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134-505: As with other elementary particles, photons are best explained by quantum mechanics and exhibit wave–particle duality , their behavior featuring properties of both waves and particles . The modern photon concept originated during the first two decades of the 20th century with the work of Albert Einstein , who built upon the research of Max Planck . While Planck was trying to explain how matter and electromagnetic radiation could be in thermal equilibrium with one another, he proposed that

268-480: A {\displaystyle a} larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle. As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that

402-418: A Hermitian operator . In 1924, Satyendra Nath Bose derived Planck's law of black-body radiation without using any electromagnetism, but rather by using a modification of coarse-grained counting of phase space . Einstein showed that this modification is equivalent to assuming that photons are rigorously identical and that it implied a "mysterious non-local interaction", now understood as the requirement for

536-587: A camel 's neck, but this has been criticized as contrived, and it is more likely that the letter is derived from an Egyptian hieroglyph representing a club or throwing stick . In Archaic Greece , the shape of gamma was closer to a classical lambda (Λ), while lambda retained the Phoenician L -shape ( 𐌋 ‎). Letters that arose from the Greek gamma include Etruscan (Old Italic) 𐌂, Roman C and G , Runic kaunan ᚲ , Gothic geuua 𐌲 ,

670-520: A point-like particle since it is absorbed or emitted as a whole by arbitrarily small systems, including systems much smaller than its wavelength, such as an atomic nucleus (≈10 m across) or even the point-like electron . While many introductory texts treat photons using the mathematical techniques of non-relativistic quantum mechanics, this is in some ways an awkward oversimplification, as photons are by nature intrinsically relativistic. Because photons have zero rest mass , no wave function defined for

804-498: 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. It was later used by Lene Hau to slow, and then completely stop, light in 1999 and 2001. The modern view on this

938-541: A velar nasal /ŋ/ . A double gamma γγ (e.g., άγγελος, "angel") represents the sequence /ŋɡ/ (phonetically varying [ŋɡ~ɡ] ) or /ŋɣ/ . Lowercase Greek gamma is used in the Americanist phonetic notation and Uralic Phonetic Alphabet to indicate voiced consonants. The gamma was also added to the Latin alphabet, as Latin gamma , in the following forms: majuscule Ɣ, minuscule ɣ, and superscript modifier letter ˠ. In

1072-511: A voiced velar fricative IPA: [ɣ] , except before either of the two front vowels (/e/, /i/), where it represents a voiced palatal fricative IPA: [ʝ] ; while /g/ in foreign words is instead commonly transcribed as γκ). In the International Phonetic Alphabet and other modern Latin-alphabet based phonetic notations , it represents the voiced velar fricative. The Greek letter Gamma Γ

1206-589: 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. At the same time, investigations of black-body radiation carried out over four decades (1860–1900) by various researchers culminated in Max Planck 's hypothesis that

1340-511: 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, ranging from radio waves to gamma rays . Photons can also be emitted when a particle and its corresponding antiparticle are annihilated (for example, electron–positron annihilation ). In empty space, the photon moves at c (the speed of light ) and its energy and momentum are related by E = pc , where p

1474-466: A choice of measuring either one of two "canonically conjugate" quantities, like the position and the momentum of a particle. According to the uncertainty principle, no matter how the particle is prepared, it is not possible to make a precise prediction for both of the two alternative measurements: if the outcome of the position measurement is made more certain, the outcome of the momentum measurement becomes less so, and vice versa. A coherent state minimizes

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1608-466: A definite prediction of what the quantum state ψ ( t ) {\displaystyle \psi (t)} will be at any later time. Some wave functions produce probability distributions that are independent of time, such as eigenstates of the Hamiltonian . Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example,

1742-510: A family of unitary operators parameterized by a variable t {\displaystyle t} . Under the evolution generated by A {\displaystyle A} , any observable B {\displaystyle B} that commutes with A {\displaystyle A} will be conserved. Moreover, if B {\displaystyle B} is conserved by evolution under A {\displaystyle A} , then A {\displaystyle A}

1876-499: A gauge boson , below.) 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. However, before Compton's experiment showed that photons carried momentum proportional to their wave number (1922), most physicists were reluctant to believe that electromagnetic radiation itself might be particulate. (See, for example,

2010-515: A geometric sum. However, Debye's approach failed to give the correct formula for the energy fluctuations of black-body radiation, which were derived by Einstein in 1909. In 1925, Born , Heisenberg and Jordan reinterpreted Debye's concept in a key way. As may be shown classically, the Fourier modes of the electromagnetic field —a complete set of electromagnetic plane waves indexed by their wave vector k and polarization state—are equivalent to

