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| {{Quantum mechanics|cTopic=Fundamental concepts}}
| | Hello, I'm Barney, a 27 year old from Webster Springs, United States.<br>My hobbies include (but are not limited to) Petal collecting and pressing, Skateboarding and watching Arrested Development.<br><br>My blog - [http://Sexhave.com/index.php?do=/blog/1266/fifa-15-coin-generator/ how to get free fifa 15 coins] |
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| '''Wave–particle duality''' is a theory that proposes that all [[matter]] exhibits the properties of not only [[particle]]s, which have mass, but also waves, which transfer energy. A central concept of [[quantum mechanics]], this duality addresses the inability of classical concepts like "particle" and "wave" to fully describe the behavior of quantum-scale 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, second-order 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 serves as one aspect of the concept of [[complementarity (physics)|complementarity]], that one can view phenomena in one way or in another, but not both simultaneously.<ref name=Kumar2011>{{cite book | last =Kumar | first =Manjit | title =Quantum: Einstein, Bohr, and the Great Debate about the Nature of Reality
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| | publisher =W. W. Norton & Company | edition =Reprint edition | year =2011 | isbn =978-0393339888}}</ref>{{rp|242, 375–376}}
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| ==Origin of theory==
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| The idea of duality originated in a debate over the nature of [[light]] and [[matter]] that dates back to the 17th century, when [[Christiaan Huygens]] and [[Isaac Newton]] proposed competing theories of light: 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).<ref>{{cite book | title = Quantum Mechanics: An Introduction | author = Walter Greiner | publisher = Springer | year = 2001 | isbn = 3-540-67458-6 | url = http://books.google.com/?id=7qCMUfwoQcAC&pg=PA29&dq=wave-particle+all-particles }}</ref> This phenomenon has been verified not only for elementary particles, but also for compound particles like atoms and even molecules. For [[macroscopic]] particles, because of their extremely short wavelengths, wave properties usually cannot be detected.<ref>{{cite book|title=Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles|author=R. Eisberg and R. Resnick|publisher=John Wiley & Sons|edition=2nd|year=1985|isbn=047187373X|pages=59–60|quote=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.}}</ref>
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| ==Brief history of wave and particle viewpoints==
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| [[Aristotle]] was one of the first to publicly hypothesize about the nature of light, proposing that light is a disturbance in the element [[Classical element#Classical elements in Greece|air]] (that is, it is a wave-like phenomenon). On the other hand, [[Democritus]]—the original [[atomic theory|''atomist'']]—argued that all things in the universe, including light, are composed of indivisible sub-components (light being some form of solar atom).<ref>Nathaniel Page Stites, M.A./M.S. "Light I: Particle or Wave?," Visionlearning Vol. PHY-1 (3), 2005.
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| http://www.visionlearning.com/library/module_viewer.php?mid=132</ref> At the beginning of the 11th Century, the Arabic scientist [[Alhazen]] wrote the first comprehensive [[Book of Optics|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 the opposing wave description in his [[The World (Descartes)|treatise on light]], showing that the behavior of light could be re-created by modeling wave-like disturbances in a universal medium ("plenum"). Beginning in 1670 and progressing over three decades, [[Isaac Newton]] developed and championed his [[Opticks|corpuscular hypothesis]], arguing that the perfectly straight lines of [[Reflection (physics)|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 [[Christiaan Huygens]]—and later [[Augustin-Jean 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 medium-dependent propagation of light waves. The resulting [[Huygens–Fresnel principle]] was extremely successful at reproducing light's behavior and, subsequently supported by [[Thomas Young (scientist)|Thomas Young]]'s discovery of [[double-slit]] interference, was the beginning of the end for the particle light camp.<ref>[http://www.juliantrubin.com/bigten/youngdoubleslit.html Thomas Young: The Double Slit Experiment]</ref>
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| [[Image:Young Diffraction.png|right|thumb|200px|Thomas Young's sketch of two-slit diffraction of waves, 1803]]
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| The final blow against corpuscular theory came when [[James Clerk Maxwell]] discovered that he could combine [[Maxwell's equations|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.
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| 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 [[law of definite proportions|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<sub>2</sub> 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<sub>2</sub>O, not the simpler and more intuitive HO—thus the [[atomic weight]]s 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 half-atoms 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 [[Avogadro's law|postulating]] that equal volumes of elemental gas contain an equal number of atoms, he was able to show that H<sub>2</sub>O was formed from two parts H<sub>2</sub> and one part O<sub>2</sub>. 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 [[Mendeleev|Dimitri Mendeleev]] saw an order in [[period (periodic table)|recurring chemical properties]], and created a [[periodic table|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.
