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Encyclopedia > Photon
Composition: Photon Photons emitted in a coherent beam from a laser Elementary particle Boson Gauge boson Electromagnetic Albert Einstein (1905–17) $gamma$ or $hnu$ 0[1] Stable[2] 0 1[1]

$E = frac{hc}{lambda},$

where h is Planck's constant, c is the speed of light, and λ is its wavelength. This is different from a classical wave, which may gain or lose arbitrary amounts of energy. For visible light the energy carried by a single photon would be around a tiny $4times10^{-19}$ joules; this energy is just sufficient to excite a single molecule in a photoreceptor cell of an eye,[citation needed] thus contributing to vision.[4] A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... A line showing the speed of light on a scale model of Earth and the Moon, about 1. ... The joule (IPA pronunciation: or ) (symbol: J) is the SI unit of energy. ... Rods and Cones redirects here. ... A human eye Eyes are organs of vision that detect light. ... This does not cite any references or sources. ...

Apart from energy a photon also carries momentum and has a polarization. It follows the laws of quantum mechanics, which means that often these properties do not have a well-defined value for a given photon. Rather, they are defined as a probability to measure a certain polarization, position, or momentum. For example, although a photon can excite a single molecule, it is often impossible to predict beforehand which molecule will be excited. In classical mechanics, momentum (pl. ... In electrodynamics, polarization (also spelled polarisation) is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. ... Fig. ...

The above description of a photon as a carrier of electromagnetic radiation is commonly used by physicists. However, in theoretical physics, a photon can be considered as a mediator for any type of electromagnetic interactions, including magnetic fields and electrostatic repulsion between like charges.

The modern concept of the photon was developed gradually (1905–17) by Albert Einstein[5][6][7][8] to explain experimental observations that did not fit the classical wave model of light. In particular, the photon model accounted for the frequency dependence of light's energy, and explained the ability of matter and radiation to be in thermal equilibrium. Other physicists sought to explain these anomalous observations by semiclassical models, in which light is still described by Maxwell's equations, but the material objects that emit and absorb light are quantized. Although these semiclassical models contributed to the development of quantum mechanics, further experiments proved Einstein's hypothesis that light itself is quantized; the quanta of light are photons. â€œEinsteinâ€ redirects here. ... Lasers used for visual effects during a musical performance. ... In physics, matter is commonly defined as the substance of which physical objects are composed, not counting the contribution of various energy or force-fields, which are not usually considered to be matter per se (though they may contribute to the mass of objects). ... Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ... In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann-distribution. ... In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ... Fig. ... In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ... In physics, a quantum (plural: quanta) is an indivisible entity of energy. ...

The photon concept has led to momentous advances in experimental and theoretical physics, such as lasers, Bose–Einstein condensation, quantum field theory, and the probabilistic interpretation of quantum mechanics. According to the Standard Model of particle physics, photons are responsible for producing all electric and magnetic fields, and are themselves the product of requiring that physical laws have a certain symmetry at every point in spacetime. The intrinsic properties of photons — such as charge, mass and spin — are determined by the properties of this gauge symmetry. Experiment with a laser (US Military) In physics, a laser is a device that emits light through a specific mechanism for which the term laser is an acronym: Light Amplification by Stimulated Emission of Radiation. ... A Boseâ€“Einstein condensate is a phase of matter formed by bosons cooled to temperatures very near to absolute zero. ... Quantum field theory (QFT) is the quantum theory of fields. ... In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... Thousands of particles explode from the collision point of two relativistic (100 GeV per ion) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... Magnetic field lines shown by iron filings In physics, a magnetic field is a solenoidal vector field in the space surrounding moving electric charges and magnetic dipoles, such as those in electric currents and magnets. ... Sphere symmetry group o. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... The invariant mass or intrinsic mass or proper mass or just mass is a measurement or calculation of the mass of an object that is the same for all frames of reference. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ...

The concept of photons is applied to many areas such as photochemistry, high-resolution microscopy, and measurements of molecular distances. Recently, photons have been studied as elements of quantum computers and for sophisticated applications in optical communication such as quantum cryptography. Photochemistry is the study of the interaction of light and chemicals. ... Two-photon excitation microscopy is a technique that allows imaging living tissue up to a depth of one millimeter. ... Fluorescent proteins localize the guanosine 5-triphosphate hydrolase ARF in the Golgi apparatus of a living macrophage. ... The Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers. ... Optical communication is any form of telecommunication that uses light as the transmission medium. ... Quantum cryptography is an approach based on quantum physics for secure communications. ...

The photon was originally called a “light quantum” (das Lichtquant) by Albert Einstein.[5] The modern name “photon” derives from the Greek word for light, φῶς, (transliterated phôs), and was coined in 1926 by the physical chemist Gilbert N. Lewis, who published a speculative theory[9] in which photons were “uncreatable and indestructible”. Although Lewis' theory was never accepted — being contradicted by many experiments — his new name, photon, was adopted immediately by most physicists. â€œEinsteinâ€ redirects here. ... Greek ( IPA: (with a palatalized k as a rule) or simply IPA: â€” Hellenic) is an Indo-European language with a documented history of 3,500 years, the longest of any single language in that language family. ... Lewis in the Berkeley Lab Gilbert Newton Lewis (October 23, 1875-March 23, 1946) was a famous American physical chemist. ...

In physics, a photon is usually denoted by the symbol $gamma!$, the Greek letter gamma. This symbol for the photon probably derives from gamma rays, which were discovered and named in 1900 by Villard[10][11] and shown to be a form of electromagnetic radiation in 1914 by Rutherford and Andrade.[12] In chemistry and optical engineering, photons are usually symbolized by $h nu !$, the energy of a photon, where $h !$ is Planck's constant and the Greek letter $nu !$ (nu) is the photon's frequency. Much less commonly, the photon can be symbolized by hf, where its frequency is denoted by f. The Greek alphabet is an alphabet that has been used to write the Greek language since about the 9th century BCE. It was the first alphabet in the narrow sense, that is, a writing system using a separate symbol for each vowel and consonant alike. ... Gamma (uppercase Î“, lowercase Î³) is the third letter of the Greek alphabet. ... This article is about electromagnetic radiation. ... Paul Ulrich Villard (1860 - 13 January 1934) was a French chemist and physicist, born near Lyon, France. ... Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ... Ernest Rutherford, 1st Baron Rutherford of Nelson OM PC FRS (30 August 1871 - 19 October 1937), widely referred to as Lord Rutherford, was a nuclear physicist who became known as the father of nuclear physics. ... Edward Neville Da Costa Andrade (December 27, 1887 - June 6, 1971), was an English physicist, writer and poet. ... Chemistry - the study of interactions of chemical substances with one another and energy based on the structure of atoms, molecules and other kinds of aggregrates Chemistry (from Egyptian kÄ“me (chem), meaning earth[1]) is the science concerned with the reactions, transformations and aggregations of matter, as well as accompanying... Optical engineering is the field of study which focuses on applications of optics. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... The Greek alphabet is an alphabet that has been used to write the Greek language since about the 9th century BCE. It was the first alphabet in the narrow sense, that is, a writing system using a separate symbol for each vowel and consonant alike. ... For other uses, see Nu. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ...

## Physical properties

A Feynman diagram of the exchange of a virtual photon (symbolized by a wavy-line and a gamma, $gamma ,$) between a positron and an electron.

The photon is massless,[3] has no electric charge[13] and does not decay spontaneously in empty space. A photon has two possible polarization states and is described by exactly three continuous parameters: the components of its wave vector, which determine its wavelength $lambda !$ and its direction of propagation. The photon is the gauge boson for electromagnetism, and therefore all other quantum numbers — such as lepton number, baryon number, or strangeness — are exactly zero. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... In this Feynman diagram, an electron and positron annihilate and become a quark-antiquark pair. ... The first detection of the positron in 1932 by Carl D. Anderson The positron is the antiparticle or the antimatter counterpart of the electron. ... e- redirects here. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... The invariant mass or intrinsic mass or proper mass or just mass is a measurement or calculation of the mass of an object that is the same for all frames of reference. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... In electrodynamics, polarization (also spelled polarisation) is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. ... A wave vector is a vector that represents two properties of a wave: the magnitude of the vector represents wavenumber (inversely related to wavelength), and the vector points in the direction of wave propagation. ... Gauge bosons are bosonic particles which act as carriers of the fundamental forces of Nature. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... In high energy physics, the lepton number is the number of leptons minus the number of antileptons. ... In particle physics, the baryon number is an approximate conserved quantum number. ... In particle physics, strangeness, denoted as , is a property of particles, expressed as a quantum number for describing decay of particles in strong and electro-magnetic reactions, which occur in a short period of time. ...

