Nomenclature

Standard model of particle physics

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In 1900, Max Planck was working on black-body radiation and suggested that the energy in electromagnetic waves could only be released in "packets" of energy; he called these quanta (singular quantum). Later, in 1905 Albert Einstein went further by suggesting that EM waves could only exist in these discrete wave-packets.^[2] He called such a wave-packet the light quantum (German: das Lichtquant). The name photon derives from the Greek word for light, φως (transliterated phôs), and was coined in 1926 by the physical chemist Gilbert Lewis, who published a speculative theory in which photons were "uncreatable and indestructible".^[3] Although Lewis' theory was never accepted—being contradicted by many experiments—his new name, photon, was adopted immediately by most physicists. Isaac Asimov credits Arthur Compton with defining quanta of energy as photons in 1927.^[4][5]

In physics, a photon is usually denoted by the symbol γ (the Greek letter gamma). This symbol for the photon probably derives from gamma rays, which were discovered and named in 1900 by Paul Villard^[6][7] and shown to be a form of electromagnetic radiation in 1914 by Ernest Rutherford and Edward Andrade.^[8] In chemistry and optical engineering, photons are usually symbolized by hν, the energy of a photon, where h is Planck's constant and the Greek letter ν (nu) is the photon's frequency. Much less commonly, the photon can be symbolized by hf, where its frequency is denoted by f.

Physical properties

See also: Special relativity

The photon is massless,^{[Notes 1]} has no electric charge,^[9] and does not decay spontaneously in empty space. A photon has two possible polarization (disambiguation) states and is described by exactly three continuous parameters: the components of its wave vector, which determine its wavelength λ and its direction of propagation. The photon is the gauge boson for electromagnetism,^[10] and therefore all other quantum numbers of the photon (such as lepton number, baryon number, and flavour quantum numbers) are zero.^[11]

Photons are emitted in many natural processes. For example, when a charge is accelerated it emits synchrotron radiation. During a molecular, atomic or nuclear transition to a lower energy level, photons of various energy will be emitted, from infrared light to gamma rays. A photon can also be emitted when a particle and its corresponding antiparticle are annihilated (see Electron-positron annihilation for an example).

In empty space, the photon moves at c (the speed of light) and its energy and momentum are related by E = pc, where p is the magnitude of the momentum vector p. For comparison, the corresponding equation for particles with mass m is:^[12]

The energy and momentum of a photon depend only on its frequency (ν) or equivalently, its wavelength (λ):

where k is the wave vector (with the wave number k = 2π/λ as its magnitude), ω = 2πν is the angular frequency, and ħ = h/2π is the reduced Planck constant.^[13]

Since p points in the direction of the photon's propagation, the magnitude of the momentum is

The photon also carries spin angular momentum that does not depend on its frequency.^[14] The magnitude of its spin is and the component measured along its direction of motion, its helicity, must be ±ħ. These two possible helicities, called right-handed and left-handed, correspond to the two possible circular polarization states of the photon.^[15]

To illustrate the significance of these formulae, the annihilation of a particle with its antiparticle in free space 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 (since it is determined, as we have seen, only by the photon's frequency or wavelength - which cannot be zero). Hence, conservation of momentum (or equivalently, translational invariance) requires that at least two photons are created, with zero net momentum. (However it is possible if the system interacts with another particle or field for annihilation to produce one photon, as when a positron annihilates with a bound atomic electron, it is possible for only one photon to be emitted, as the nuclear Coulomb field breaks translational symmetry.) 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.^[16] That process is the reverse of "annihilation to one photon" allowed in the electric field of an atomic nucleus.

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.^[17]

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. One of the earliest particle theories was described in the Book of Optics (1021) by Alhazen, who held light rays to be streams of minute particles that "lack all sensible qualities except energy."^[18] Since particle models cannot easily account for the refraction, diffraction and birefringence of light, wave theories of light were proposed by René Descartes (1637),^[19] Robert Hooke (1665),^[20] and Christian Huygens (1678);^[21] however, particle models remained dominant, chiefly due to the influence of Isaac Newton.^[22] 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.^[23] In 1865, James Clerk Maxwell's prediction^[24] that light was an electromagnetic wave—which was confirmed experimentally in 1888 by Heinrich Hertz's detection of radio waves^[25]—seemed to be the final blow to particle models of light.