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| [[File:Wave-particle duality.ogv|thumb|upright=1.5|center|Animation showing the wave-particle duality with a double slit experiment and effect of an observer - (increase size to see explanations in the video itself)]]
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| | caption1 = Particle impacts make visible the interference pattern of waves.
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| | caption2 = A quantum particle is represented by a wave packet.
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| | image3 = Interference of a quantum particle with itself.gif
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| | caption3 = Interference of a quantum particle with itself.
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| == Turn of the 20th century and the paradigm shift ==
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| ===Particles of electricity===
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| 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 [[electron]]s. 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 [[Battery (electricity)|batteries]], [[electric motors]], [[dynamo]]s, and [[arc lamps]]). More importantly, the intimate relation between electric charge and electromagnetism had been well documented following the discoveries of [[Michael Faraday]] and [[James Clerk Maxwell]]. Since electromagnetism was ''known'' to be a wave generated by a changing electric or magnetic ''field'' (a continuous, wave-like entity itself) an atomic/particle description of electricity and charge was a [[non sequitur (logic)|non sequitur]]. Furthermore, classical electrodynamics was not the only classical theory rendered incomplete.
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| ===Radiation quantization===
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| [[Black-body 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 [[Normal mode|modes]]. This worked well when describing thermal objects, whose vibrational modes were defined as the speeds of their constituent atoms, and the [[Maxwell–Boltzmann distribution|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 zero-speed 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).
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| 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 longest 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 [[Node (physics)|nodes]]. Thus there were no waves/modes with a wavelength larger than twice the length (''L'') of the cavity.
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| [[Image:Waves in Box.JPG|right|thumb|200px|Standing waves in a cavity]]
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| The first few allowable modes would therefore have wavelengths of : 2''L'', ''L'', 2''L''/3, ''L''/2, etc. (each successive wavelength adding one node to the wave). However, while the wavelength could never exceed 2''L'', 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]].
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| 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 [[Planck's constant|his constant]] ''h'', where E = hν). This was not an unsound proposal considering that macroscopic oscillators operate similarly: when studying five [[simple harmonic oscillator]]s 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 high-frequency 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 high-frequency oscillators produced successively fewer oscillators and less emitted light. And as in the [[Maxwell–Boltzmann distribution]], the low-frequency, low-energy oscillators were suppressed by the onslaught of thermal jiggling from higher energy oscillators, which necessarily increased their energy and frequency.
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| 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.
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| ===Photoelectric effect illuminated===
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| 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 wave-like 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]], the phenomenon where electrons are emitted from atoms when they absorb energy from light. Ever since the discovery of electrons eight years previously, electrons had been ''the'' thing to study in physics laboratories worldwide.
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| In 1902 [[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 low-frequency light upon a metal, a few low energy electrons are ejected. If one now shines a very intense beam of low-frequency 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 high-frequency 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.<ref>{{cite news|first = Willis E.|last = Lamb|author-link = Willis Lamb|last2 = Scully|first2 = Marlan O.|title = The photoelectric effect without photons|url = http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19680009569_1968009569.pdf|year = 1968}}</ref>
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| 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. Low-frequency light only ejects low-energy electrons because each electron is excited by the absorption of a single photon. Increasing the intensity of the low-frequency 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 undergraduate-level labs.<ref>http://dx.doi.org/10.1119/1.1737397</ref> This phenomenon could only be explained via photons, and not through any semi-classical 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 relativity|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.