2144-475: A higher energy E i {\displaystyle E_{i}} is proportional to the number N j {\displaystyle N_{j}} of atoms with energy E j {\displaystyle E_{j}} and to the energy density ρ ( ν ) {\displaystyle \rho (\nu )} of ambient photons of that frequency, where B j i {\displaystyle B_{ji}}

2278-504: A higher energy E i {\displaystyle E_{i}} to a lower energy E j {\displaystyle E_{j}} is where A i j {\displaystyle A_{ij}} is the rate constant for emitting a photon spontaneously , and B i j {\displaystyle B_{ij}} is the rate constant for emissions in response to ambient photons ( induced or stimulated emission ). In thermodynamic equilibrium,

2412-452: A light beam may have mixtures of these two values; a linearly polarized light beam will act as if it were composed of equal numbers of the two possible angular momenta. The spin angular momentum of light does not depend on its frequency, and was experimentally verified by C. V. Raman and S. Bhagavantam in 1931. The collision of a particle with its antiparticle can create photons. In free space at least two photons must be created since, in

2546-421: A link between the rates at which atoms emit and absorb photons. The condition follows from the assumption that functions of the emission and absorption of radiation by the atoms are independent of each other, and that thermal equilibrium is made by way of the radiation's interaction with the atoms. Consider a cavity in thermal equilibrium with all parts of itself and filled with electromagnetic radiation and that

2680-471: A loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system. Just as density matrices specify the state of a subsystem of a larger system, analogously, positive operator-valued measures (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory. As described above, entanglement

2814-426: A mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples. In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector ψ {\displaystyle \psi } belonging to a ( separable ) complex Hilbert space H {\displaystyle {\mathcal {H}}} . This vector

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2948-417: A measurement of its position and also at the same time for a measurement of its momentum . Another consequence of the mathematical rules of quantum mechanics is the phenomenon of quantum interference , which is often illustrated with the double-slit experiment . In the basic version of this experiment, a coherent light source , such as a laser beam, illuminates a plate pierced by two parallel slits, and

3082-598: A paper in which he proposed that many light-related phenomena—including black-body radiation and the photoelectric effect —would be better explained by modelling electromagnetic waves as consisting of spatially localized, discrete energy quanta. He called these a light quantum (German: ein Lichtquant ). The name photon derives from the Greek word for light, φῶς (transliterated phôs ). Arthur Compton used photon in 1928, referring to Gilbert N. Lewis , who coined

3216-457: A photon can have all the properties familiar from wave functions in non-relativistic quantum mechanics. In order to avoid these difficulties, physicists employ the second-quantized theory of photons described below, quantum electrodynamics , in which photons are quantized excitations of electromagnetic modes. Another difficulty is finding the proper analogue for the uncertainty principle , an idea frequently attributed to Heisenberg, who introduced

3350-399: A photon is calculated by equations that describe waves. This combination of aspects is known as wave–particle duality . For example, the probability distribution for the location at which a photon might be detected displays clearly wave-like phenomena such as diffraction and interference . A single photon passing through a double slit has its energy received at a point on the screen with

3484-471: A probability amplitude. Applying the Born rule to these amplitudes gives a probability density function for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The Schrödinger equation relates the collection of probability amplitudes that pertain to one moment of time to

3618-400: A probability distribution given by its interference pattern determined by Maxwell's wave equations . However, experiments confirm that the photon is not a short pulse of electromagnetic radiation; a photon's Maxwell waves will diffract, but photon energy does not spread out as it propagates, nor does this energy divide when it encounters a beam splitter . Rather, the received photon acts like

3752-409: A relatively simple assumption. He decomposed the electromagnetic field in a cavity into its Fourier modes , and assumed that the energy in any mode was an integer multiple of h ν {\displaystyle h\nu } , where ν {\displaystyle \nu } is the frequency of the electromagnetic mode. Planck's law of black-body radiation follows immediately as

3886-444: A semiclassical approach, and, in 1927, succeeded in deriving all the rate constants from first principles within the framework of quantum theory. 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 ; earlier quantum mechanical treatments only treat material particles as quantum mechanical, not

4020-460: A set of uncoupled simple harmonic oscillators . Treated quantum mechanically, the energy levels of such oscillators are known to be E = n h ν {\displaystyle E=nh\nu } , where ν {\displaystyle \nu } is the oscillator frequency. The key new step was to identify an electromagnetic mode with energy E = n h ν {\displaystyle E=nh\nu } as

4154-405: A single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus , whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an s orbital ( Fig. 1 ). Analytic solutions of

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4288-551: A single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy: The general solution of the Schrödinger equation is given by which is a superposition of all possible plane waves e i ( k x − ℏ k 2 2 m t ) {\displaystyle e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}} , which are eigenstates of