Photons are emitted in many natural processes, e.g., when a charge is accelerated, during a molecular, atomic or nuclear transition to a lower energy level, or when a particle and its antiparticle are annihilated. Photons are absorbed in the time-reversed processes which correspond to those mentioned above: for example, in the production of particle–antiparticle pairs or in molecular, atomic or nuclear transitions to a higher energy level. Electron-positron annihilation is the process that occurs when an electron (which is matter) and a positron (which is antimatter) collide. ... T-symmetry is the symmetry of physical laws under a time-reversal transformationâ€” The universe is not symmetric under time reversal, although in restricted contexts one may find this symmetry. ... Pair production refers to the creation of an elementary particle and its antiparticle, usually from a photon (or another neutral boson). ...

In empty space, the photon moves at $c !$ (the speed of light) and its energy $E !$ and momentum $mathbf{p}$ are related by $E = c , p !$, where $p !$ is the magnitude of the momentum. For comparison, the corresponding equation for particles with a mass $m !$ would be $E^{2} = c^{2} p^{2} + m^{2} c^{4} !$, as shown in special relativity. A line showing the speed of light on a scale model of Earth and the Moon, about 1. ... In classical mechanics, momentum (pl. ... This article or section is in need of attention from an expert on the subject. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest...

The energy and momentum of a photon depend only on its frequency $nu !$ or, equivalently, its wavelength $lambda !$ FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ... The wavelength is the distance between repeating units of a wave pattern. ...

$E = hbaromega = hnu = frac{h c}{lambda}$
$mathbf{p} = hbarmathbf{k}$

and consequently the magnitude of the momentum is

$p = hbar k = frac{h}{lambda} = frac{hnu}{c}$

where $hbar = h/2pi !$ (known as Dirac's constant or Planck's reduced constant); $mathbf{k}$ is the wave vector (with the wave number $k = 2pi/lambda !$ as its magnitude) and $omega = 2pinu !$ is the angular frequency. Notice that $mathbf{k}$ points in the direction of the photon's propagation. The photon also carries spin angular momentum that does not depend on its frequency. The magnitude of its spin is $sqrt{2} hbar$ and the component measured along its direction of motion, its helicity, must be $pmhbar$. These two possible helicities correspond to the two possible circular polarization states of the photon (right-handed and left-handed). A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... A wave vector is a vector that represents two properties of a wave: the magnitude of the vector represents wavenumber (inversely related to wavelength), and the vector points in the direction of wave propagation. ... It has been suggested that this article or section be merged into Angular velocity. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... In particle physics, helicity is the projection of the angular momentum to the direction of motion: Because the angular momentum with respect to an axis has discrete values, helicity is discrete, too. ... In electrodynamics, circular polarization of electromagnetic radiation is a polarization such that the tip of the electric field vector, at a fixed point in space, describes a circle as time progresses. ...

To illustrate the significance of these formulae, the annihilation of a particle with its antiparticle must result in the creation of at least two photons for the following reason. In the center of mass frame, the colliding antiparticles have no net momentum, whereas a single photon always has momentum. Hence, conservation of momentum requires that at least two photons are created, with zero net momentum. The energy of the two photons — or, equivalently, their frequency — may be determined from conservation of four-momentum. Seen another way, the photon can be considered as its own antiparticle. The reverse process, pair production, is the dominant mechanism by which high-energy photons such as gamma rays lose energy while passing through matter. Electron-positron annihilation is the process that occurs when an electron (which is matter) and a positron (which is antimatter) collide. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ... This article or section is in need of attention from an expert on the subject. ... In classical mechanics, momentum (pl. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... Pair production refers to the creation of an elementary particle and its antiparticle, usually from a photon (or another neutral boson). ... This article is about electromagnetic radiation. ...

The classical formulae for the energy and momentum of electromagnetic radiation can be re-expressed in terms of photon events. For example, the pressure of electromagnetic radiation on an object derives from the transfer of photon momentum per unit time and unit area to that object, since pressure is force per unit area and force is the change in momentum per unit time. Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ... Radiation pressure is the pressure exerted upon any surface exposed to electromagnetic radiation. ... In classical mechanics, momentum (pl. ...

## Historical development

Main article: Light
Thomas Young's double-slit experiment in 1805 showed that light can act as a wave, helping to defeat early particle theories of light.

In most theories up to the eighteenth century, light was pictured as being made up of particles. Since particle models cannot easily account for the refraction, diffraction and birefringence of light, wave theories of light were proposed by René Descartes (1637),[14] Robert Hooke (1665),[15] and Christian Huygens (1678);[16] however, particle models remained dominant, chiefly due to the influence of Isaac Newton.[17] In the early nineteenth century, Thomas Young and August Fresnel clearly demonstrated the interference and diffraction of light and by 1850 wave models were generally accepted.[18] In 1865, James Clerk Maxwell's prediction[19] that light was an electromagnetic wave — which was confirmed experimentally in 1888 by Heinrich Hertz's detection of radio waves[20] — seemed to be the final blow to particle models of light. This article does not cite any references or sources. ... Image File history File links Young_Diffraction. ... Image File history File links Young_Diffraction. ... Thomas Young, English scientist Thomas Young (June 13, 1773-May 10, 1829) was an English polymath, contributing to the scientific understanding of vision, light, solid mechanics, energy, physiology, and Egyptology. ... Double-slit diffraction and interference pattern The double-slit experiment consists of letting light diffract through two slits, which produces fringes or wave-like interference patterns on a screen. ... A wave is a disturbance that propagates through space or spacetime, transferring energy and momentum and sometimes angular momentum. ... In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not made up of smaller particles. ... The straw seems to be broken, due to refraction of light as it emerges into the air. ... The intensity pattern formed on a screen by diffraction from a square aperture Diffraction refers to various phenomena associated with wave propagation, such as the bending, spreading and interference of waves passing by an object or aperture that disrupts the wave. ... A calcite crystal laid upon a paper with some letters showing the double refraction Birefringence, or double refraction, is the decomposition of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through certain types of material, such as calcite crystals, depending on... RenÃ© Descartes (French IPA: ) (March 31, 1596 â€“ February 11, 1650), also known as Renatus Cartesius (latinized form), was a highly influential French philosopher, mathematician, scientist, and writer. ... Robert Hooke, FRS (July 18, 1635 â€“ March 3, 1703) was an English polymath who played an important role in the scientific revolution, through both experimental and theoretical work. ... Christiaan Huygens Christiaan Huygens (approximate pronunciation: HOW-khens; SAMPA /h9yGEns/ or /[email protected]@ns/) (April 14, 1629&#8211;July 8, 1695), was a Dutch mathematician and physicist; born in The Hague as the son of Constantijn Huygens. ... Sir Isaac Newton (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1726][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Thomas Young, English scientist Thomas Young (June 13, 1773-May 10, 1829) was an English polymath, contributing to the scientific understanding of vision, light, solid mechanics, energy, physiology, and Egyptology. ... Augustin Fresnel Augustin-Jean Fresnel (pronounced [] in AmE (or fray-NELL), [] in French) (May 10, 1788 â€“ July 14, 1827), was a French physicist who contributed significantly to the establishment of the theory of wave optics. ... Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ... James Clerk Maxwell (13 June 1831 â€“ 5 November 1879) was a Scottish mathematician and theoretical physicist. ... In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ... Heinrich Rudolf Hertz (February 22, 1857 - January 1, 1894) was the German physicist and mechanician for whom the hertz, an SI unit, is named. ... For other uses, see Radio (disambiguation). ...

In 1900, Maxwell's theoretical model of light as oscillating electric and magnetic fields seemed complete. However, several observations could not be explained by any wave model of electromagnetic radiation, leading to the idea that light-energy was packaged into quanta described by E=hν. Later experiments showed that these light-quanta also carry momentum and, thus, can be considered particles: the photon concept was born, leading to a deeper understanding of the electric and magnetic fields themselves.