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| == Developmental milestones ==
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| ===Huygens and Newton===
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| The earliest comprehensive theory of [[light]] was advanced by [[Christiaan Huygens]], who proposed a [[Huygens–Fresnel principle|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 (physics)|reflection]]. With considerably more difficulty, he could also explain [[refraction]] through a [[lens (optics)|lens]], and the splitting of sunlight into a [[rainbow]] by a [[dispersive prism|prism]]. Newton's particle viewpoint went essentially unchallenged for over a century.<ref>"[http://www.bartleby.com/65/li/light.html light]", The Columbia Encyclopedia, Sixth Edition. 2001–05.</ref>
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| ===Young, Fresnel, and Maxwell===
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| In the early 19th century, the [[double-slit experiment]]s by [[Thomas Young (scientist)|Young]] and [[Augustin-Jean Fresnel|Fresnel]] provided [[scientific evidence|evidence]] for Huygens' wave theories. The double-slit experiments showed that when light is sent through a grid, a characteristic [[Interference (wave propagation)|interference pattern]] is observed, very similar to the pattern resulting from the interference of [[ripple tank|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.<ref>{{Cite book | last=Buchwald | first=Jed | author-link= | year=1989 | publication-date=1989 | title=The Rise of the Wave Theory of Light: Optical Theory and Experiment in the Early Nineteenth Century | edition= | place= | publication-place=Chicago | publisher=University of Chicago Press | isbn=0-226-07886-8 | oclc=18069573 59210058 | postscript=<!--None-->}}</ref>
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| In the late 19th century, [[James Clerk Maxwell]] explained light as the propagation of [[electromagnetic wave]]s according to the [[Maxwell equations]]. These equations were verified by experiment by [[Heinrich Hertz]] in 1887, and the wave theory became widely accepted.
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| ===Planck's formula for black-body radiation===
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| {{Main|Planck's law}}
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| In 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.
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| === Einstein's explanation of the photoelectric effect ===
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| {{Main|Photoelectric effect}}
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| [[Image:Photoelectric effect.svg|thumb|right|222px|The photoelectric effect. Incoming photons on the left strike a metal plate (bottom), and eject electrons, depicted as flying off to the right.]]
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| In 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 [[photon]]s, [[quantum|quanta]] of light energy with particulate qualities.
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| In the [[photoelectric effect]], it was observed that shining a light on certain metals would lead to an [[electric current]] in a [[electrical network|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.
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| Einstein explained this conundrum by [[axiom|postulating]] that the electrons can receive energy from electromagnetic field only in discrete portions (quanta that were called [[photon]]s): an amount of [[energy]] ''E'' that was related to the [[frequency]] ''f'' of the light by
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| :<math>E = h f\,</math>
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| where ''h'' is [[Planck's constant]] (6.626 × 10<sup>−34</sup> 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. Intensity-dependent phenomena have now been studied in detail with such lasers.<ref>{{cite journal|doi=10.1016/0375-9601(96)00259-9|title=Intensity dependence of the photoelectric effect induced by a circularly polarized laser beam|year=1996|last1=Zhang|first1=Q|journal=Physics Letters A|volume=216|issue=1-5|pages=125|bibcode = 1996PhLA..216..125Z }}</ref>
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| Einstein was awarded the [[Nobel Prize in Physics]] in 1921 for his discovery of the law of the photoelectric effect.
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| === De Broglie's wavelength ===
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| {{Main|Matter wave}}
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| [[File:Propagation of a de broglie wave.svg|290px|"290px"|right|thumb|Propagation of [[matter wave|de Broglie waves]] in 1d—real part of the [[complex number|complex]] amplitude is blue, imaginary part is green. The probability (shown as the colour [[opacity (optics)|opacity]]) of finding the particle at a given point ''x'' is spread out like a waveform; there is no definite position of the particle. As the amplitude increases above zero the [[curvature]] decreases, so the amplitude decreases again, and vice versa—the result is an alternating amplitude: a wave. Top: [[Plane wave]]. Bottom: [[Wave packet]].]]
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| In 1924, [[Louis-Victor de Broglie]] formulated the [[Matter wave|de Broglie hypothesis]], claiming that ''all'' matter,<ref>Donald H Menzel, "''Fundamental formulas of Physics''", volume 1, page 153; Gives the de Broglie wavelengths for composite particles such as protons and neutrons.</ref><ref>[[Brian Greene]], [[The Elegant Universe]], page 104 "all matter has a wave-like character"</ref> not just light, has a wave-like nature; he related [[wavelength]] (denoted as ''[[λ]]''), and [[momentum]] (denoted as ''p''):
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| :<math>\lambda = \frac{h}{p}</math>
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| This is a generalization of Einstein's equation above, since the momentum of a photon is given by ''p'' = <math>\tfrac{E}{c}</math> and the wavelength (in a vacuum) by ''λ'' = <math>\tfrac{c}{f}</math>, where ''c'' is the [[speed of light]] in vacuum.
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| De Broglie's formula was confirmed three years later for [[electron]]s (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]] [[Davisson-Germer experiment|guided their beam through a crystalline grid.]]
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| 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.