4422-423: A state with n {\displaystyle n} photons, each of energy h ν {\displaystyle h\nu } . This approach gives the correct energy fluctuation formula. Dirac took this one step further. 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

4556-425: A symbol for: The lowercase Latin gamma ɣ can also be used in contexts (such as chemical or molecule nomenclature) where gamma must not be confused with the letter y , which can occur in some computer typefaces. The uppercase letter Γ {\displaystyle \Gamma } is used as a symbol for: These characters are used only as mathematical symbols. Stylized Greek text should be encoded using

4690-412: A unit related to the illumination of the eye and the resulting sensation of light and was used later in a physiological context. Although Wolfers's and Lewis's theories were contradicted by many experiments and never accepted, the new name was adopted by most physicists very soon after Compton used it. In physics, a photon is usually denoted by the symbol γ (the Greek letter gamma ). This symbol for

4824-473: Is and this provides the lower bound on the product of standard deviations: Another consequence of the canonical commutation relation is that the position and momentum operators are Fourier transforms of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of

4958-604: Is a grapheme derived from the Phoenician letter 𐤂 ‎ ( gīml ) which was rotated from the right-to-left script of Canaanite to accommodate the Greek language's writing system of left-to-right. The Canaanite grapheme represented the /g/ phoneme in the Canaanite language , and as such is cognate with gimel ג of the Hebrew alphabet . Based on its name, the letter has been interpreted as an abstract representation of

5092-478: Is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as quantum decoherence . This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic. There are many mathematically equivalent formulations of quantum mechanics. One of

5226-424: Is a valid joint state that is not separable. States that are not separable are called entangled . If the state for a composite system is entangled, it is impossible to describe either component system A or system B by a state vector. One can instead define reduced density matrices that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes

5360-405: Is conserved under the evolution generated by B {\displaystyle B} . This implies a quantum version of the result proven by Emmy Noether in classical ( Lagrangian ) mechanics: for every differentiable symmetry of a Hamiltonian, there exists a corresponding conservation law . The simplest example of a quantum system with a position degree of freedom is a free particle in

5494-1066: Is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the action principle in classical mechanics. The Hamiltonian H {\displaystyle H} is known as the generator of time evolution, since it defines a unitary time-evolution operator U ( t ) = e − i H t / ℏ {\displaystyle U(t)=e^{-iHt/\hbar }} for each value of t {\displaystyle t} . From this relation between U ( t ) {\displaystyle U(t)} and H {\displaystyle H} , it follows that any observable A {\displaystyle A} that commutes with H {\displaystyle H} will be conserved : its expectation value will not change over time. This statement generalizes, as mathematically, any Hermitian operator A {\displaystyle A} can generate

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5628-448: Is given by The operator U ( t ) = e − i H t / ℏ {\displaystyle U(t)=e^{-iHt/\hbar }} is known as the time-evolution operator, and has the crucial property that it is unitary . This time evolution is deterministic in the sense that – given an initial quantum state ψ ( 0 ) {\displaystyle \psi (0)} – it makes

5762-406: Is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ⟨ ψ , P λ ψ ⟩ {\displaystyle \langle \psi ,P_{\lambda }\psi \rangle } , where P λ {\displaystyle P_{\lambda }} is the projector onto its associated eigenspace. In

5896-726: Is known as wave–particle duality . In addition to light, electrons , atoms , and molecules are all found to exhibit the same dual behavior when fired towards a double slit. Another non-classical phenomenon predicted by quantum mechanics is quantum tunnelling : a particle that goes up against a potential barrier can cross it, even if its kinetic energy is smaller than the maximum of the potential. In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling radioactive decay , nuclear fusion in stars, and applications such as scanning tunnelling microscopy , tunnel diode and tunnel field-effect transistor . When quantum systems interact,

6030-444: Is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states. Instead, we can consider a Gaussian wave packet : which has Fourier transform, and therefore momentum distribution We see that as we make a {\displaystyle a} smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making

6164-483: Is not quantized, but matter appears to obey the laws of quantum mechanics . Although the evidence from chemical and physical experiments for the existence of photons was overwhelming by the 1970s, this evidence could not be considered as absolutely definitive; since it relied on the interaction of light with matter, and a sufficiently complete theory of matter could in principle account for the evidence. Nevertheless, all semiclassical theories were refuted definitively in

6298-628: Is not sufficient for describing them at very small submicroscopic (atomic and subatomic ) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation, valid at large (macroscopic/microscopic) scale. Quantum systems have bound states that are quantized to discrete values of energy , momentum , angular momentum , and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both particles and waves ( wave–particle duality ), and there are limits to how accurately

6432-815: Is part of quantum communication protocols, such as quantum key distribution and superdense coding . Contrary to popular misconception, entanglement does not allow sending signals faster than light , as demonstrated by the no-communication theorem . Another possibility opened by entanglement is testing for " hidden variables ", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory provides. A collection of results, most significantly Bell's theorem , have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of local hidden variables, then