The Maxwell wave theory, however, does not account for all properties of light. The Maxwell theory predicts that the energy of a light wave depends only on its intensity, not on its frequency; nevertheless, several independent types of experiments show that the energy imparted by light to atoms depends only on the light's frequency, not on its intensity. For example, some chemical reactions are provoked only by light of frequency higher than a certain threshold; light of frequency lower than the threshold, no matter how intense, does not initiate the reaction. Similarly, electrons can be ejected from a metal plate by shining light of sufficiently high frequency on it (the photoelectric effect); the energy of the ejected electron is related only to the light's frequency, not to its intensity. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... James Clerk Maxwell (13 June 1831 â€“ 5 November 1879) was a Scottish mathematician and theoretical physicist. ... In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ... In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... Magnetic field lines shown by iron filings In physics, a magnetic field is a solenoidal vector field in the space surrounding moving electric charges and magnetic dipoles, such as those in electric currents and magnets. ... Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ... In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not made up of smaller particles. ... Lasers used for visual effects during a musical performance. ... In physics, intensity is a measure of the time-averaged energy flux. ... FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ... Photochemistry is the study of the interaction of light and chemicals. ... A diagram illustrating the emission of electrons from a metal plate, requiring energy gained from an incoming photon to be more than the work function of the material. ...

At the same time, investigations of blackbody radiation carried out over four decades (1860–1900) by various researchers[21] culminated in Max Planck's hypothesis[22][23] that the energy of any system that absorbs or emits electromagnetic radiation of frequency ν is an integer multiple of an energy quantum E = hν. As shown by Albert Einstein,[5][6] some form of energy quantization must be assumed to account for the thermal equilibrium observed between matter and electromagnetic radiation. As the temperature decreases, the peak of the black body radiation curve moves to lower intensities and longer wavelengths. ... Max Karl Ernst Ludwig Planck (April 23, 1858 in Kiel, Germany â€“ October 4, 1947 in GÃ¶ttingen, Germany) was a German physicist. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... â€œEinsteinâ€ redirects here. ... Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ...

Since the Maxwell theory of light allows for all possible energies of electromagnetic radiation, most physicists assumed initially that the energy quantization resulted from some unknown constraint on the matter that absorbs or emits the radiation. In 1905, Einstein was the first to propose that energy quantization was a property of electromagnetic radiation itself.[5] Although he accepted the validity of Maxwell's theory, Einstein pointed out that many anomalous experiments could be explained if the energy of a Maxwellian light wave were localized into point-like quanta that move independently of one another, even if the wave itself is spread continuously over space.[5] In 1909[6] and 1916,[8] Einstein showed that, if Planck's law of black-body radiation is accepted, the energy quanta must also carry momentum p = h / λ, making them full-fledged particles. This photon momentum was observed experimentally[24] by Arthur Compton, for which he received the Nobel Prize in 1927. The pivotal question was then: how to unify Maxwell's wave theory of light with its experimentally observed particle nature? The answer to this question occupied Albert Einstein for the rest of his life,[25] and was solved in quantum electrodynamics and its successor, the Standard Model. In physics, the spectral intensity of electromagnetic radiation from a black body at temperature T is given by the Plancks law of black body radiation: where: I(&#957;) is the amount of energy per unit time per unit surface area per unit solid angle per unit frequency. ... In classical mechanics, momentum (pl. ... In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not made up of smaller particles. ... Arthur Holly Compton (September 10, 1892 â€“ March 15, 1962) won the Nobel Prize in Physics (1927) for discovery of the Compton effect named in his honor. ... Nobel Prize medal. ... â€œEinsteinâ€ redirects here. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ...

## Early objections

Up to 1923, most physicists were reluctant to accept that electromagnetic radiation itself was quantized. Instead, they tried to account for photon behavior by quantizing matter, as in the Bohr model of the hydrogen atom (shown here). Although all semiclassical models have been disproved by experiment, these early atomic models led to quantum mechanics.

Even after Compton's experiment, Bohr, Hendrik Kramers and John Slater made one last attempt to preserve the Maxwellian continuous electromagnetic field model of light, the so-called BKS model.[27] To account for the then-available data, two drastic hypotheses had to be made: Hendrik Anthony Kramers (Rotterdam, February 2, 1894 â€“ Oegstgeest, April 24, 1952) was a Dutch physicist. ... John Clark Slater (1900-1976) was a major physicist and theoretical chemist. ...

• Energy and momentum are conserved only on the average in interactions between matter and radiation, not in elementary processes such as absorption and emission. This allows one to reconcile the discontinuously changing energy of the atom (jump between energy states) with the continuous release of energy into radiation.

However, refined Compton experiments showed that energy-momentum is conserved extraordinarily well in elementary processes; and also that the jolting of the electron and the generation of a new photon in Compton scattering obey causality to within 10 ps. Accordingly, Bohr and his co-workers gave their model “as honorable a funeral as possible“.[25] Nevertheless, the BKS model inspired Werner Heisenberg in his development[28] of quantum mechanics. Spontaneous emission is the process by which a molecule in an excited state drops to the ground state, resulting in the creation of a photon. ... In optics, stimulated emission is the process by which, when perturbed by a photon, matter may lose energy resulting in the creation of another photon. ... In physics, Compton scattering or the Compton effect, is the decrease in energy (increase in wavelength) of an X-ray or gamma ray photon, when it interacts with matter. ... A picosecond is an SI unit of time equal to 10-12 of a second. ... Werner Karl Heisenberg (December 5, 1901 â€“ February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics, and acknowledged to be one of the most important physicists of the twentieth century. ... Fig. ...

A few physicists persisted[29] in developing semiclassical models in which electromagnetic radiation is not quantized, but matter obeys the laws of quantum mechanics. Although the evidence for photons from chemical and physical experiments was overwhelming by the 1970s, this evidence could not be considered as absolutely definitive; since it relied on the interaction of light with matter, a sufficiently complicated theory of matter could in principle account for the evidence. Nevertheless, all semiclassical theories were refuted definitively in the 1970s and 1980s by elegant photon-correlation experiments.[30] Hence, Einstein's hypothesis that quantization is a property of light itself is considered to be proven. Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ... Fig. ...

## Wave–particle duality and uncertainty principles

Photons, like all quantum objects, exhibit both wave-like and particle-like properties. Their dual wave–particle nature can be difficult to visualize. The photon displays clearly wave-like phenomena such as diffraction and interference on the length scale of its wavelength. For example, a single photon passing through a double-slit experiment lands on the screen with a probability distribution given by its interference pattern determined by Maxwell's equations.[31] However, experiments confirm that the photon is not a short pulse of electromagnetic radiation; it does not spread out as it propagates, nor does it divide when it encounters a beam splitter. Rather, the photon seems like a point-like particle, since it is absorbed or emitted as a whole by arbitrarily small systems, systems much smaller than its wavelength, such as an atomic nucleus (≈10–15 m across) or even the point-like electron. Nevertheless, the photon is not a point-like particle whose trajectory is shaped probabilistically by the electromagnetic field, as conceived by Einstein and others; that hypothesis was also refuted by the photon-correlation experiments cited above.[30] According to our present understanding, the electromagnetic field itself is produced by photons, which in turn result from a local gauge symmetry and the laws of quantum field theory (see the Second quantization and Gauge boson sections below). In physics and chemistry, wave-particle duality is a conceptualization that all objects in our universe exhibit properties of both waves and of particles. ... In physics, a squeezed coherent state a in the Hilbert space of quantum mechanics that saturates the uncertainty principle that is the product of the corresponding two operators takes on its minimum value: The simplest such state is the ground state of the quantum harmonic oscillator. ... In quantum physics, the Heisenberg uncertainty principle is a mathematical property of a pair of canonical conjugate quantities - usually stated in a form of reciprocity of spans of their spectra. ... The intensity pattern formed on a screen by diffraction from a square aperture Diffraction refers to various phenomena associated with wave propagation, such as the bending, spreading and interference of waves passing by an object or aperture that disrupts the wave. ... Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ... Double-slit diffraction and interference pattern The double-slit experiment consists of letting light diffract through two slits, which produces fringes or wave-like interference patterns on a screen. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ... A beam splitter is an optical device, that splits a beam of light in two. ... A point particle (or point-like, often spelt pointlike) is an idealized object heavily used in physics. ... e- redirects here. ... The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ... â€œEinsteinâ€ redirects here. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... Quantum field theory (QFT) is the quantum theory of fields. ... In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ... In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ...

Heisenberg's thought experiment for locating an electron (shown in blue) with a high-resolution gamma-ray microscope. The incoming gamma ray (shown in green) is scattered by the electron up into the microscope's aperture angle θ. The scattered gamma ray is shown in red. Classical optics shows that the electron position can be resolved only up to an uncertainty Δx that depends on θ and the wavelength λ of the incoming light.