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| === Heisenberg's uncertainty principle ===
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| {{Main|Heisenberg uncertainty principle}}
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| In his work on formulating quantum mechanics, [[Werner Heisenberg]] postulated his [[uncertainty principle]], which states:
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| :<math>\Delta x \Delta p \ge \frac{\hbar}{2} </math>
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| where
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| :<math>\Delta</math> here indicates [[standard deviation]], a measure of spread or uncertainty;
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| :'''x''' and '''p''' are a particle's position and [[linear momentum]] respectively.
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| :''<math>\hbar</math>'' is the [[Planck constant|reduced Planck's constant]] (Planck's constant divided by 2<math>\pi</math>).
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| Heisenberg originally explained this as a consequence of the process of measuring: Measuring position accurately would disturb momentum and vice-versa, offering an example (the "gamma-ray 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.
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| In fact, the modern explanation of the uncertainty principle, extending the [[Copenhagen interpretation]] first put forward by [[Niels Bohr|Bohr]] and [[Werner Heisenberg|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 pulse-like and has a very ill-defined 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 ill-defined position.
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| === de Broglie–Bohm theory ===
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| De Broglie himself had proposed a [[pilot wave]] construct to explain the observed wave–particle duality. In this view, each particle has a well-defined 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 non-local effects when applied to systems involving more than one particle. Non-locality, however, soon became established as an integral feature of [[Quantum mechanics|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,<ref>[http://plato.stanford.edu/entries/qm-bohm/ Bohmian Mechanics], ''Stanford Encyclopedia of Philosophy.''</ref> 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]].
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| ==Wave behavior of large objects==
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| Since the demonstrations of wave-like properties in [[photon]]s and [[electron]]s, similar experiments have been conducted with [[neutron]]s and [[proton]]s. Among the most famous experiments are those of [[Estermann]] and [[Otto Stern]] in 1929.<ref>{{cite journal|last = Estermann
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| | first = I.
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| | authorlink =
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| | coauthors = Stern O.
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| | title = Beugung von Molekularstrahlen
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| | journal = Zeitschrift für Physik
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| | volume = 61
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| | issue =
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| 1-2| pages = 95–125
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| | publisher =
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| | location =
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| | year = 1930
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| | url =
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| | doi = 10.1007/BF01340293
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| Authors of similar recent experiments with atoms and molecules, described below, claim that these larger particles also act like waves.
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| 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]].<ref>R. Colella, A. W. Overhauser and S. A. Werner, Observation of Gravitationally Induced Quantum Interference, ''Phys. Rev. Lett.'' '''34''', 1472–1474 (1975).</ref> 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 quantum-mechanical 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 [[Pound-Rebka falling photon experiment]]), the self-interference of the quantum mechanical wave of a massive fermion in a gravitational field had never been experimentally confirmed before.