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.^[26] .^{[Notes 2]}

At the same time, investigations of blackbody radiation carried out over four decades (1860–1900) by various researchers^[27] culminated in Max Planck's hypothesis^[28][29] 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,^[2][30] some form of energy quantization must be assumed to account for the thermal equilibrium observed between matter and electromagnetic radiation; for this explanation of the photoelectric effect, Einstein received the 1921 Nobel Prize in physics.^[31]

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.^[2] 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.^[2] In 1909^[30] and 1916,^[32] 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^[33] 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,^[34] and was solved in quantum electrodynamics and its successor, the Standard Model (see Second quantization and The photon as a gauge boson, below).

Early objections

Up to 1923, most physicists were reluctant to accept that light itself was quantized. Instead, they tried to explain photon behavior by quantizing only matter, as in the Bohr model of the hydrogen atom (shown here). Even though these semiclassical models were only a first approximation, they were accurate for simple systems and they led to quantum mechanics.

Einstein's 1905 predictions were verified experimentally in several ways in the first two decades of the 20th century, as recounted in Robert Millikan's Nobel lecture.^[35] However, before Compton's experiment^[33] showing that photons carried momentum proportional to their wave number (or frequency) (1922), most physicists were reluctant to believe that electromagnetic radiation itself might be particulate. (See, for example, the Nobel lectures of Wien,^[27] Planck^[29] and Millikan.^[35]). Instead, there was a widespread belief that energy quantization resulted from some unknown constraint on the matter that absorbs or emits radiation. Attitudes changed over gradually. In part, the change can be traced to experiments such as Compton scattering, where it was much more difficult not to ascribe quantization to light itself to explain the observed results.^[36]

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.^[37] To account for the then-available data, two drastic hypotheses had to be made:

1. 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 (at the time believed to be) discontinuously changing energy of the atom (jump between energy states) with the continuous release of energy into radiation. (It is now known that this is actually a continuous process, the combined atom-field system evolving in time according to Schroedinger's equation.)
2. Causality is abandoned. For example, spontaneous emissions are merely emissions induced by a "virtual" electromagnetic field.

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".^[34] Nevertheless, the failures of the BKS model inspired Werner Heisenberg in his development of matrix mechanics.^[38]

A few physicists persisted^[39] in developing semiclassical models in which electromagnetic radiation is not quantized, but matter appears to obey 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 photon-correlation experiments.^{[Notes 3]} Hence, Einstein's hypothesis that quantization is a property of light itself is considered to be proven.

Wave–particle duality and uncertainty principles

See also: Wave–particle duality, Squeezed coherent state, and Uncertainty principle

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 exhibiting interference phenomena but only if no measure was made on the actual slit being run across. To account for the particle interpretation that phenomena is called probability distribution but behaves according to the Maxwell's equations.^[40] 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"^[41]. Rather, the photon seems to be 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. 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).

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.

A key element of quantum mechanics is Heisenberg's uncertainty principle, which forbids the simultaneous measurement of the position and momentum of a particle along the same direction. Remarkably, the uncertainty principle for charged, material particles requires the quantization of light into photons, and even the frequency dependence of the photon's energy and momentum. An elegant illustration is Heisenberg's thought experiment for locating an electron with an ideal microscope.^[42] The position of the electron can be determined to within the resolving power of the microscope, which is given by a formula from classical optics

where θ is the aperture angle of the microscope. Thus, the position uncertainty Δx can be made arbitrarily small by reducing the wavelength λ. 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

giving the product , 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.^[43]

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

ΔnΔφ > 1

See coherent state and squeezed coherent state for more details.

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.^[44][45] 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.^[46] Being massless, they cannot be localized without being destroyed; technically, photons cannot have a position eigenstate , and, thus, the normal Heisenberg uncertainty principle ΔxΔp > h / 2 does not pertain to photons. A few substitute wave functions have been suggested for the photon,^{[47][48][49][50]} 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.