 Wave–particle duality

Quantum mechanics
Introduction
Mathematical formulationsFundamental conceptsQuantum state · Wave function
Superposition · Entanglement
Complementarity · Duality
Uncertainty · Measurement
Exclusion · Decoherence
Ehrenfest theorem
Tunnelling · NonlocalityExperimentsDoubleslit · Davisson–Germer
Stern–Gerlach · Bell's inequality
Popper · Schrödinger's cat
Elitzur–Vaidman bomb tester
Quantum eraser
Delayed choice quantum eraser
Wheeler's delayed choiceFormulationsEquationsInterpretationsde Broglie–Bohm
Consciousnesscaused
Consistent histories · Copenhagen
Ensemble · Hidden variables
Manyworlds · Objective collapse
Pondicherry · Quantum logic
Relational · Stochastic
TransactionalScientistsBell · Bohm · Bohr · Born · Bose
de Broglie · Dirac · Ehrenfest
Everett · Feynman · Heisenberg
Jordan · Kramers · von Neumann
Pauli · Planck · Schrödinger
Sommerfeld · Wien · WignerWave–particle duality postulates that all particles exhibit both wave and particle properties. A central concept of quantum mechanics, this duality addresses the inability of classical concepts like "particle" and "wave" to fully describe the behavior of quantumscale objects. Standard interpretations of quantum mechanics explain this paradox as a fundamental property of the Universe, while alternative interpretations explain the duality as an emergent, secondorder consequence of various limitations of the observer. This treatment focuses on explaining the behavior from the perspective of the widely used Copenhagen interpretation, in which wave–particle duality is one aspect of the concept of complementarity, that a phenomenon can be viewed in one way or in another, but not both simultaneously.
The idea of duality originated in a debate over the nature of light and matter that dates back to the 17th century, when competing theories of light were proposed by Christiaan Huygens and Isaac Newton: light was thought either to consist of waves (Huygens) or of particles (Newton). Through the work of Max Planck, Albert Einstein, Louis de Broglie, Arthur Compton, Niels Bohr, and many others, current scientific theory holds that all particles also have a wave nature (and vice versa).^{[1]} This phenomenon has been verified not only for elementary particles, but also for compound particles like atoms and even molecules. In fact, according to traditional formulations of nonrelativistic quantum mechanics, wave–particle duality applies to all objects, even macroscopic ones; but because of their small wavelengths, the wave properties of macroscopic objects cannot be detected.^{[2]}
Contents
Brief history of wave and particle viewpoints
Aristotle was one of the first to publicly hypothesize about the nature of light, proposing that light is a disturbance in the element air. (That is, it is a wavelike phenomenon). On the other hand, Democritus – the original atomist – argued that all things in the universe, including light, are composed of indivisible subcomponents (light being some form of solar atom).^{[3]} At the beginning of the 11th Century, the Arabic scientist Alhazen wrote the first comprehensive treatise on optics; describing refraction, reflection, and the operation of a pinhole lens via rays of light traveling from the point of emission to the eye. He asserted that these rays were composed of particles of light. In 1630, René Descartes popularized and accredited in the West the opposing wave description in his treatise on light, showing that the behavior of light could be recreated by modeling wavelike disturbances in his universal medium ("plenum"). Beginning in 1670 and progressing over three decades, Isaac Newton developed and championed his corpuscular hypothesis, arguing that the perfectly straight lines of reflection demonstrated light's particle nature; only particles could travel in such straight lines. He explained refraction by positing that particles of light accelerated laterally upon entering a denser medium. Around the same time, Newton's contemporaries Robert Hooke and Christian Huygens – and later AugustinJean Fresnel – mathematically refined the wave viewpoint, showing that if light traveled at different speeds in different media (such as water and air), refraction could be easily explained as the mediumdependent propagation of light waves. The resulting Huygens–Fresnel principle was extremely successful at reproducing light's behavior and, subsequently supported by Thomas Young's discovery of doubleslit interference, was the beginning of the end for the particle light camp.^{[4]}
The final blow against corpuscular theory came when James Clerk Maxwell discovered that he could combine four simple equations, which had been previously discovered, along with a slight modification to describe self propagating waves of oscillating electric and magnetic fields. When the propagation speed of these electromagnetic waves was calculated, the speed of light fell out. It quickly became apparent that visible light, ultraviolet light, and infrared light (phenomena thought previously to be unrelated) were all electromagnetic waves of differing frequency. The wave theory had prevailed – or at least it seemed to.