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| In 1999, the diffraction of C<sub>60</sub> [[fullerenes]] by researchers from the [[University of Vienna]] was reported.<ref>{{cite journal | first = Markus | last = Arndt | coauthors = O. Nairz, [[Julian Voss-Andreae|J. Voss-Andreae]], C. Keller, G. van der Zouw, [[Anton Zeilinger|A. Zeilinger]] |date=14 October 1999 | title = Wave–particle duality of C<sub>60</sub> | journal = Nature | volume = 401 | pages = 680–682 | url = http://www.nature.com/nature/journal/v401/n6754/abs/401680a0.html | doi = 10.1038/44348 | pmid = 18494170 | issue = 6754 | bibcode=1999Natur.401..680A}}</ref> Fullerenes are comparatively large and massive objects, having an atomic mass of about 720 [[unified atomic mass unit|u]]. The [[Matter wave|de Broglie wavelength]] is 2.5 [[picometre|pm]], whereas the diameter of the molecule is about 1 [[nanometre|nm]], about 400 times larger. In 2012, these far-field diffraction experiments could be extended to 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 single molecule sensitivity.<ref name="Nano-20120325">{{cite journal |author=Juffmann, Thomas et al |title=Real-time single-molecule imaging of quantum interference |url=http://www.nature.com/nnano/journal/vaop/ncurrent/full/nnano.2012.34.html |date=25 March 2012 |publisher=[[Nature Nanotechnology]] |accessdate=27 March 2012 }}</ref><ref>{{cite web|last=Quantumnanovienna|title=Single molecules in a quantum interference movie|url=http://www.youtube.com/watch?v=vCiOMQIRU7I|accessdate=2012-04-21}}</ref>
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| In 2003, the Vienna group also demonstrated the wave nature of [[tetraphenylporphyrin]]<ref name="fn_4">{{cite journal | first = Lucia | last = Hackermüller | coauthors = Stefan Uttenthaler, Klaus Hornberger, Elisabeth Reiger, Björn Brezger, Anton Zeilinger and Markus Arndt | year = 2003 | title = The wave nature of biomolecules and fluorofullerenes | journal = Phys. Rev. Lett. | volume = 91 | pages = 090408 | url = http://www.nature.com/cgi-taf/DynaPage.taf?file=/nature/journal/v401/n6754/full/401680a0_fs.html&content_filetype=pdf | doi = 10.1103/PhysRevLett.91.090408 | pmid = 14525169 | issue = 9 | bibcode=2003PhRvL..91i0408H|arxiv = quant-ph/0309016 }}</ref>—a flat biodye with an extension of about 2 nm and a mass of 614 u. For this demonstration they employed a near-field [[Talbot Lau interferometer]].<ref>{{cite journal | first = John F. | last = Clauser | coauthors = S. Li | year = 1994 | title = Talbot von Lau interefometry with cold slow potassium atoms. | journal = Phys. Rev. A | volume = 49 | issue = 4 | pages = R2213–17| url = http://ojps.aip.org | doi = 10.1103/PhysRevA.49.R2213 |bibcode = 1994PhRvA..49.2213C | pmid=9910609}}</ref><ref>{{cite journal | first = Björn | last = Brezger | coauthors = Lucia Hackermüller, Stefan Uttenthaler, Julia Petschinka, Markus Arndt and Anton Zeilinger | year = 2002 | title = Matter-wave interferometer for large molecules | journal = Phys. Rev. Lett. | volume = 88 | pages = 100404| url = http://ojps.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PRLTAO000088000010100404000001&idtype=cvips | doi = 10.1103/PhysRevLett.88.100404 | pmid = 11909334 | issue = 10 | bibcode=2002PhRvL..88j0404B|arxiv = quant-ph/0202158 }}</ref> In the same interferometer they also found interference fringes for C<sub>60</sub>F<sub>48.</sub>, a fluorinated [[buckyball]] with a mass of about 1600 u, composed of 108 atoms.<ref name="fn_4" /> Large molecules are already so complex that they give experimental access to some aspects of the quantum-classical interface, i.e., to certain [[decoherence]] mechanisms.<ref>{{cite journal | first = Klaus | last = Hornberger | coauthors = Stefan Uttenthaler,Björn Brezger, Lucia Hackermüller, Markus Arndt and Anton Zeilinger | year = 2003 | title = Observation of Collisional Decoherence in Interferometry | journal = Phys. Rev. Lett. | volume = 90 | pages = 160401 | url = http://ojps.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PRLTAO000090000016160401000001&idtype=cvips | doi = 10.1103/PhysRevLett.90.160401 | pmid = 12731960 | issue = 16 | bibcode=2003PhRvL..90p0401H|arxiv = quant-ph/0303093 }}</ref><ref>{{cite journal | first = Lucia | last = Hackermüller | coauthors = Klaus Hornberger, Björn Brezger, Anton Zeilinger and Markus Arndt | year = 2004 | title = Decoherence of matter waves by thermal emission of radiation| journal = Nature | volume = 427 | pages = 711–714 | url = http://www.nature.com/cgi-taf/DynaPage.taf?file=/nature/journal/v427/n6976/full/nature02276_fs.html&content_filetype=PDF | doi = 10.1038/nature02276 | pmid = 14973478 | issue = 6976 |arxiv = quant-ph/0402146 |bibcode = 2004Natur.427..711H }}</ref> In 2011, the interference of molecules as heavy as 6910 u could be demonstrated in a Kapitza–Dirac–Talbot–Lau interferometer. These are the largest objects that so far showed de Broglie matter-wave interference.<ref>{{cite journal|last=Gerlich|first=Stefan|coauthors=et al.|title=Quantum interference of large organic molecules|journal=[[Nature Communications]]|year=2011|volume=2|issue=263|doi=10.1038/ncomms1263|bibcode = 2011NatCo...2E.263G|pmid=21468015|pmc=3104521}}</ref> In 2013, the interference of molecules beyond 10,000 u has been demonstrated.<ref name="Eibenberger2013">{{cite doi|10.1039/c3cp51500a.}}</ref>
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| 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.<ref>Peter Gabriel Bergmann, ''The Riddle of Gravitation'', Courier Dover Publications, 1993 ISBN 0-486-27378-4 [http://books.google.com/books?vid=ISBN0486273784&id=WYxkrwMidp0C&pg=PR10&lpg=PR10&dq=wavelength+%22planck+mass%22+%22Schwarzschild+radius%22&sig=Lwt9PWQoB7R4RWxQXYE_tyRVp9A online]</ref>