$Delta x sim frac{lambda}{sin theta}$

where θ is the aperture angle of the microscope. Thus, the position uncertainty Δx can be made arbitrarily small by reducing the wavelength. The momentum of the electron is uncertain, since it received a “kick” Δp from the light scattering from it into the microscope. If light were not quantized into photons, the uncertainty Δp could be made arbitrarily small by reducing the light's intensity. In that case, since the wavelength and intensity of light can be varied independently, one could simultaneously determine the position and momentum to arbitrarily high accuracy, violating the uncertainty principle. By contrast, Einstein's formula for photon momentum preserves the uncertainty principle; since the photon is scattered anywhere within the aperture, the uncertainty of momentum transferred equals The angular aperture of a lens is the apparent angle of the lens aperture as seen from the focal point: where is the focal length is the diameter of the aperture See also f-number numerical aperture Categories: Optics | Angle ... In quantum physics, the Heisenberg uncertainty principle is a mathematical property of a pair of canonical conjugate quantities - usually stated in a form of reciprocity of spans of their spectra. ...

$Delta p sim p_{mathrm{photon}} sintheta = frac{h}{lambda} sintheta$

giving the product $Delta x Delta p , sim , h$, which is Heisenberg's uncertainty principle. Thus, the entire world is quantized; both matter and fields must obey a consistent set of quantum laws, if either one is to be quantized.

The analogous uncertainty principle for photons forbids the simultaneous measurement of the number n of photons (see Fock state and the Second quantization section below) in an electromagnetic wave and the phase φ of that wave A Fock state, in quantum mechanics, is any state of the Fock space with a well-defined number of particles in each state. ... In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ...

ΔnΔφ > 1

See coherent state and squeezed coherent state for more details. In quantum mechanics a coherent state is a specific kind of quantum state of the quantum harmonic oscillator whose dynamics most closely resemble the oscillating behaviour of a classical harmonic oscillator system. ... In physics, a squeezed coherent state a in the Hilbert space of quantum mechanics that saturates the uncertainty principle that is the product of the corresponding two operators takes on its minimum value: The simplest such state is the ground state of the quantum harmonic oscillator. ...

Both photons and material particles such as electrons create analogous interference patterns when passing through a double-slit experiment. For photons, this corresponds to the interference of a Maxwell light wave whereas, for material particles, this corresponds to the interference of the Schrödinger wave equation. Although this similarity might suggest that Maxwell's equations are simply Schrödinger's equation for photons, most physicists do not agree.[33][34] For one thing, they are mathematically different; most obviously, Schrödinger's one equation solves for a complex field, whereas Maxwell's four equations solve for real fields. More generally, the normal concept of a Schrödinger probability wave function cannot be applied to photons.[35] Being massless, they cannot be localized without being destroyed; technically, photons cannot have a position eigenstate $|mathbf{r} rangle$, and, thus, the normal Heisenberg uncertainty principle $Delta x Delta p , > , h/2$ does not pertain to photons. A few substitute wave functions have been suggested for the photon,[36][37][38][39] but they have not come into general use. Instead, physicists generally accept the second-quantized theory of photons described below, quantum electrodynamics, in which photons are quantized excitations of electromagnetic modes. Interference of two circular waves - Wavelength (decreasing bottom to top) and Wave centers distance (increasing to the right). ... Double-slit diffraction and interference pattern The double-slit experiment consists of letting light diffract through two slits, which produces fringes or wave-like interference patterns on a screen. ... Lasers used for visual effects during a musical performance. ... For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... The magnitude of an electric field surrounding two equally charged (repelling) particles. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... The magnitude of an electric field surrounding two equally charged (repelling) particles. ... In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity. ... A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ...

## Bose–Einstein model of a photon gas

Main articles: Bose gas, Bose–Einstein statistics, and Spin-statistics theorem

In 1924, Satyendra Nath Bose derived Planck's law of black-body radiation without using any electromagnetism, but rather a modification of coarse-grained counting of phase space.[40] Einstein showed that this modification is equivalent to assuming that photons are rigorously identical and that it implied a “mysterious non-local interaction”,[41][42] now understood as the requirement for a symmetric quantum mechanical state. This work led to the concept of coherent states and the development of the laser. In the same papers, Einstein extended Bose's formalism to material particles (bosons) and predicted that they would condense into their lowest quantum state at low enough temperatures; this Bose–Einstein condensation was observed experimentally in 1995.[43] An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. ... For other topics related to Einstein see Einstein (disambiguation). ... The spin-statistics theorem in quantum mechanics relates the spin of a particle to the statistics obeyed by that particle. ... Satyendra Nath Bose Bengali: ) (January 1, 1894 â€“ February 4, 1974) was an Indian physicist, specializing in mathematical physics. ... In physics, the spectral intensity of electromagnetic radiation from a black body at temperature T is given by the Plancks law of black body radiation: where: I(&#957;) is the amount of energy per unit time per unit surface area per unit solid angle per unit frequency. ... Phase space of a dynamical system with focal stability. ... Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. ... In quantum mechanics a coherent state is a specific kind of quantum state of the quantum harmonic oscillator whose dynamics most closely resemble the oscillating behaviour of a classical harmonic oscillator system. ... In particle physics, bosons, named after Satyendra Nath Bose, are particles having integer spin. ... A Boseâ€“Einstein condensate (BEC) is a state of matter formed by a system of bosons confined in an external potential and cooled to temperatures very near to absolute zero (0 kelvin or -273. ...

Photons must obey Bose–Einstein statistics if they are to allow the superposition principle of electromagnetic fields, the condition that Maxwell's equations are linear. All particles are divided into bosons and fermions, depending on whether they have integer or half-integer spin, respectively. The spin-statistics theorem shows that all bosons obey Bose–Einstein statistics, whereas all fermions obey Fermi-Dirac statistics or, equivalently, the Pauli exclusion principle, which states that at most one particle can occupy any given state. Thus, if the photon were a fermion, only one photon could move in a particular direction at a time. This is inconsistent with the experimental observation that lasers can produce coherent light of arbitrary intensity, that is, with many photons moving in the same direction. Hence, the photon must be a boson and obey Bose–Einstein statistics. For other topics related to Einstein see Einstein (disambiguation). ... In linear algebra, the principle of superposition states that, for a linear system, a linear combination of solutions to the system is also a solution to the same linear system. ... The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ... In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ... In particle physics, bosons, named after Satyendra Nath Bose, are particles having integer spin. ... In particle physics, fermions are particles with half-integer spin, such as protons and electrons. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... The spin-statistics theorem in quantum mechanics relates the spin of a particle to the statistics obeyed by that particle. ... Fermi-Dirac distribution as a function of Îµ/Î¼ plotted for 4 different temperatures. ... The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ...

## Stimulated and spontaneous emission

Main articles: Stimulated emission and Laser
Stimulated emission (in which photons “clone” themselves) was predicted by Einstein in his kinetic derivation of E=hν, and led to the development of the laser. Einstein's derivation also provoked further developments in the quantum treatment of light, the semiclassical model and quantum electrodynamics (see below).

In 1916, Einstein showed that Planck's quantum hypothesis E = hν could be derived from a kinetic rate equation.[7] Consider a cavity in thermal equilibrium and filled with electromagnetic radiation and systems that can emit and absorb that radiation. Thermal equilibrium requires that the number density ρ(ν) of photons with frequency ν is constant in time; hence, the rate of emitting photons of that frequency must equal the rate of absorbing them. In optics, stimulated emission is the process by which, when perturbed by a photon, matter may lose energy resulting in the creation of another photon. ... Experiment with a laser (US Military) In physics, a laser is a device that emits light through a specific mechanism for which the term laser is an acronym: Light Amplification by Stimulated Emission of Radiation. ... Image File history File links Stimulated emission File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Stimulated emission File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In optics, stimulated emission is the process by which, when perturbed by a photon, matter may lose energy resulting in the creation of another photon. ... Experiment with a laser (US Military) In physics, a laser is a device that emits light through a specific mechanism for which the term laser is an acronym: Light Amplification by Stimulated Emission of Radiation. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ... In thermodynamics, a thermodynamic system is in thermodynamic equilibrium if its energy distribution equals a Maxwell-Boltzmann-distribution. ... Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ...

Einstein hypothesized that the rate Rji for a system to absorb a photon of frequency ν and transition from a lower energy Ej to a higher energy Ei was proportional to the number Nj of molecules with energy Ej and to the number density ρ(ν) of ambient photons with that frequency

$R_{ji} = N_{j} B_{ji} rho(nu) !$

where Bji is the rate constant for absorption. In chemical kinetics a reaction rate constant quantifies the speed of a chemical reaction. ...