While the 19th century had seen the success of the wave theory at describing light, it had also witnessed the rise of the atomic theory at describing matter. In 1789, Antoine Lavoisier securely differentiated chemistry from alchemy by introducing rigor and precision into his laboratory techniques; allowing him to deduce the conservation of mass and categorize many new chemical elements and compounds. However, the nature of these essential chemical elements remained unknown. In 1799, Joseph Louis Proust advanced chemistry towards the atom by showing that elements combined in definite proportions. This led John Dalton to resurrect Democritus' atom in 1803, when he proposed that elements were invisible sub components; which explained why the varying oxides of metals (e.g. stannous oxide and cassiterite, SnO and SnO_{2} respectively) possess a 1:2 ratio of oxygen to one another. But Dalton and other chemists of the time had not considered that some elements occur in monatomic form (like Helium) and others in diatomic form (like Hydrogen), or that water was H_{2}O, not the simpler and more intuitive HO – thus the atomic weights presented at the time were varied and often incorrect. Additionally, the formation of HO by two parts of hydrogen gas and one part of oxygen gas would require an atom of oxygen to split in half (or two halfatoms of hydrogen to come together). This problem was solved by Amedeo Avogadro, who studied the reacting volumes of gases as they formed liquids and solids. By postulating that equal volumes of elemental gas contain an equal number of atoms, he was able to show that H_{2}O was formed from two parts H_{2} and one part O_{2}. By discovering diatomic gases, Avogadro completed the basic atomic theory, allowing the correct molecular formulae of most known compounds – as well as the correct weights of atoms – to be deduced and categorized in a consistent manner. The final stroke in classical atomic theory came when Dimitri Mendeleev saw an order in recurring chemical properties, and created a table presenting the elements in unprecedented order and symmetry. But there were holes in Mendeleev's table, with no element to fill them in. His critics initially cited this as a fatal flaw, but were silenced when new elements were discovered that perfectly fit into these holes. The success of the periodic table effectively converted any remaining opposition to atomic theory; even though no single atom had ever been observed in the laboratory, chemistry was now an atomic science.
The turn of the 19th century and the paradigm shift
Particles of electricity?
At the close of the 19th century, the reductionism of atomic theory began to advance into the atom itself; determining, through physics, the nature of the atom and the operation of chemical reactions. Electricity, first thought to be a fluid, was now understood to consist of particles called electrons. This was first demonstrated by J. J. Thomson in 1897 when, using a cathode ray tube, he found that an electrical charge would travel across a vacuum (which would possess infinite resistance in classical theory). Since the vacuum offered no medium for an electric fluid to travel, this discovery could only be explained via a particle carrying a negative charge and moving through the vacuum. This electron flew in the face of classical electrodynamics, which had successfully treated electricity as a fluid for many years (leading to the invention of batteries, electric motors, dynamos, and arc lamps). More importantly, the intimate relation between electric charge and electromagnetism had been well documented following the discoveries of Michael Faraday and Clerk Maxwell. Since electromagnetism was known to be a wave generated by a changing electric or magnetic field (a continuous, wavelike entity itself) an atomic/particle description of electricity and charge was a non sequitur. And classical electrodynamics was not the only classical theory rendered incomplete.
Radiation quantization
Blackbody radiation, the emission of electromagnetic energy due to an object's heat, could not be explained from classical arguments alone. The equipartition theorem of classical mechanics, the basis of all classical thermodynamic theories, stated that an object's energy is partitioned equally among the object's vibrational modes. This worked well when describing thermal objects, whose vibrational modes were defined as the speeds of their constituent atoms, and the speed distribution derived from egalitarian partitioning of these vibrational modes closely matched experimental results. Speeds much higher than the average speed were suppressed by the fact that kinetic energy is quadratic – doubling the speed requires four times the energy – thus the number of atoms occupying high energy modes (high speeds) quickly drops off because the constant, equal partition can excite successively fewer atoms. Low speed modes would ostensibly dominate the distribution, since low speed modes would require ever less energy, and prima facie a zerospeed mode would require zero energy and its energy partition would contain an infinite number of atoms. But this would only occur in the absence of atomic interaction; when collisions are allowed, the low speed modes are immediately suppressed by jostling from the higher energy atoms, exciting them to higher energy modes. An equilibrium is swiftly reached where most atoms occupy a speed proportional to the temperature of the object (thus defining temperature as the average kinetic energy of the object).
