Photon
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| Photon
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In modern physics, the photon is the elementary particle responsible for electromagnetic phenomena. It mediates electromagnetic interactions and is the fundamental constituent of all forms of electromagnetic radiation, that is, light. The photon has zero rest mass and, in empty space, travels at a constant speed c; in the presence of matter, it can be slowed or even absorbed, transferring energy and momentum proportional to its frequency. The photon has both wave and particle properties; it exhibits wave-particle duality.
The modern concept of the photon was developed gradually (1905–17) by Albert Einstein[1][2][3][4] to explain experimental observations that seemed anomalous by the classical wave model of light. In particular, the photon model captured the frequency dependence of light's energy and momentum, 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, experiments eventually proved Einstein's hypothesis that light itself is particulate.
The photon concept has led to many 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. Photons have many applications in technology such as photochemistry, CCD cameras, medical imaging, 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.
[edit] Nomenclature
The photon was originally called a "light quantum" (das Lichtquant) by Albert Einstein.[1] The modern name "photon" derives from the Greek word for light, φῶς, and was coined in 1926 by the physical chemist Gilbert N. Lewis, who published a speculative theory[5] 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.
In physics, a photon is usually denoted by the symbol 
, the Greek letter gamma. In chemistry and optical engineering, photons are usually symbolized by 
, the energy of a photon, where 
is Planck's constant and the Greek letter 
(nu) is the photon's frequency. Much less commonly, the photon can be symbolized by hf, where its frequency is denoted by f.
[edit] Physical properties of the photon
- See also: Special relativity
- Some readers may wish to skip this section on first reading. The definitions and descriptions are clearly written, and necessary for a quantitative understanding of the photon. However, their interpretation may require some prior knowledge of physics.
The photon is massless,[6] has no electric charge[7] and does not decay spontaneously in empty space. A photon has two possible polarization states and is described by three continuous parameters: the components of its wave vector, which determine its wavelength 
and its direction of propagation. Photons are emitted in many natural processes, e.g., when a charge is accelerated, when an atom or a nucleus jumps from a higher to lower energy level, or when a particle and its antiparticle are annihilated. Photons are absorbed in the time-reversed processes, e.g., in the production of particle–antiparticle pairs or in atomic or nuclear transitions to a higher energy level.
Since the photon is massless, the photon moves at 
(the speed of light in empty space) and its energy 
and momentum 
are related by 
, where 
is the magnitude of the momentum. For comparison, the corresponding equation for particles with an invariant mass 
would be 
, as shown in special relativity.
The energy and momentum of a photon depend only on its frequency or, equivalently, its wavelength
and consequently the magnitude of the momentum is
where 
(known as Dirac's constant or Planck's reduced constant); 
is the wave vector (with the wave number 
as its magnitude) and 
is the angular frequency. Notice that 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 
and the component measured along its direction of motion, its helicity, must be 
. These two possible helicities correspond to the two possible circular polarization states of the photon (right-handed and left-handed).
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. The reverse process, pair production, is the dominant mechanism by which high-energy photons such as gamma rays lose energy while passing through matter.
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.
[edit] Historical development of the photon concept
- Main article: 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),[8] Robert Hooke (1665),[9] and Christian Huygens (1678);[10] however, particle models remained dominant, chiefly due to the influence of Isaac Newton.[11] 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.[12] In 1865, James Clerk Maxwell's prediction[13] that light was an electromagnetic wave — which was confirmed experimentally in 1888 by Heinrich Hertz's detection of radio waves[14] — seemed to be the final blow to particle models of light.
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 can be provoked only by light of frequency higher than a certain threshold; light of lower frequency, no matter how intense, is incapable of exciting 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.
At the same time, investigations of blackbody radiation carried out over four decades (1860–1900) by various researchers[15] culminated in Max Planck's hypothesis[16][17] that the energy of any system that absorbs or emits electromagnetic radiation of frequency is an integer multiple of an energy quantum 
. As shown by Albert Einstein,[1][2] some form of energy quantization must be assumed to account for the thermal equilibrium observed between matter and electromagnetic radiation.