More daringly, Einstein hypothesized that the reverse rate Rij for a system to emit a photon of frequency ν and transition from a higher energy Ei to a lower energy Ej was composed of two terms:

$R_{ij} = N_{i} A_{ij} + N_{i} B_{ij} rho(nu) !$

where Aij is the rate constant for emitting a photon spontaneously, and Bij is the rate constant for emitting it in response to ambient photons (induced or stimulated emission). Einstein showed that Planck's energy formula E = hν is a necessary consequence of these hypothesized rate equations and the basic requirements that the ambient radiation be in thermal equilibrium with the systems that absorb and emit the radiation and independent of the systems' material composition. Spontaneous emission is the process by which a molecule in an excited state drops to the ground state, resulting in the creation of a photon. ... In optics, stimulated emission is the process by which, when perturbed by a photon, matter may lose energy resulting in the creation of another photon. ...

This simple kinetic model was a powerful stimulus for research. Einstein was able to show that Bij = Bji (i.e., the rate constants for induced emission and absorption are equal) and, perhaps more remarkably,

$A_{ij} = frac{8 pi h nu^{3}}{c^{3}} B_{ij}.$

Einstein did not attempt to justify his rate equations but noted that Aij and Bij should be derivable from a “mechanics and electrodynamics modified to accommodate the quantum hypothesis”. This prediction was borne out in quantum mechanics and quantum electrodynamics, respectively; both are required to derive Einstein's rate constants from first principles. Paul Dirac derived the Bij rate constants in 1926 using a semiclassical approach,[44] and, in 1927, succeeded in deriving all the rate constants from first principles.[45][46] Dirac's work was the foundation of quantum electrodynamics, i.e., the quantization of the electromagnetic field itself. Dirac's approach is also called second quantization or quantum field theory;[47][48][49] the earlier quantum mechanics (the quantization of material particles moving in a potential) represents the “first quantization”. Fig. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dÉªrÃ¦k]) (August 8, 1902 â€“ October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Quantum field theory (QFT) is the quantum theory of fields. ...

Einstein was troubled by the fact that his theory seemed incomplete, since it did not determine the direction of a spontaneously emitted photon. A probabilistic nature of light-particle motion was first considered by Newton in his treatment of birefringence and, more generally, of the splitting of light beams at interfaces into a transmitted beam and a reflected beam. Newton hypothesized that hidden variables in the light particle determined which path it would follow.[17] Similarly, Einstein hoped for a more complete theory that would leave nothing to chance, beginning his separation[25] from quantum mechanics. Ironically, Max Born's probabilistic interpretation of the wave function[50][51] was inspired by Einstein's later work searching for a more complete theory.[52] Sir Isaac Newton (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1726][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... A calcite crystal laid upon a paper with some letters showing the double refraction Birefringence, or double refraction, is the decomposition of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through certain types of material, such as calcite crystals, depending on... Max Born (December 11, 1882 in Breslau â€“ January 5, 1970 in GÃ¶ttingen) was a mathematician and physicist. ... In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity. ... A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ...

## Second quantization

Main article: Quantum field theory
Different electromagnetic modes (such as those depicted here) can be treated as independent simple harmonic oscillators. A photon corresponds to a unit of energy E=hν in its electromagnetic mode.

In 1910, Peter Debye derived Planck's law of black-body radiation from a relatively simple assumption.[53] He correctly decomposed the electromagnetic field in a cavity into its Fourier modes, and assumed that the energy in any mode was an integer multiple of $hnu !$, where $nu !$ is the frequency of the electromagnetic mode. Planck's law of black-body radiation follows immediately as a geometric sum. However, Debye's approach failed to give the correct formula for the energy fluctuations of blackbody radiation, which were derived by Einstein in 1909.[6] Quantum field theory (QFT) is the quantum theory of fields. ... Image File history File links Visible_EM_modes. ... Image File history File links Visible_EM_modes. ... The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ... Petrus Josephus Wilhelmus Debije (March 24, 1884 â€“ November 2, 1966) was a Dutch physical chemist. ... In physics, the spectral intensity of electromagnetic radiation from a black body at temperature T is given by the Plancks law of black body radiation: where: I(&#957;) is the amount of energy per unit time per unit surface area per unit solid angle per unit frequency. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...

In 1925, Born, Heisenberg and Jordan reinterpreted Debye's concept in a key way.[54] As may be shown classically, the Fourier modes of the electromagnetic field — a complete set of electromagnetic plane waves indexed by their wave vector $mathbf{k}$ and polarization state — are equivalent to a set of uncoupled simple harmonic oscillators. Treated quantum mechanically, the energy levels of such oscillators are known to be $E = nhnu !$, where $nu !$ is the oscillator frequency. The key new step was to identify an electromagnetic mode with energy $E = nhnu !$ as a state with $n !$ photons, each of energy $hnu !$. This approach gives the correct energy fluctuation formula. Max Born (December 11, 1882 in Breslau â€“ January 5, 1970 in GÃ¶ttingen) was a mathematician and physicist. ... Werner Karl Heisenberg (December 5, 1901 â€“ February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics, and acknowledged to be one of the most important physicists of the twentieth century. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... Electromagnetic potential is . ...

In quantum field theory, probabilities of events are computed by summing over all possible ways in which they can happen, as in the Feynman diagram shown here.

Dirac took this one step further.[45][46] He treated the interaction between a charge and an electromagnetic field as a small perturbation that induces transitions in the photon states, changing the numbers of photons in the modes, while conserving energy and momentum overall. Dirac was able to derive Einstein's $A_{ij} !$ and $B_{ij} !$ coefficients from first principles, and showed that the Bose–Einstein statistics of photons is a natural consequence of quantizing the electromagnetic field correctly (Bose's reasoning went in the opposite direction; he derived Planck's law of black body radiation by assuming BE statistics). In Dirac's time, it was not yet known that all bosons, including photons, must obey BE statistics. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity. ... In this Feynman diagram, an electron and positron annihilate and become a quark-antiquark pair. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dÉªrÃ¦k]) (August 8, 1902 â€“ October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Black body spectrum In physics, Plancks law of black body radiation predicts the spectral intensity of electromagnetic radiation at all wavelengths from a black body at temperature  : where the following table provides the definition and SI units of measure for each symbol: The wavelength is related to the frequency...

Dirac's second-order perturbation theory can involve virtual photons, transient intermediate states of the electromagnetic field; the static electric and magnetic interactions are mediated by such virtual photons. In such quantum field theories, the probability amplitude of observable events is calculated by summing over all possible intermediate steps, even ones that are unphysical; hence, virtual photons are not constrained to satisfy $E = p , c !$, and may have extra polarization states; depending on the gauge used, virtual photons may have three or four polarization states, instead of the two states of real photons. Although these transient virtual photons can never be observed, they contribute measurably to the probabilities of observable events. Indeed, such second-order and higher-order perturbation calculations can give apparently infinite contributions to the sum. Such unphysical results are corrected for using the technique of renormalization. Other virtual particles may contribute to the summation as well; for example, two photons may interact indirectly through virtual electron-positron pairs. In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. ... In physics, a virtual particle is a particle which exists for such a short time and space that its energy and momentum do not have to obey the usual relationship. ... Coulombs torsion balance In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrostatic force that one stationary, electrically charged object of small dimensions (ideally, a point source) exerts on another. ... Magnetic lines of force of a bar magnet shown by iron filings on paper In physics, magnetism is one of the phenomena by which materials exert attractive or repulsive forces on other materials. ... Quantum field theory (QFT) is the quantum theory of fields. ... In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity. ... In electrodynamics, polarization (also spelled polarisation) is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. ... In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. ... The infinity symbol âˆž in several typefaces. ... Figure 1. ... e- redirects here. ... The first detection of the positron in 1932 by Carl D. Anderson The positron is the antiparticle or the antimatter counterpart of the electron. ... Pair production refers to the creation of an elementary particle and its antiparticle, usually from a photon (or another neutral boson). ...

In modern physics notation, the quantum state of the electromagnetic field is written as a Fock state, a tensor product of the states for each electromagnetic mode A quantum state is any possible state in which a quantum mechanical system can be. ... A Fock state, in quantum mechanics, is any state of the Fock space with a well-defined number of particles in each state. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ...

$|n_{k_0}rangleotimes|n_{k_1}rangleotimesdotsotimes|n_{k_n}rangledots$

where $|n_{k_i}rangle$ represents the state in which $, n_{k_i}$ photons are in the mode $, k_i$. In this notation, the creation of a new photon in mode $, k_i$ (e.g., emitted from an atomic transition) is written as $|n_{k_i}rangle rightarrow |n_{k_i}+1rangle$. This notation merely expresses the concept of Born, Heisenberg and Jordan described above, and does not add any physics.