But applying the same reasoning to the electromagnetic emission of such a thermal object was not so successful. It had been long known that thermal objects emit light. Hot metal glows red, and upon further heating, white (this is the underlying principle of the incandescent bulb). Since light was known to be waves of electromagnetism, physicists hoped to describe this emission via classical laws. This became known as the black body problem. Since the equipartition theorem worked so well in describing the vibrational modes of the thermal object itself, it was trivial to assume that it would perform equally well in describing the radiative emission of such objects. But a problem quickly arose when determining the vibrational modes of light. To simplify the problem (by limiting the vibrational modes) a lowest allowable wavelength was defined by placing the thermal object in a cavity. Any electromagnetic mode at equilibrium (i.e. any standing wave) could only exist if it used the walls of the cavities as nodes. Thus there were no waves/modes with a wavelength larger than twice the length (L) of the cavity.
The first few allowable modes would therefore have wavelengths of : 2L, L, 2L/3, L/2, etc. (each successive wavelength adding one node to the wave). However, while the wavelength could never exceed 2L, there was no such limit on decreasing the wavelength, and adding nodes to reduce the wavelength could proceed ad infinitum. Suddenly it became apparent that the short wavelength modes completely dominated the distribution, since ever shorter wavelength modes could be crammed into the cavity. If each mode received an equal partition of energy, the short wavelength modes would consume all the energy. This became clear when plotting the Rayleigh–Jeans law which, while correctly predicting the intensity of long wavelength emissions, predicted infinite total energy as the intensity diverges to infinity for short wavelengths. This became known as the ultraviolet catastrophe.
The solution arrived in 1900 when Max Planck hypothesized that the frequency of light emitted by the black body depended on the frequency of the oscillator that emitted it, and the energy of these oscillators increased linearly with frequency (according to his constant h, where E = hν). This was not an unsound proposal considering that macroscopic oscillators operate similarly: when studying five simple harmonic oscillators of equal amplitude but different frequency, the oscillator with the highest frequency possesses the highest energy (though this relationship is not linear like Planck's). By demanding that highfrequency light must be emitted by an oscillator of equal frequency, and further requiring that this oscillator occupy higher energy than one of a lesser frequency, Planck avoided any catastrophe; giving an equal partition to highfrequency oscillators produced successively fewer oscillators and less emitted light. And as in the Maxwell–Boltzmann distribution, the lowfrequency, lowenergy oscillators were suppressed by the onslaught of thermal jiggling from higher energy oscillators, which necessarily increased their energy and frequency.
The most revolutionary aspect of Planck's treatment of the black body is that it inherently relies on an integer number of oscillators in thermal equilibrium with the electromagnetic field. These oscillators give their entire energy to the electromagnetic field, creating a quantum of light, as often as they are excited by the electromagnetic field, absorbing a quantum of light and beginning to oscillate at the corresponding frequency. Planck had intentionally created an atomic theory of the black body, but had unintentionally generated an atomic theory of light, where the black body never generates quanta of light at a given frequency with an energy less than hν. However, once realizing that he had quantized the electromagnetic field, he denounced particles of light as a limitation of his approximation, not a property of reality.
The photoelectric effect illuminated
Yet while Planck had solved the ultraviolet catastrophe by using atoms and a quantized electromagnetic field, most physicists immediately agreed that Planck's "light quanta" were unavoidable flaws in his model. A more complete derivation of black body radiation would produce a fully continuous, fully wavelike electromagnetic field with no quantization. However, in 1905 Albert Einstein took Planck's black body model in itself and saw a wonderful solution to another outstanding problem of the day: the photoelectric effect. Ever since the discovery of electrons eight years previously, electrons had been the thing to study in physics laboratories worldwide. Nikola Tesla discovered in 1901 that when a metal was illuminated by highfrequency light (e.g. ultraviolet light), electrons were ejected from the metal at high energy. This work was based on the previous knowledge that light incident upon metals produces a current, but Tesla was the first to describe it as a particle phenomenon.