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| Recently Couder, Fort, ''et al.'' showed<ref>http://www.youtube.com/watch?v=W9yWv5dqSKk - You Tube video - Yves Couder Explains Wave/Particle Duality via Silicon Droplets</ref> that we can use macroscopic oil droplets on a vibrating surface as a model of wave–particle duality—localized droplet creates periodical waves around and interaction with them leads to quantum-like phenomena: interference in double-slit experiment,<ref>Y. Couder, E. Fort, ''Single-Particle Diffraction and Interference at a Macroscopic Scale'', PRL 97, 154101 (2006) [https://hekla.ipgp.fr/IMG/pdf/Couder-Fort_PRL_2006.pdf online]</ref> unpredictable tunneling<ref>A. Eddi, E. Fort, F. Moisy, Y. Couder, ''Unpredictable Tunneling of a Classical Wave–Particle Association'', [http://prl.aps.org/abstract/PRL/v102/i24/e240401 PRL 102, 240401 (2009)]</ref> (depending in complicated way on practically hidden state of field), orbit quantization<ref>E. Fort, A. Eddi, A. Boudaoud, J. Moukhtar, Y. Couder, ''Path-memory induced quantization of classical orbits'', [http://www.pnas.org/content/107/41/17515 PNAS October 12, 2010 vol. 107 no. 41 17515-17520]</ref> (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) and [[Zeeman effect]].<ref>http://prl.aps.org/abstract/PRL/v108/i26/e264503 - Level Splitting at Macroscopic Scale</ref>
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| ==Treatment in modern quantum mechanics==
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| Wave–particle duality is deeply embedded into the foundations of [[quantum mechanics]]. In the [[formalism (mathematics)|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]]). For particles with mass this equation has solutions that follow the form of the wave equation. Propagation of such waves leads to wave-like phenomena such as interference and diffraction. Particles without mass, like photons, has no solutions of the Schrödinger equation so have another wave.
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| The particle-like behavior is most evident due to phenomena associated with [[measurement in quantum mechanics]]. Upon measuring the location of the particle, the particle will be forced into a more localized state as given by the uncertainty principle. When viewed through this formalism, the measurement of the wave function will randomly "[[wave function collapse|collapse]]", or rather "[[quantum decoherence|decohere]]", to a sharply peaked function at some location. For particles with mass the likelihood of detecting the particle at any particular location is equal to the squared amplitude of the wave function there. The measurement will return a well-defined position, (subject to [[Heisenberg's uncertainty principle|uncertainty]]), a property traditionally associated with particles. It is important to note that a measurement is only a particular type of interaction where some data is recorded and the measured quantity is forced into a particular [[eigenstate]]. The act of measurement is therefore not fundamentally different from any other interaction.
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| Following the development of [[quantum field theory]] the ambiguity disappeared. The field permits solutions that follow the wave equation, which are referred to as the wave functions. The term particle is used to label the irreducible representations of the [[Lorentz group]] that are permitted by the field. An interaction as in a [[Feynman diagram]] is accepted as a calculationally convenient approximation where the outgoing legs are known to be simplifications of the propagation and the internal lines are for some order in an expansion of the field interaction. Since the field is non-local and quantized, the phenomena which previously were thought of as paradoxes are explained. Within the limits of the wave-particle duality the quantum field theory gives the same results.
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| ===Visualization===
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| Below is an illustration of wave–particle duality as it relates to De Broglie's hypothesis and Heisenberg's uncertainty principle (above), in terms of the [[position and momentum space]] [[wavefunction]]s for one spinless particle with mass in one dimension. These wavefunctions are [[Fourier transform]]s of each other.
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| The more localized the position-space wavefunction, the more likely the particle is to be found with the position coordinates in that region, and correspondingly the momentum-space wavefunction is less localized so the possible momentum components the particle could have are more widespread.
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| Conversely the more localized the momentum-space wavefunction, the more likely the particle is to be found with those values of momentum components in that region, and correspondingly the less localized the position-space wavefunction, so the position coordinates the particle could occupy are more widespread.