## The photon as a gauge boson

Main article: Gauge theory

The electromagnetic field can be understood as a gauge theory, i.e., as a field that results from requiring that symmetry hold independently at every position in spacetime.[55] For the electromagnetic field, this gauge symmetry is the Abelian U(1) symmetry of a complex number, which reflects the ability to vary the phase of a complex number without affecting real numbers made from it, such as the energy or the Lagrangian. In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, the unitary group of degree n, denoted U(n), is the group of nÃ—n unitary matrices with complex entries, with the group operation that of matrix multiplication. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In mathematics, complex geometry is the application of complex numbers to plane geometry. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ...

The quanta of an Abelian gauge field must be massless, uncharged bosons, as long as the symmetry is not broken; hence, the photon is predicted to be massless, and to have zero electric charge and integer spin. The particular form of the electromagnetic interaction specifies that the photon must have spin ±1; thus, its helicity must be $pm hbar$. These two spin components correspond to the classical concepts of right-handed and left-handed circularly polarized light. However, the transient virtual photons of quantum electrodynamics may also adopt unphysical polarization states.[55] In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... Electromagnetic interaction is a fundamental force of nature and is felt by charged leptons and quarks. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... In particle physics, helicity is the projection of the angular momentum to the direction of motion: Because the angular momentum with respect to an axis has discrete values, helicity is discrete, too. ... In electrodynamics, circular polarization of electromagnetic radiation is a polarization such that the tip of the electric field vector, at a fixed point in space, describes a circle as time progresses. ... In the description of the interaction between elementary particles in quantum field theory, a virtual particle is a temporary elementary particle, used to describe an intermediate stage in the interaction. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ...

In the prevailing Standard Model of physics, the photon is one of four gauge bosons in the electroweak interaction; the other three are denoted W+, W and Z0 and are responsible for the weak interaction. Unlike the photon, these gauge bosons have invariant mass, owing to a mechanism that breaks their SU(2) gauge symmetry. The unification of the photon with W and Z gauge bosons in the electroweak interaction was accomplished by Sheldon Glashow, Abdus Salam and Steven Weinberg, for which they were awarded the 1979 Nobel Prize in physics.[56][57][58] Physicists continue to hypothesize grand unified theories that connect these four gauge bosons with the eight gluon gauge bosons of quantum chromodynamics; however, key predictions of these theories, such as proton decay, have not been observed experimentally. The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... Gauge bosons are bosonic particles which act as carriers of the fundamental forces of Nature. ... This article or section does not cite its references or sources. ... In physics, the W and Z bosons are the elementary particles that mediate the weak nuclear force. ... The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four fundamental interactions of nature. ... The invariant mass or intrinsic mass or proper mass or just mass is a measurement or calculation of the mass of an object that is the same for all frames of reference. ... The Higgs mechanism or Anderson-Higgs mechanism, originally proposed by the British physicist Peter Higgs based on a suggestion by Philip Anderson, is the mechanism that gives mass to all elementary particles in particle physics. ... In mathematics, the special unitary group of degree n, denoted SU(n), is the group of nÃ—n unitary matrices with unit determinant. ... Professor Sheldon Lee Glashow (born December 5, 1932) is an American physicist. ... Abdus Salam at Nobel Prize ceremony with the King of Sweden Dr. Abdus Salam (Urdu: Ø¹Ø¨Ø¯ Ø§Ù„Ø³Ù„Ø§Ù…) (January 29, 1926 at Santokdas, Sahiwal in Punjab â€“ 21 November 1996 in Oxford, England) was a Pakistani theoretical physicist who received the Nobel Prize in Physics in 1979 for his work in electroweak theory which... Steven Weinberg (born May 3, 1933) is an American physicist. ... Nobel Prize medal. ... It has been suggested that this article or section be merged into Unified field theory. ... Gauge bosons are bosonic particles which act as carriers of the fundamental forces of Nature. ... In particle physics, gluons are subatomic particles that cause quarks to interact, and are indirectly responsible for the binding of protons and neutrons together in atomic nuclei. ... Quantum chromodynamics (abbreviated as QCD) is the theory of the strong interaction (color force), a fundamental force describing the interactions of the quarks and gluons found in hadrons (such as the proton, neutron or pion). ... In particle physics, proton decay is a hypothetical form of radioactive decay in which the proton decays into lighter subatomic particles, usually a neutral pion and a positron. ...

## Photon structure

According to Quantum Chromodynamics, a real photon can interact both as a point-like particle, or as a collection of quarks and gluons, i.e., like a hadron. The structure of the photon is determined not by the traditional valence quark distributions as in a proton, but by fluctuations of the point-like photon into a collection of partons.[59] Quantum chromodynamics (abbreviated as QCD) is the theory of the strong interaction (color force), a fundamental force describing the interactions of the quarks and gluons found in hadrons (such as the proton, neutron or pion). ... Quantum chromodynamics (abbreviated as QCD) is the theory of the strong interaction (color force), a fundamental force describing the interactions of the quarks and gluons found in hadrons (such as the proton, neutron or pion). ... The six flavours of quarks and their most likely decay modes. ... In particle physics, gluons are subatomic particles that cause quarks to interact, and are indirectly responsible for the binding of protons and neutrons together in atomic nuclei. ... A hadron, in particle physics, is a subatomic particle which experiences the nuclear force. ... In physics, the proton (Greek proton = first) is a subatomic particle with an electric charge of one positive fundamental unit (1. ... In particle physics, the parton was a hypothetical fundamental particle considered, in the parton model of strong interactions, to be a constituent of the hadron. ...

## Contributions to the mass of a system

The energy of a system that emits a photon is decreased by the energy E of the photon as measured in the rest frame of the emitting system, which may result in a reduction in mass in the amount E / c2. Similarly, the mass of a system that absorbs a photon is increased by a corresponding amount. The term mass in special relativity can be used in different ways, occasionally leading to confusion. ... â€œGravityâ€ redirects here. ...

This concept is applied in a key prediction of QED, the theory of quantum electrodynamics begun by Dirac (described above). QED is able to predict the magnetic dipole moment of leptons to extremely high accuracy; experimental measurements of these magnetic dipole moments have agreed with these predictions perfectly. The predictions, however, require counting the contributions of virtual photons to the mass of the lepton. Another example of such contributions verified experimentally is the QED prediction of the Lamb shift observed in the hyperfine structure of bound lepton pairs, such as muonium and positronium. Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dÉªrÃ¦k]) (August 8, 1902 â€“ October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... In quantum electrodynamics, anomalous magnetic moment of a particle is a contribution of effects of quantum mechanics, expressed by Feynman diagrams with loops, to the magnetic moment of that particle. ... In physics, a lepton is a particle with spin-1/2 (a fermion) that does not experience the strong interaction (that is, the strong nuclear force). ... In physics, the Lamb shift, named after Willis Lamb, is a small difference in energy between two energy levels and of the hydrogen atom in quantum mechanics. ... In atomic physics, hyperfine structure is a small perturbation in the energy levels (or spectra) of atoms or molecules due to the magnetic dipole-dipole interaction, arising from the interaction of the nuclear magnetic dipole with the magnetic field of the electron. ... A muonium particle is an exotic atom made up of a positive muon and an electron, and is given the symbol Mu or Î¼+eâ€“. During the muons 2 microsecond lifetime, muonium can enter into compounds such as muonium chloride (MuCl) or sodium muonide (NaMu). ... Positronium (Ps) is a quasi-stable system consisting of an electron and its anti-particle, a positron, bound together into an exotic atom. The orbit of the two particles and the set of energy levels is similar to that of the hydrogen atom (electron and proton). ...

Since photons contribute to the stress-energy tensor, they exert a gravitational attraction on other objects, according to the theory of general relativity. Conversely, photons are themselves affected by gravity; their normally straight trajectories may be bent by warped spacetime, as in gravitational lensing, and their frequencies may be lowered by moving to a higher gravitational potential, as in the Pound-Rebka experiment. However, these effects are not specific to photons; exactly the same effects would be predicted for classical electromagnetic waves. This article is in need of attention from an expert on the subject. ... Gravity is a force of attraction that acts between bodies that have mass. ... An illustration of a rotating black hole at the center of a galaxy General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... A gravitational lens is formed when the light from a very distant, bright source (such as a quasar) is bent around a massive object (such as a massive galaxy) between the source object and the observer. ... This article or section is in need of attention from an expert on the subject. ... {{Portal|Energy}Potential energy is the energy available within a physical system due to an objects position in conjunction with a conservative force which acts upon it (such as the gravitational force or Coulomb force). ... The Pound-Rebka experiment is a well known experiment in general relativity. ... Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ...