The following year, Philipp Lenard discovered that (within the range of the experimental parameters he was using) the energy of these ejected electrons did not depend on the intensity of the incoming light, but on its frequency. So if one shines a little lowfrequency light upon a metal, a few low energy electrons are ejected. If one now shines a very intense beam of lowfrequency light upon the same metal, a whole slew of electrons are ejected; however they possess the same low energy, there are merely more of them. In order to get high energy electrons, one must illuminate the metal with highfrequency light. The more light there is, the more electrons are ejected. Like blackbody radiation, this was at odds with a theory invoking continuous transfer of energy between radiation and matter. However, it can still be explained using a fully classical description of light, as long as matter is quantum mechanical in nature.^{[5]}
If one used Planck's energy quanta, and demanded that electromagnetic radiation at a given frequency could only transfer energy to matter in integer multiples of an energy quantum hν, then the photoelectric effect could be explained very simply. Lowfrequency light only ejects lowenergy electrons because each electron is excited by the absorption of a single photon. Increasing the intensity of the lowfrequency light (increasing the number of photons) only increases the number of excited electrons, not their energy, because the energy of each photon remains low. Only by increasing the frequency of the light, and thus increasing the energy of the photons, can one eject electrons with higher energy. Thus, using Planck's constant h to determine the energy of the photons based upon their frequency, the energy of ejected electrons should also increase linearly with frequency; the gradient of the line being Planck's constant. These results were not confirmed until 1915, when Robert Andrews Millikan, who had previously determined the charge of the electron, produced experimental results in perfect accord with Einstein's predictions. While the energy of ejected electrons reflected Planck's constant, the existence of photons was not explicitly proven until the discovery of the photon antibunching effect, of which a modern experiment can be performed in undergraduatelevel labs.^{[6]} This phenomenon could only be explained via photons, and not through any semiclassical theory (which could alternatively explain the photoelectric effect). When Einstein received his Nobel Prize in 1921, it was not for his more difficult and mathematically laborious special and general relativity, but for the simple, yet totally revolutionary, suggestion of quantized light. Einstein's "light quanta" would not be called photons until 1925, but even in 1905 they represented the quintessential example of wave–particle duality. Electromagnetic radiation propagates following linear wave equations, but can only be emitted or absorbed as discrete elements, thus acting as a wave and a particle simultaneously.
Developmental milestones
Huygens and Newton
The earliest comprehensive theory of light was advanced by Christiaan Huygens, who proposed a wave theory of light, and in particular demonstrated how waves might interfere to form a wavefront, propagating in a straight line. However, the theory had difficulties in other matters, and was soon overshadowed by Isaac Newton's corpuscular theory of light. That is, Newton proposed that light consisted of small particles, with which he could easily explain the phenomenon of reflection. With considerably more difficulty, he could also explain refraction through a lens, and the splitting of sunlight into a rainbow by a prism. Newton's particle viewpoint went essentially unchallenged for over a century.^{[7]}
Young, Fresnel, and Maxwell
In the early 19th century, the doubleslit experiments by Young and Fresnel provided evidence for Huygens' wave theories. The doubleslit experiments showed that when light is sent through a grid, a characteristic interference pattern is observed, very similar to the pattern resulting from the interference of water waves; the wavelength of light can be computed from such patterns. The wave view did not immediately displace the ray and particle view, but began to dominate scientific thinking about light in the mid 19th century, since it could explain polarization phenomena that the alternatives could not.^{[8]}
In the late 19th century, James Clerk Maxwell explained light as the propagation of electromagnetic waves according to the Maxwell equations. These equations were verified by experiment by Heinrich Hertz in 1887, and the wave theory became widely accepted.
Planck's formula for blackbody radiation
Main article: Planck's lawIn 1901, Max Planck published an analysis that succeeded in reproducing the observed spectrum of light emitted by a glowing object. To accomplish this, Planck had to make an ad hoc mathematical assumption of quantized energy of the oscillators (atoms of the black body) that emit radiation. It was Einstein who later proposed that it is the electromagnetic radiation itself that is quantized, and not the energy of radiating atoms.