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| [[File:Quantum mechanics travelling wavefunctions wavelength.svg|center|thumb|502px|Position ''x'' and momentum ''p'' wavefunctions corresponding to quantum particles. The colour opacity (%) of the particles corresponds to the probability density of finding the particle with position ''x'' or momentum component ''p''.<br/>
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| '''Top:''' If wavelength ''λ'' is unknown, so are momentum ''p'', wave-vector ''k'' and energy ''E'' (de Broglie relations). As the particle is more localized in position space, Δ''x'' is smaller than for Δ''p<sub>x</sub>''.<br/>
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| '''Bottom:''' If ''λ'' is known, so are ''p'', ''k'', and ''E''. As the particle is more localized in momentum space, Δ''p'' is smaller than for Δ''x''.]]
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| {{-}}
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| ==Alternative views==
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| Wave–particle duality is an ongoing conundrum in modern physics. Most physicists accept wave-particle duality as the best explanation for a broad range of observed phenomena; however, it is not without controversy. Alternative views are also presented here. These views are not generally accepted by mainstream physics, but serve as a basis for valuable discussion within the community.
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| ===Both-particle-and-wave view===
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| 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 a system exhibits both particle properties and wave properties simultaneously, and particles are guided, in a [[determinism|deterministic]] fashion, by the 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.<ref>(Buchanan pp. 29–31)</ref>
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| At least one physicist considers the "wave-duality" a misnomer, as L. Ballentine, ''Quantum Mechanics, A Modern Development'', p. 4, explains:
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| <blockquote>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.</blockquote>
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| [[Afshar experiment]]<ref>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</ref> (2007) has demonstrated that it is possible to simultaneously observe both wave and particle properties of photons.
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| ===Wave-only view===
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| At least one scientist proposes that the duality can be replaced by a "wave-only" view. In his book ''Collective Electrodynamics: Quantum Foundations of Electromagnetism'' (2000), [[Carver Mead]] purports to analyze the behavior of electrons and photons purely in terms of electron wave functions, and attributes the apparent particle-like behavior to quantization effects and eigenstates. According to reviewer David Haddon:<ref>{{cite web | title = Recovering Rational Science | author = David Haddon | work = Touchstone | accessdate = 2007-09-12 | url = http://www.touchstonemag.com/archives/article.php?id=16-07-044-b}}</ref>
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| <blockquote>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 self-propagating electrical currents of superconductors, and the [[Bose–Einstein condensate]] of atoms.</blockquote>
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| [[Albert Einstein]], who, in his search for a [[Unified Field Theory]], did not accept wave-particle duality, wrote:<ref>[[Paul Arthur Schilpp]], ed, ''Albert Einstein: Philosopher-Scientist'', Open Court (1949), ISBN 0-87548-131-7 {{Please check ISBN|reason=Check digit (7) does not correspond to calculated figure.}}, p 51.</ref>
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| <blockquote>This double nature of radiation (and of material corpuscles)...has been interpreted by quantum-mechanics in an ingenious and amazingly successful fashion. This interpretation...appears to me as only a temporary way out...</blockquote>
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| The [[many-worlds interpretation]] (MWI) is sometimes presented as a waves-only theory, including by its originator, [[Hugh Everett]] who referred to MWI as "the wave interpretation".<ref>See section VI(e) of Everett's thesis: ''The Theory of the Universal Wave Function'', in [[Bryce Seligman DeWitt]], [[R. Neill Graham]], eds, ''The Many-Worlds Interpretation of Quantum Mechanics'', Princeton Series in Physics, [[Princeton University Press]] (1973), ISBN 0-691-08131-X, pp 3–140.</ref>
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| The ''{{visible anchor|Three Wave Hypothesis}}'' of R. Horodecki relates the particle to wave.<ref>{{Cite journal |last=Horodecki |first=R. |title=De broglie wave and its dual wave |journal=[[Physics Letters|Phys. Lett.]] A |volume=87 |issue=3 |pages=95–97 |year=1981 |doi=10.1016/0375-9601(81)90571-5 |bibcode = 1981PhLA...87...95H }}</ref><ref>{{Cite journal |last=Horodecki |first=R. |title=Superluminal singular dual wave |journal=Lett. Novo Cimento |volume=38 |issue= |pages=509–511 |year=1983 }}</ref> The hypothesis implies that a massive particle is an intrinsically spatially as well as temporally extended wave phenomenon by a nonlinear law.