## Photons in matter

$v = frac{domega}{dk} = frac{dE}{dp}$
Retinal straightens after absorbing a photon γ of the correct wavelength

Photons can also be absorbed by nuclei, atoms or molecules, provoking transitions between their energy levels. A classic example is the molecular transition of retinal (C20H28O, Figure at right), which is responsible for vision, as discovered in 1958 by Nobel laureate biochemist George Wald and co-workers. As shown here, the absorption provokes a cis-trans isomerization that, in combination with other such transitions, is transduced into nerve impulses. The absorption of photons can even break chemical bonds, as in the photodissociation of chlorine; this is the subject of photochemistry. In physics, absorption is the process by which the energy of a photon is taken up by another entity, for example, by an atom whose valence electrons make a transition between two electronic energy levels. ... A quantum mechanical system can only be in certain states, so that only certain energy levels are possible. ... Retinal, technically called retinene1 or retinaldehyde, is a light-sensitive retinene molecule found in the photoreceptor cells of the retina. ... This does not cite any references or sources. ... George Wald (November 18, 1906â€“April 12, 1997) was an American scientist who is best known for his work with pigments in the retina. ... Photodissociation is the breakup of molecules caused by exposure to photons. ... General Name, Symbol, Number chlorine, Cl, 17 Chemical series halogens Group, Period, Block 17, 3, p Appearance yellowish green Standard atomic weight 35. ... Photochemistry is the study of the interaction of light and chemicals. ...

## Technological applications

Photons have many applications in technology. These examples are chosen to illustrate applications of photons per se, rather than general optical devices such as lenses, etc. that could operate under a classical theory of light. The laser is an extremely important application and is discussed above under stimulated emission. In optics, stimulated emission is the process by which, when perturbed by a photon, matter may lose energy resulting in the creation of another photon. ...

Individual photons can be detected by several methods. The classic photomultiplier tube exploits the photoelectric effect; a photon landing on a metal plate ejects an electron, initiating an ever-amplifying avalanche of electrons. Charge-coupled device chips use a similar effect in semiconductors; an incident photon generates a charge on a microscopic capacitor that can be detected. Other detectors such as Geiger counters use the ability of photons to ionize gas molecules, causing a detectable change in conductivity. Photomultipliers, or photomultiplier tubes (PMT) are extremely sensitive detectors of light in the ultraviolet, visible and near infrared. ... A diagram illustrating the emission of electrons from a metal plate, requiring energy gained from an incoming photon to be more than the work function of the material. ... A specially developed CCD used for ultraviolet imaging in a wire bonded package. ... A semiconductor is a fuckin solid whose electrical conductivity is in between that of a metal and that of an insulator, and can be controlled over a wide range, either permanently or dynamically. ... This article or section does not cite any references or sources. ...

Planck's energy formula E = hν is often used by engineers and chemists in design, both to compute the change in energy resulting from a photon absorption and to predict the frequency of the light emitted for a given energy transition. For example, the emission spectrum of a fluorescent light bulb can be designed using gas molecules with different electronic energy levels and adjusting the typical energy with which an electron hits the gas molecules within the bulb. A materials emission spectrum is the amount of electromagnetic radiation of each frequency it emits when it is heated (or more generally when it is excited). ... Fluorescent lamps in Shinbashi, Tokyo, Japan Assorted types of fluorescent lamps. ...

Under some conditions, an energy transition can be excited by two photons that individually would be insufficient. This allows for higher resolution microscopy, because the sample absorbs energy only in the region where two beams of different colors overlap significantly, which can be made much smaller than the excitation volume of a single beam (see two-photon excitation microscopy). Moreover, these photons cause less damage to the sample, since they are of lower energy. Two-photon excitation microscopy is a technique that allows imaging living tissue up to a depth of one millimeter. ...

In some cases, two energy transitions can be coupled so that, as one system absorbs a photon, another nearby system "steals" its energy and re-emits a photon of a different frequency. This is the basis of fluorescence resonance energy transfer, which is used to measure molecular distances. Fluorescent proteins localize the guanosine 5-triphosphate hydrolase ARF in the Golgi apparatus of a living macrophage. ...

## Recent research

The fundamental nature of the photon is believed to be understood theoretically; the prevailing Standard Model predicts that the photon is a gauge boson of spin 1, without mass and without charge, that results from a local U(1) gauge symmetry and mediates the electromagnetic interaction. However, physicists continue to check for discrepancies between experiment and the Standard Model predictions, in the hope of finding clues to physics beyond the Standard Model. In particular, experimental physicists continue to set ever better upper limits on the charge and mass of the photon; a non-zero value for either parameter would be a serious violation of the Standard Model. However, all experimental data hitherto are consistent with the photon having zero charge[13] and mass.[60] The best universally accepted upper limits on the photon charge and mass are 5×10−52 C (or 3×10−33 times the elementary charge) and 1.8×10−50 kg, respectively. [citation needed] Quantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... In mathematics, the unitary group of degree n, denoted U(n), is the group of nÃ—n unitary matrices with complex entries, with the group operation that of matrix multiplication. ... The coulomb (symbol: C) is the SI unit of electric charge. ... The elementary charge (symbol e or sometimes q) is the electric charge carried by a single proton, or equivalently, the negative of the electric charge carried by a single electron. ... The U.S. National Prototype Kilogram, which currently serves as the primary standard for measuring mass in the U.S. It was assigned to the United States in 1889 and is periodically recertified and traceable to the primary international standard, The Kilogram, held at the Bureau International des Poids et...

Much research has been devoted to applications of photons in the field of quantum optics. Photons seem well-suited to be elements of an ultra-fast quantum computer, and the quantum entanglement of photons is a focus of research. Nonlinear optical processes are another active research area, with topics such as two-photon absorption, self-phase modulation and optical parametric oscillators. However, such processes generally do not require the assumption of photons per se; they may often be modeled by treating atoms as nonlinear oscillators. The nonlinear process of spontaneous parametric down conversion is often used to produce single-photon states. Finally, photons are essential in some aspects of optical communication, especially for quantum cryptography. Quantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter. ... The Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers. ... It has been suggested that Quantum coherence be merged into this article or section. ... Nonlinear optics is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization P responds nonlinearly to the electric field E of the light. ... Self-phase modulation (SPM) is a nonlinear optical effect of light-matter interaction. ... An optical parametric oscillator (OPO) converts a input laser wave (called pump) into two output waves of lower frequency () by means of nonlinear optical interaction. ... Spontaneous parametric down-conversion is an important process in quantum optics. ... Optical communication is any form of telecommunication that uses light as the transmission medium. ... Quantum cryptography is an approach based on quantum physics for secure communications. ...

 Physics Portal

Image File history File links Portal. ... This article does not cite any references or sources. ... Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ... Quantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter. ... This article or section is in need of attention from an expert on the subject. ... Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. ... Photography [fÓ™tÉ‘grÓ™fi:],[foÊŠtÉ‘grÓ™fi:] is the process of recording pictures by means of capturing light on a light-sensitive medium, such as a film or sensor. ... Experiment with a laser (US Military) In physics, a laser is a device that emits light through a specific mechanism for which the term laser is an acronym: Light Amplification by Stimulated Emission of Radiation. ...