Einstein's explanation of the photoelectric effect
Main article: Photoelectric effectIn 1905, Albert Einstein provided an explanation of the photoelectric effect, a hitherto troubling experiment that the wave theory of light seemed incapable of explaining. He did so by postulating the existence of photons, quanta of light energy with particulate qualities.
In the photoelectric effect, it was observed that shining a light on certain metals would lead to an electric current in a circuit. Presumably, the light was knocking electrons out of the metal, causing current to flow. However, using the case of potassium as an example, it was also observed that while a dim blue light was enough to cause a current, even the strongest, brightest red light available with the technology of the time caused no current at all. According to the classical theory of light and matter, the strength or amplitude of a light wave was in proportion to its brightness: a bright light should have been easily strong enough to create a large current. Yet, oddly, this was not so.
Einstein explained this conundrum by postulating that the electrons can receive energy from electromagnetic field only in discrete portions (quanta that were called photons): an amount of energy E that was related to the frequency f of the light by
where h is Planck's constant (6.626 × 10^{−34} J seconds). Only photons of a high enough frequency (above a certain threshold value) could knock an electron free. For example, photons of blue light had sufficient energy to free an electron from the metal, but photons of red light did not. More intense light above the threshold frequency could release more electrons, but no amount of light (using technology available at the time) below the threshold frequency could release an electron. To "violate" this law would require extremely high intensity lasers which had not yet been invented. Intensitydependent phenomena have now been studied in detail with such lasers.^{[9]}
Einstein was awarded the Nobel Prize in Physics in 1921 for his discovery of the law of the photoelectric effect.
De Broglie's wavelength
Main article: Matter waveIn 1924, LouisVictor de Broglie formulated the de Broglie hypothesis, claiming that all matter,^{[10]}^{[11]} not just light, has a wavelike nature; he related wavelength (denoted as λ), and momentum (denoted as p):
This is a generalization of Einstein's equation above, since the momentum of a photon is given by p = and the wavelength (in a vacuum) by λ = , where c is the speed of light in vacuum.
De Broglie's formula was confirmed three years later for electrons (which differ from photons in having a rest mass) with the observation of electron diffraction in two independent experiments. At the University of Aberdeen, George Paget Thomson passed a beam of electrons through a thin metal film and observed the predicted interference patterns. At Bell Labs Clinton Joseph Davisson and Lester Halbert Germer guided their beam through a crystalline grid.
De Broglie was awarded the Nobel Prize for Physics in 1929 for his hypothesis. Thomson and Davisson shared the Nobel Prize for Physics in 1937 for their experimental work.
Heisenberg's uncertainty principle
Main article: Heisenberg uncertainty principleIn his work on formulating quantum mechanics, Werner Heisenberg postulated his uncertainty principle, which states:
where
 Δ here indicates standard deviation, a measure of spread or uncertainty;
 x and p are a particle's position and linear momentum respectively.
 is the reduced Planck's constant (Planck's constant divided by 2π).
Heisenberg originally explained this as a consequence of the process of measuring: Measuring position accurately would disturb momentum and viceversa, offering an example (the "gammaray microscope") that depended crucially 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 principle, extending the Copenhagen interpretation first put forward by Bohr and Heisenberg, depends even more centrally on the wave nature of a particle: Just as it is nonsensical 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 have perfectly precise momenta (which corresponds to the inverse of wavelength). Moreover, when position is relatively well defined, the wave is pulselike and has a very illdefined wavelength (and thus momentum). And conversely, when momentum (and thus wavelength) is relatively well defined, the wave looks long and sinusoidal, and therefore it has a very illdefined position.
De Broglie himself had proposed a pilot wave construct to explain the observed wave–particle duality. In this view, each particle has a welldefined position and momentum, but is guided by a wave function derived from Schrödinger's equation. The pilot wave theory was initially rejected because it generated nonlocal effects when applied to systems involving more than one particle. Nonlocality, however, soon became established as an integral feature of quantum theory (see EPR paradox), and David Bohm extended de Broglie's model to explicitly include it. In the resulting representation, also called the de Broglie–Bohm theory or Bohmian mechanics,^{[12]}, the wave–particle duality is not a property of matter itself, but an appearance generated by the particle's motion subject to a guiding equation or quantum potential.