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| ===Neither-wave-nor-particle view===
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| It has been argued that there are never exact particles or waves, but only some compromise or intermediate between them. One consideration
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| is that zero dimensional [[point (geometry)|mathematical points]] cannot be observed. Another is that the formal representation of such points, the [[Kronecker delta function]] is unphysical, because it cannot be [[Normalisable wave function|normalized]]. Parallel arguments apply to pure wave states. [[Roger Penrose]] states:<ref>(R Penrose, Road to Reality p521 s21.11)</ref>
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| <blockquote>
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| "Such positions states are idealised wavefunctions [...] Whereas the momentum states
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| are infinitely spread out, the position states are infinitely concentrated. Neither is normaliseable [...]"
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| </blockquote>
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| ===Relational approach to wave–particle duality===
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| [[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.<ref>http://www.quantum-relativity.org/Quantum-Relativity.pdf. See Q. Zheng and T. Kobayashi, ''Quantum Optics as a Relativistic Theory of Light''; Physics Essays 9 (1996) 447. Annual Report, Department of Physics, School of Science, University of Tokyo (1992) 240.</ref>
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| ===Semantic artifact view===
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| [[Robert Anton Wilson]] suggests that many of the so-called quantum paradoxes represent semantic artifacts arising from poor language choice when reporting observed phenomena in Science.<ref>{{cite book|last=Wilson|first=Robert Anton|title=Quantum Psychology|year=June 1990|publisher=New Falcon Publications|isbn=978-1561840717}}</ref> Wilson frequently used the wave-particle duality example to illustrate his idea. To wit, let us propose one group of scientists performs a set of experiments and reports, "One experiment demonstrates light is a wave while another demonstrates light is a particle." Let us also propose that another group of scientists halfway around the world also perform an identical set of experiments and reports, "When constrained by one experiment, light behaved as a wave and while constrained by another experiment, light behaved as a particle." Although both groups of scientists aim to report similar empirical observations, the first group makes an existential conclusion about the 'is-ness' of light. The second group reports their observations operationally, describing what they actually observed light doing rather than jumping to an existential conclusion about what light 'is'.
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| ==Applications==
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| 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.
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| * 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.
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| * Similarly, [[neutron diffraction]] uses neutrons with a wavelength of about 0.1 [[nanometer|nm]], the typical spacing of atoms in a solid, to determine the structure of solids.
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| ==See also==
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| * [[Arago spot]]
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| * [[Afshar experiment]]
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| * [[Basic concepts of quantum mechanics]]
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| * [[Complementarity (physics)]]
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| * [[Electron wave-packet interference]]
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| * [[Faraday wave]]
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| * [[Hanbury Brown and Twiss effect]]
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| * [[Photon polarization]]
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| * [[Scattering theory]]
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| * [[Wavelet]]
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| * [[Wheeler's delayed choice experiment]]
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| ==Notes and references==
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| {{Reflist|colwidth=30em}}
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| ==External links==
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| * [http://www.toutestquantique.fr/#dualite Animation, applications and research linked to the wave-particle duality and other basic quantum phenomena] (Université Paris Sud)
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| *{{cite arxiv | author=H. Nikolic | eprint=quant-ph/0609163 | title=Quantum mechanics: Myths and facts }}
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| *{{cite web | author = Young & Geller | url=http://wps.aw.com/aw_young_physics_11 | title=College Physics }}
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| *{{cite web | author=B. Crowell | url=http://www.lightandmatter.com/html_books/6mr/ch03/ch03.html | title=Light as a Particle | format=Web page | accessdate=December 10, 2006 }}
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| * E.H. Carlson, [http://www.physnet.org/modules/pdf_modules/m246.pdf ''Wave–Particle Duality: Light ''] on [http://www.physnet.org Project PHYSNET]
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| *{{cite web | author=R. Nave | url=http://hyperphysics.phy-astr.gsu.edu/hbase/mod1.html | title=Wave–Particle Duality | format=Web page | work=HyperPhysics | publisher=Georgia State University, Department of Physics and Astronomy | accessdate=December 12, 2005 }}
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| *{{cite web | author=Juffmann, Thomas et al| url=http://www.quantumnano.at/far-field-more.3953.html | title=Real-time single-molecule imaging of quantum interference |date=25 March 2012 | publisher=[[Nature Nanotechnology]] | accessdate=21 January 2014 }}
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| {{DEFAULTSORT:Wave Particle Duality}}
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| [[Category:Foundational quantum physics]]
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| [[Category:Duality theories]]
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| [[Category:Dichotomies]]
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| {{Link FA|eu}}
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