## References and footnotes

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2. ^ Official particle table for gauge and Higgs bosons Retrieved October 24, 2006
3. ^ a b The mass of the photon is believed to be exactly zero, based on experiment and theoretical considerations described in the article. Some sources also refer to the "relativistic mass" concept, which is just the energy scaled to units of mass. For a photon with wavelength λ or energy E, this is h/λc or E/c2. This usage for the term "mass" is no longer common in scientific literature.
4. ^ Vimal, R. L. P., Pokorny, J., Smith, V. C., & Shevell, S. K. (1989). Foveal cone thresholds. Vision Res, 29(1), 61-78.http://www.geocities.com/vri98/Vimal-foveal-cone-ratio-VR-1989
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6. ^ a b c d Einstein, A (1909). "Über die Entwicklung unserer Anschauungen über das Wesen und die Konstitution der Strahlung (trans. The Development of Our Views on the Composition and Essence of Radiation)". Physikalische Zeitschrift 10: 817–825.  (German). An English translation is available from Wikisource.
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27. ^ Bohr, N; Kramers HA and Slater JC (1924). "The Quantum Theory of Radiation". Philosophical Magazine 47: 785–802.  Also Zeitschrift für Physik, 24, 69 (1924).
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29. ^ Mandel, L (1976). "The case for and against semiclassical radiation theory". Progress in Optics XIII: 27–69.
30. ^ a b These experiments produce results that cannot be explained by any classical theory of light, since they involve anticorrelations that result from the quantum measurement process. In 1974, the first such experiment was carried out by Clauser, who reported a violation of a classical Cauchy–Schwarz inequality. In 1977, Kimble et al. demonstrated an analogous anti-bunching effect of photons interacting with a beam splitter; this approach was simplified and sources of error eliminated in the photon-anticorrelation experiment of Grangier et al. (1986). This work is reviewed and simplified further in Thorn et al. (2004). (These references are listed below under Additional references.)
31. ^ Taylor, GI (1909). "Interference fringes with feeble light". Proceedings of the Cambridge Philosophical Society 15: 114–115.
32. ^ Heisenberg, W (1927). "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik". Zeitschrift für Physik 43: 172–198.  (German)
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34. ^ Bohm, D (1954). Quantum Theory. London: Constable.
35. ^ Newton, TD; Wigner EP (1949). "Localized states for elementary particles". Reviews of Modern Physics 21: 400–406.
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38. ^ Bialynicki-Birula, I (1996). "Photon wave function". Progress in Optics 36: 245–294.
39. ^ Scully, MO; Zubairy MS (1997). Quantum Optics. Cambridge: Cambridge University Press.
40. ^ Bose, SN (1924). "Plancks Gesetz und Lichtquantenhypothese". Zeitschrift für Physik 26: 178–181.  (German)
41. ^ Einstein, A (1924). "Quantentheorie des einatomigen idealen Gases". Sitzungsberichte der Preussischen Akademie der Wissenschaften (Berlin), Physikalisch-mathematische Klasse 1924: 261–267.  (German)
42. ^ Einstein, A (1925). "Quantentheorie des einatomigen idealen Gases, Zweite Abhandlung". Sitzungsberichte der Preussischen Akademie der Wissenschaften (Berlin), Physikalisch-mathematische Klasse 1925: 3–14.  (German)
43. ^ Anderson, MH; Ensher JR, Matthews MR, Wieman CE, and Cornell EA (1995). "Observation of Bose–Einstein Condensation in a Dilute Atomic Vapor". Science 269: 198–201.
44. ^ Dirac, PAM (1926). "On the Theory of Quantum Mechanics". Proc. Roy. Soc. A 112: 661–677.
45. ^ a b Dirac, PAM (1927a). "The Quantum Theory of the Emission and Absorption of Radiation". Proc. Roy. Soc. A 114: 243–265.
46. ^ a b Dirac, PAM (1927b). "The Quantum Theory of Dispersion". Proc. Roy. Soc. A 114: 710–728.
47. ^ Heisenberg, W; Pauli W (1929). "Zur Quantentheorie der Wellenfelder". Zeitschrift für Physik 56: 1.  (German)
48. ^ Heisenberg, W; Pauli W (1930). "Zur Quantentheorie der Wellenfelder". Zeitschrift für Physik 59: 139.  (German)
49. ^ Fermi, E (1932). "Quantum Theory of Radiation". Reviews of Modern Physics 4: 87.
50. ^ Born, M (1926a). "Zur Quantenmechanik der Stossvorgänge". Zeitschrift für Physik 37: 863–867.  (German)
51. ^ Born, M (1926b). "Zur Quantenmechanik der Stossvorgänge". Zeitschrift für Physik 38: 803.  (German)
52. ^ Pais, A (1986). Inward Bound: Of Matter and Forces in the Physical World. Oxford University Press.  Specifically, Born claimed to have been inspired by Einstein's never-published attempts to develop a “ghost-field” theory, in which point-like photons are guided probabilistically by ghost fields that follow Maxwell's equations.
53. ^ Debye, P (1910). "Der Wahrscheinlichkeitsbegriff in der Theorie der Strahlung". Annalen der Physik 33: 1427–34.  (German)
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55. ^ a b Ryder, LH (1996). Quantum field theory, 2nd edition, Cambridge University Press. ISBN 0-521-47814-6.
56. ^ Sheldon Glashow Nobel lecture, delivered 8 December 1979.
57. ^ Abdus Salam Nobel lecture, delivered 8 December 1979.
58. ^ Steven Weinberg Nobel lecture, delivered 8 December 1979.
59. ^ QCD and Two-Photon Physics, in Linear Collider Physics Resource Book for Snowmass 2001, Chapter 7, LC-REV-2001-074-US.
60. ^ (a) Goldhaber, AS (1971). "Terrestrial and Extraterrestrial Limits on The Photon Mass". Reviews of Modern Physics 43: 277–96..
(b) Fischbach, E; Kloor H, Langel RA, Lui ATY, and Peredo M (1994). "New Geomagnetic Limits on the Photon Mass and on Long-Range Forces Coexisting with Electromagnetism". Physical Review Letters 73: 514–17..
(c) Official particle table for gauge and Higgs bosons S. Eidelman et al. (Particle Data Group) Physics Letters B 592, 1 (2004)
(d) Davis, L; Goldhaber AS and Nieto MM (1975). "Limit on Photon Mass Deduced from Pioneer-10 Observations of Jupiter's Magnetic Field". Physical Review Letters 35: 1402–1405..
(e) Luo, J; Shao CG, Liu ZZ, and Hu ZK (1999). "Determination of the limit of photon mass and cosmic magnetic vector with rotating torsion balance". Physical Review A 270: 288–292..
(f) Schaeffer, BE (1999). "Severe limits on variations of the speed of light with frequency". Physical Review Letters 82: 4964–4966..
(g) Luo, J; Tu LC, Hu ZK, and Luan EJ (2003). "New experimental limit on the photon rest mass with a rotating torsion balance". Physical Review Letters 90: Art. No. 081801.
(h) Williams, ER; Faller JE and Hill HA (1971). "New Experimental Test of Coulomb's Law: A Laboratory Upper Limit on the Photon Rest Mass". Physical Review Letters 26: 721–724.
(i) Lakes, R (1998). "Experimental Limits on the Photon Mass and Cosmic Magnetic Vector Potential". Physical Review Letters 80: 1826.
(j) 2006 PDG listing for photon W.-M. Yao et al. (Particle Data Group) Journal of Physics G 33, 1 (2006).
(k) Adelberger, E; Dvali, G and Gruzinov, A. "Photon Mass Bound Destroyed by Vortices". Physical Review Letters 98: Art. No. 010402.

• Clauser, JF. (1974). "Experimental distinction between the quantum and classical field-theoretic predictions for the photoelectric effect". Phys. Rev. D 9: 853–860.
• Kimble, HJ; Dagenais M, and Mandel L. (1977). "Photon Anti-bunching in Resonance Fluorescence". Phys. Rev. Lett. 39: 691–695.  article web link
• Grangier, P; Roger G, and Aspect A. (1986). "Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences". Europhysics Letters 1: 501–504.
• Thorn, JJ; Neel MS, Donato VW, Bergreen GS, Davies RE and Beck M. (2004). "Observing the quantum behavior of light in an undergraduate laboratory". American Journal of Physics 72: 1210–1219.  http://people.whitman.edu/~beckmk/QM/grangier/grangier.html
• Pais, A. (1982). Subtle is the Lord: The Science and the Life of Albert Einstein. Oxford University Press.  An excellent history of the photon's early development.
• Ray Glauber's Nobel Lecture, “100 Years of Light Quanta”. Delivered 8 December 2005. Another history of the photon, summarized by a key physicist who developed the concepts of coherent states of photons.
• Lamb, WE (1995). "Anti-photon". Applied Physics B 60: 77–84.  Feisty, fun and sometimes snarky history of the photon, with a strong argument for allowing only its second-quantized definition, by Willis Lamb, the 1955 Nobel laureate in Physics.
• Special supplemental issue of Optics and Photonics News (vol. 14, October 2003)
• Roychoudhuri, C; Rajarshi R. "The nature of light: what is a photon?". Optics and Photonics News 14: S1 (Supplement).
• Zajonc, A. "Light reconsidered". Optics and Photonics News 14: S2–S5 (Supplement).
• Loudon, R. "What is a photon?". Optics and Photonics News 14: S6–S11 (Supplement).
• Finkelstein, D. "What is a photon?". Optics and Photonics News 14: S12–S17 (Supplement).
• Muthukrishnan, A; Scully MO, Zubairy MS. "The concept of the photon — revisited". Optics and Photonics News 14: S18–S27 (Supplement).
• Mack, H; Schleich WP. "A photon viewed from Wigner phase space". Optics and Photonics News 14: S28–S35 (Supplement).

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