Wave behavior of large objects
Since the demonstrations of wavelike properties in photons and electrons, similar experiments have been conducted with neutrons and protons. Among the most famous experiments are those of Estermann and Otto Stern in 1929.^{[13]} Authors of similar recent experiments with atoms and molecules, described below, claim that these larger particles also act like waves.
A dramatic series of experiments emphasizing the action of gravity in relation to wave–particle duality were conducted in the 1970s using the neutron interferometer.^{[14]} Neutrons, one of the components of the atomic nucleus, provide much of the mass of a nucleus and thus of ordinary matter. In the neutron interferometer, they act as quantummechanical waves directly subject to the force of gravity. While the results were not surprising since gravity was known to act on everything, including light (see tests of general relativity and the PoundRebka falling photon experiment), the selfinterference of the quantum mechanical wave of a massive fermion in a gravitational field had never been experimentally confirmed before.
In 1999, the diffraction of C_{60} fullerenes by researchers from the University of Vienna was reported.^{[15]} Fullerenes are comparatively large and massive objects, having an atomic mass of about 720 u. The de Broglie wavelength is 2.5 pm, whereas the diameter of the molecule is about 1 nm, about 400 times larger. As of 2005, this is the largest object for which quantummechanical wavelike properties have been directly observed in farfield diffraction.
In 2003 the Vienna group also demonstrated the wave nature of tetraphenylporphyrin^{[16]}—a flat biodye with an extension of about 2 nm and a mass of 614 u. For this demonstration they employed a nearfield Talbot Lau interferometer.^{[17]}^{[18]} In the same interferometer they also found interference fringes for C_{60}F_{48.}, a fluorinated buckyball with a mass of about 1600 u, composed of 108 atoms.^{[16]} Large molecules are already so complex that they give experimental access to some aspects of the quantumclassical interface, i.e. to certain decoherence mechanisms.^{[19]}^{[20]}
Whether objects heavier than the Planck mass (about the weight of a large bacterium) have a de Broglie wavelength is theoretically unclear and experimentally unreachable; above the Planck mass a particle's Compton wavelength would be smaller than the Planck length and its own Schwarzschild radius, a scale at which current theories of physics may break down or need to be replaced by more general ones.^{[21]}
Recently Couder, Fort et al. showed that we can use macroscopic oil droplets on vibrating surface as a model of waveparticle duality  localized droplet creates periodical waves around and interaction with them leads to quantumlike phenomena: interference in doubleslit experiment,^{[22]} unpredictable tunneling^{[23]} (depending in complicated way on practically hidden state of field) and orbit quantization^{[24]} (that particle has to 'find a resonance' with field perturbations it creates  after one orbit, its internal phase has to return to the initial state).
Treatment in modern quantum mechanics
Wave–particle duality is deeply embedded into the foundations of quantum mechanics, so well that modern practitioners rarely discuss it as such. In the formalism of the theory, all the information about a particle is encoded in its wave function, a complex valued function roughly analogous to the amplitude of a wave at each point in space. This function evolves according to a differential equation (generically called the Schrödinger equation), and this equation gives rise^{[clarification needed]} to wavelike phenomena such as interference and diffraction.
The particlelike behavior is most evident due to phenomena associated with measurement in quantum mechanics. Upon measuring the location of the particle, the wavefunction will randomly "collapse," or rather, "decoheres" to a sharply peaked function at some location, with the likelihood of any particular location equal to the squared amplitude of the wavefunction there.^{[clarification needed]} The measurement will return a welldefined position, (subject to uncertainty), a property traditionally associated with particles.
Although this picture is somewhat simplified (to the nonrelativistic case), it is adequate^{[citation needed]} to capture the essence of current thinking on the phenomena historically called "wave–particle duality".^{[clarification needed]} (See also: Particle in a box, Mathematical formulation of quantum mechanics.)