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.[1] 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.[1] In 1909[2] and 1916,[4] Einstein showed that, if Planck's law of black-body radiation is accepted, the energy quanta must also carry momentum 
, making them full-fledged particles. This photon momentum was observed experimentally[18] 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, and was solved in quantum electrodynamics and its successor, the Standard Model.
[edit] Early objections to the photon hypothesis
Einstein's 1905 predictions were verified experimentally in several ways within the first two decades of the 20th century, as recounted in Robert Millikan's Nobel lecture.[19] However, before Compton's experiment[18] showing that photons carried momentum proportional to their frequency (1922), most physicists were reluctant to believe that electromagnetic radiation itself might be particulate. (See, for example, the Nobel lectures of Wien,[15] Planck[17] and Millikan.[19]) This reluctance is understandable, given the success and plausibility of Maxwell's electromagnetic wave model of light. Therefore, most physicists assumed rather that energy quantization resulted from some unknown constraint on the matter that absorbs or emits radiation. Niels Bohr, Arnold Sommerfeld and others developed atomic models with discrete energy levels that could account qualitatively for the sharp spectral lines and energy quantization observed in the emission and absorption of light by atoms; their models agreed excellently with the spectrum of hydrogen, but not with those of other atoms. It was only the Compton scattering of a photon by a free electron (which can have no energy levels, since it has no internal structure) that convinced most physicists that light itself was quantized.
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.[20] To account for the then-available data, two drastic hypotheses had to be made:
- 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.
- 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".[21] Nevertheless, the BKS model inspired Werner Heisenberg in his development[22] of quantum mechanics.
A few physicists persisted[23] 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 1970's and 1980's by elegant photon-correlation experiments.[24] Hence, Einstein's hypothesis that quantization is a property of light itself is considered to be proven.
[edit] Wave–particle duality and uncertainty principles
- See also: Wave–particle duality, Squeezed coherent state, and Uncertainty principle
Photons exhibit both wave-like and particle-like properties, and 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.[25] 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.[24] 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).
The quantum mechanics of material particles features an uncertainty principle that forbids the simultaneous measurement of the position and momentum of a particle in the same direction. An analogous principle for photons forbids the simultaneous measurement of the number 
of photons (see Fock state and the Second quantization section below) in an electromagnetic wave and the phase 
of that wave
See coherent state and squeezed coherent state for more details.
Remarkably, the quantization of light into photons and even the frequency dependence of the photon's energy and momentum can be seen as necessary consequences of the quantum mechanics of charged, material particles such as the electron. An elegant illustration is Werner Heisenberg's thought experiment for locating an electron with an ideal microscope.[26] 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 
is the aperture angle of the microscope. Thus, the position uncertainty 
can be made arbitrarily small by reducing the wavelength. The momentum of the electron is uncertain, since it received a "kick" 
from the light scattering from it into the microscope. If light were not quantized into photons, the uncertainty 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, all the world is quantized; both matter and fields must obey a consistent set of quantum laws, if either one is to be quantized.
[edit] Bose–Einstein model of a photon gas
- Main article: Bose–Einstein statistics
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.[27] Einstein showed that this modification is equivalent to assuming that photons are rigorously identical and that it implied a "mysterious non-local interaction",[28][29] 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[30].
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.
[edit] Stimulated and spontaneous emission
- Main article: Stimulated emission
In 1916, Einstein showed that Planck's quantum hypothesis 
could be derived from a kinetic rate equation.[3] 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 
of photons with frequency 
is constant in time; hence, the rate of emitting photons of that frequency must equal the rate of absorbing them.
Einstein hypothesized that the rate 
for a system to absorb a photon of frequency and transition from a lower energy 
to a higher energy 
was proportional to the number 
of molecules with energy and to the number density of ambient photons with that frequency
where 
is the rate constant for absorption.
More daringly, Einstein hypothesized that the reverse rate 
for a system to emit a photon of frequency and transition from a higher energy