)

[19] Estermann, I.; Stern O. (1930). “Beugung vonMolekularstrahlen”. Zeitschrift für Physik 61(1-2): 95–125. Bibcode:1930ZPhy...61...95E.doi:10.1007/BF01340293.

[20] R. Colella, A. W. Overhauser and S. A. Werner, Obser-vation of Gravitationally Induced Quantum Interference,Phys. Rev. Lett. 34, 1472–1474 (1975).

[21] Arndt, Markus; 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.

[22] Juffmann, Thomas et al (25 March 2012). “Real-time single-molecule imaging of quantum interference”.Nature Nanotechnology. Retrieved 27 March 2012.

[23] Quantumnanovienna. “Single molecules in a quantum in-terference movie”. Retrieved 2012-04-21.

[24] Hackermüller, Lucia; Stefan Uttenthaler, KlausHornberger, Elisabeth Reiger, Björn Brezger, An-ton Zeilinger and Markus Arndt (2003). “Thewave nature of biomolecules and fluorofullerenes”.Phys. Rev. Lett. 91 (9): 090408. arXiv:quant-ph/0309016. Bibcode:2003PhRvL..91i0408H.doi:10.1103/PhysRevLett.91.090408. PMID 14525169.

11

[25] Clauser, John F.; S. Li (1994). “Talbot von Lau intere-fometry with cold slow potassium atoms.”. Phys. Rev.A 49 (4): R2213–17. Bibcode:1994PhRvA..49.2213C.doi:10.1103/PhysRevA.49.R2213. PMID 9910609.

[26] Brezger, Björn; Lucia Hackermüller, StefanUttenthaler, Julia Petschinka, Markus Arndtand Anton Zeilinger (2002). “Matter-waveinterferometer for large molecules”. Phys.Rev. Lett. 88 (10): 100404. arXiv:quant-ph/0202158. Bibcode:2002PhRvL..88j0404B.doi:10.1103/PhysRevLett.88.100404. PMID 11909334.

[27] Hornberger, Klaus; Stefan Uttenthaler,BjörnBrezger, Lucia Hackermüller, Markus Arndt andAnton Zeilinger (2003). “Observation of Col-lisional Decoherence in Interferometry”. Phys.Rev. Lett. 90 (16): 160401. arXiv:quant-ph/0303093. Bibcode:2003PhRvL..90p0401H.doi:10.1103/PhysRevLett.90.160401. PMID 12731960.

[28] Hackermüller, Lucia; Klaus Hornberger, BjörnBrezger, Anton Zeilinger and Markus Arndt (2004).“Decoherence of matter waves by thermal emission ofradiation”. Nature 427 (6976): 711–714. arXiv:quant-ph/0402146. Bibcode:2004Natur.427..711H.doi:10.1038/nature02276. PMID 14973478.

[29] Gerlich, Stefan; et al. (2011). “Quantum inter-ference of large organic molecules”. Nature Com-munications 2 (263). Bibcode:2011NatCo...2E.263G.doi:10.1038/ncomms1263. PMC 3104521. PMID21468015.

[30] Eibenberger, S.; Gerlich, S.; Arndt, M.; Mayor, M.;Tüxen, J. (2013). “Matter–wave interference of particlesselected from a molecular library with masses exceeding10 000 amu”. Physical Chemistry Chemical Physics 15(35): 14696–14700. doi:10.1039/c3cp51500a. PMID23900710.

[31] Peter Gabriel Bergmann, The Riddle of Gravitation,Courier Dover Publications, 1993 ISBN 0-486-27378-4online

[32] http://www.youtube.com/watch?v=W9yWv5dqSKk -You Tube video - Yves Couder Explains Wave/ParticleDuality via Silicon Droplets

[33] Y. Couder, E. Fort, Single-Particle Diffraction and Inter-ference at a Macroscopic Scale, PRL 97, 154101 (2006)online

[34] A. Eddi, E. Fort, F.Moisy, Y. Couder,Unpredictable Tun-neling of a Classical Wave–Particle Association, PRL 102,240401 (2009)

[35] E. Fort, A. Eddi, A. Boudaoud, J. Moukhtar, Y. Couder,Path-memory induced quantization of classical orbits,PNAS October 12, 2010 vol. 107 no. 41 17515-17520

[36] http://prl.aps.org/abstract/PRL/v108/i26/e264503 -Level Splitting at Macroscopic Scale

[37] (Buchanan pp. 29–31)

[38] Afshar S.S. et al: Paradox in Wave Particle Duality.Found. Phys. 37, 295 (2007) http://arxiv.org/abs/quant-ph/0702188 arXiv:quant-ph/0702188

[39] David Haddon. “Recovering Rational Science”. Touch-stone. Retrieved 2007-09-12.

[40] Paul Arthur Schilpp, ed, Albert Einstein: Philosopher-Scientist, Open Court (1949), ISBN 0-87548-133-7, p 51.

[41] See section VI(e) of Everett’s thesis: The Theory ofthe Universal Wave Function, in Bryce Seligman De-Witt, R. Neill Graham, eds, The Many-Worlds Interpreta-tion of Quantum Mechanics, Princeton Series in Physics,Princeton University Press (1973), ISBN 0-691-08131-X,pp 3–140.

[42] Horodecki, R. (1981). “De broglie wave andits dual wave”. Phys. Lett. A 87 (3): 95–97.Bibcode:1981PhLA...87...95H. doi:10.1016/0375-9601(81)90571-5.

[43] Horodecki, R. (1983). “Superluminal singular dualwave”. Lett. Novo Cimento 38: 509–511.

[44] Penrose, Roger (2007). The Road to Reality: A Com-plete Guide to the Laws of the Universe. Vintage. p. 521,§21.10. ISBN 978-0-679-77631-4.

[45] http://www.quantum-relativity.org/Quantum-Relativity.pdf. See Q. Zheng and T. Kobayashi, Quantum Optics asa Relativistic Theory of Light; Physics Essays 9 (1996)447. Annual Report, Department of Physics, School ofScience, University of Tokyo (1992) 240.

11 External links• Animation, applications and research linked to thewave-particle duality and other basic quantum phe-nomena (Université Paris Sud)

• H. Nikolic. “Quantum mechanics: Myths andfacts”. arXiv:quant-ph/0609163.

• Young & Geller. “College Physics”.

• B. Crowell. “Light as a Particle” (Web page). Re-trieved December 10, 2006.

• E.H. Carlson, Wave–Particle Duality: Light onProject PHYSNET

• R. Nave. “Wave–Particle Duality” (Web page). Hy-perPhysics. Georgia State University, Departmentof Physics and Astronomy. Retrieved December 12,2005.

• Juffmann, Thomas et al (25 March 2012). “Real-time single-molecule imaging of quantum interfer-ence”. Nature Nanotechnology. Retrieved 21 Jan-uary 2014.

12 12 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

12 Text and image sources, contributors, and licenses

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