Alternative views
Particleonly view
The pilot wave model, originally developed by Louis de Broglie and further developed by David Bohm into the hidden variable theory proposes that there is no duality, but rather particles are guided, in a deterministic fashion, by a pilot wave (or its "quantum potential") which will direct them to areas of constructive interference in preference to areas of destructive interference. This idea is held by a significant minority within the physics community.^{[25]}
At least one physicist considers the “waveduality” a misnomer, as L. Ballentine, Quantum Mechanics, A Modern Development, p. 4, explains:
When first discovered, particle diffraction was a source of great puzzlement. Are "particles" really "waves?" In the early experiments, the diffraction patterns were detected holistically by means of a photographic plate, which could not detect individual particles. As a result, the notion grew that particle and wave properties were mutually incompatible, or complementary, in the sense that different measurement apparatuses would be required to observe them. That idea, however, was only an unfortunate generalization from a technological limitation. Today it is possible to detect the arrival of individual electrons, and to see the diffraction pattern emerge as a statistical pattern made up of many small spots (Tonomura et al., 1989). Evidently, quantum particles are indeed particles, but whose behaviour is very different from classical physics would have us to expect.
Afshar's^{[26]} experiment (2007) has demonstrated that it is possible to simultaneously observe both wave and particle properties of photons. Biddulph (2010)^{[27]} has explained this by applying techniques from deterministic chaos to nonchaotic systems, in particular a computable version of Palmer's Universal Invariant Set proposition^{[28]} (2009), which allows the apparent weirdness of quantum phenomena to be explained as artefacts of the quantum apparatus not a fundamental property of nature. Waves are shown to be the only means of describing motion, since smooth motion on a continuum is impossible. If a particle visits every point on its trajectory then the motion is an algorithm for each point. Turing^{[29]} has shown that almost all numbers are noncomputable, which means that there is no possible algorithm, so the set of points on a trajectory is sparse. This implies that motion is either jerky or wavelike. By removing the need to load the particle with the properties of space and time, a fully deterministic, local and causal description of quantum phenomena is possible by use of a simple dynamical operator on a Universal Invariant Set.
Waveonly view
At least one scientist proposes that the duality can be replaced by a "waveonly" view. Carver Mead's Collective Electrodynamics: Quantum Foundations of Electromagnetism (2000) analyzes the behavior of electrons and photons purely in terms of electron wave functions, and attributes the apparent particlelike behavior to quantization effects and eigenstates. According to reviewer David Haddon:^{[30]}
Mead has cut the Gordian knot of quantum complementarity. He claims that atoms, with their neutrons, protons, and electrons, are not particles at all but pure waves of matter. Mead cites as the gross evidence of the exclusively wave nature of both light and matter the discovery between 1933 and 1996 of ten examples of pure wave phenomena, including the ubiquitous laser of CD players, the selfpropagating electrical currents of superconductors, and the Bose–Einstein condensate of atoms.
Albert Einstein, who, in his search for a Unified Field Theory, did not accept waveparticle duality, wrote:^{[31]}
This double nature of radiation (and of material corpuscles)...has been interpreted by quantummechanics in an ingenious and amazingly successful fashion. This interpretation...appears to me as only a temporary way out...
And theoretical physicist Mendel Sachs, who completed Einstein's unified field theory, writes:^{[32]}
Instead, one has a single, holistic continuum, wherein what were formerly called discrete, separable particles of matter are instead the infinite number of distinguishable, though correlated manifestations of this continuum, that in principle is the universe. Hence, waveparticle dualism, which is foundational for the quantum theory, is replaced by wave (continuous field) monism.
The manyworlds interpretation (MWI) is sometimes presented as a wavesonly theory, including by its originator, Hugh Everett who referred to MWI as "the wave interpretation".^{[33]}
The Three Wave Hypothesis of R. Horodecki relates the particle to wave.^{[34]}^{[35]} The hypothesis implies that a massive particle is an intrinsically spatially as well as temporally extended wave phenomenon by a nonlinear law. According to M. I. Sanduk this hypothesis is related to a hypothetical bevel gear model.^{[36]} Then both concepts of particle and wave may be attributed to an observation problem of the gear.^{[37]}
Neitherwavenorparticle view
It has been argued that there are never exact particles or waves, but only some compromise or intermediate 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 it cannot be renormalized. Parallel arguments apply to pure wave states.
"Such positions states are idealised wavefunctions [..] Whereas the momentum states are infinitely spread out, the position states are infinitely concentrated. Neither is normaliseable[..]"
^{[38]}
Relational approach to wave–particle duality
Relational quantum mechanics is developed which regards the detection event as establishing a relationship between the quantized field and the detector. The inherent ambiguity associated with applying Heisenberg's uncertainty principle and thus wave–particle duality is subsequently avoided.^{[39]}
Applications
Although it is difficult to draw a line separating wave–particle duality from the rest of quantum mechanics, it is nevertheless possible to list some applications of this basic idea.
 Wave–particle duality is exploited in electron microscopy, where the small wavelengths associated with the electron can be used to view objects much smaller than what is visible using visible light.
 Similarly, neutron diffraction uses neutrons with a wavelength of about 0.1 nm, the typical spacing of atoms in a solid, to determine the structure of solids.
See also
 Arago spot
 Afshar experiment
 Basic concepts of quantum mechanics
 Electron wavepacket interference
 Hanbury Brown and Twiss effect
 Photon polarization
 Scattering theory
 Wavelet
 Wheeler's delayed choice experiment
Notes and references
^ 39. Noel, David. The Photon Hoop Model for light. http://www.aoi.com.au/bcw/PhotonHoop/.
 ^ Walter Greiner (2001). Quantum Mechanics: An Introduction. Springer. ISBN 3540674586. http://books.google.com/?id=7qCMUfwoQcAC&pg=PA29&dq=waveparticle+allparticles.
 ^ R. Eisberg and R. Resnick (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd ed.). John Wiley & Sons. pp. 59–60. ISBN 047187373X. "For both large and small wavelengths, both matter and radiation have both particle and wave aspects.... But the wave aspects of their motion become more difficult to observe as their wavelengths become shorter.... For ordinary macroscopic particles the mass is so large that the momentum is always sufficiently large to make the de Broglie wavelength small enough to be beyond the range of experimental detection, and classical mechanics reigns supreme."
 ^ Nathaniel Page Stites, M.A./M.S. "Light I: Particle or Wave?," Visionlearning Vol. PHY1 (3), 2005. http://www.visionlearning.com/library/module_viewer.php?mid=132
 ^ Thomas Young: The Double Slit Experiment
 ^ Lamb, Willis E.; Scully, Marlan O. (1968). "The photoelectric effect without photons". http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19680009569_1968009569.pdf.
 ^ http://www.hep.princeton.edu/~mcdonald/examples/QM/thorn_ajp_72_1210_04.pdf
 ^ "light", The Columbia Encyclopedia, Sixth Edition. 2001–05.
 ^ Buchwald, Jed (1989). The Rise of the Wave Theory of Light: Optical Theory and Experiment in the Early Nineteenth Century. Chicago: University of Chicago Press. ISBN 0226078868. OCLC 59210058 18069573 59210058
 ^ Zhang, Q (1996). "Intensity dependence of the photoelectric effect induced by a circularly polarized laser beam". Physics Letters A 216 (15): 125. Bibcode 1996PhLA..216..125Z. doi:10.1016/03759601(96)002599.
 ^ Donald H Menzel, "Fundamental formulas of Physics", volume 1, page 153; Gives the de Broglie wavelengths for composite particles such as protons and neutrons.
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External links
 H. Nikolic. "Quantum mechanics: Myths and facts". arXiv:quantph/0609163.
 Young & Geller. "College Physics". http://wps.aw.com/aw_young_physics_11.
 B. Crowell. "Light as a Particle" (Web page). http://www.lightandmatter.com/html_books/6mr/ch03/ch03.html. Retrieved December 10, 2006.
 E.H. Carlson, Wave–Particle Duality: Light on Project PHYSNET
 R. Nave. "Wave–Particle Duality" (Web page). HyperPhysics. Georgia State University, Department of Physics and Astronomy. http://hyperphysics.phyastr.gsu.edu/hbase/mod1.html. Retrieved December 12, 2005.
 Markus Arndt (2006). "Interferometry and decoherence experiments with large molecules" (Web page). University of Vienna. http://www.quantum.at/typo/index.php?id=177&tx_jppageteaser_pi1=35. Retrieved May 6, 2006.^{[dead link]}
Categories: Foundational quantum physics
 Duality theories
 Dichotomies
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