6566-540: Is postulated to be normalized under the Hilbert space inner product, that is, it obeys ⟨ ψ , ψ ⟩ = 1 {\displaystyle \langle \psi ,\psi \rangle =1} , and it is well-defined up to a complex number of modulus 1 (the global phase), that is, ψ {\displaystyle \psi } and e i α ψ {\displaystyle e^{i\alpha }\psi } represent

6700-466: Is replaced by − i ℏ ∂ ∂ x {\displaystyle -i\hbar {\frac {\partial }{\partial x}}} , and in particular in the non-relativistic Schrödinger equation in position space the momentum-squared term is replaced with a Laplacian times − ℏ 2 {\displaystyle -\hbar ^{2}} . When two different quantum systems are considered together,

6834-400: Is that photons are, by virtue of their integer spin, bosons (as opposed to fermions with half-integer spin). By the spin-statistics theorem , all bosons obey Bose–Einstein statistics (whereas all fermions obey Fermi–Dirac statistics ). In 1916, Albert Einstein showed that Planck's radiation law could be derived from a semi-classical, statistical treatment of photons and atoms, which implies

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6968-504: Is the gauge boson for electromagnetism , and therefore all other quantum numbers of the photon (such as lepton number , baryon number , and flavour quantum numbers ) are zero. Also, the photon obeys Bose–Einstein statistics , and not Fermi–Dirac statistics . That is, they do not obey the Pauli exclusion principle and more than one can occupy the same bound quantum state. Photons are emitted in many natural processes. For example, when

7102-413: Is the magnitude of the momentum vector p . This derives from the following relativistic relation, with m = 0 : The energy and momentum of a photon depend only on its frequency ( ν {\displaystyle \nu } ) or inversely, its wavelength ( λ ): where k is the wave vector , where Since p {\displaystyle {\boldsymbol {p}}} points in

7236-512: Is the rate constant for absorption. For the reverse process, there are two possibilities: spontaneous emission of a photon, or the emission of a photon initiated by the interaction of the atom with a passing photon and the return of the atom to the lower-energy state. Following Einstein's approach, the corresponding rate R i j {\displaystyle R_{ij}} for the emission of photons of frequency ν {\displaystyle \nu } and transition from

7370-416: Is the photon's frequency . The photon has no electric charge , is generally considered to have zero rest mass and is a stable particle . The experimental upper limit on the photon mass is very small, on the order of 10 kg; its lifetime would be more than 10 years. For comparison the age of the universe is about 1.38 × 10 years. In a vacuum, a photon has two possible polarization states. The photon

7504-415: Is the reduced Planck constant . The constant i ℏ {\displaystyle i\hbar } is introduced so that the Hamiltonian is reduced to the classical Hamiltonian in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the correspondence principle . The solution of this differential equation

7638-469: Is then If the state for the first system is the vector ψ A {\displaystyle \psi _{A}} and the state for the second system is ψ B {\displaystyle \psi _{B}} , then the state of the composite system is Not all states in the joint Hilbert space H A B {\displaystyle {\mathcal {H}}_{AB}} can be written in this form, however, because

7772-615: Is written as a Fock state , a tensor product of the states for each electromagnetic mode Quantum mechanics Quantum mechanics is a fundamental theory that describes the behavior of nature at and below the scale of atoms . It is the foundation of all quantum physics , which includes quantum chemistry , quantum field theory , quantum technology , and quantum information science . Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary ( macroscopic and (optical) microscopic ) scale, but

7906-505: The Born rule : in the simplest case the eigenvalue λ {\displaystyle \lambda } is non-degenerate and the probability is given by | ⟨ λ → , ψ ⟩ | 2 {\displaystyle |\langle {\vec {\lambda }},\psi \rangle |^{2}} , where λ → {\displaystyle {\vec {\lambda }}}

8040-729: The Coptic Ⲅ, and the Cyrillic letters Г and Ґ . The Ancient Greek /g/ phoneme was the voiced velar stop, continuing the reconstructed proto-Indo-European *g , *ǵ . The modern Greek phoneme represented by gamma is realized either as a voiced palatal fricative ( /ʝ/ ) before a front vowel (/e/, /i/), or as a voiced velar fricative /ɣ/ in all other environments. Both in Ancient and in Modern Greek, before other velar consonants (κ, χ, ξ – that is, k, kh, ks ), gamma represents

8174-420: The International Phonetic Alphabet the minuscule letter is used to represent a voiced velar fricative and the superscript modifier letter is used to represent velarization . It is not to be confused with the character ɤ , which looks like a lowercase Latin gamma that lies above the baseline rather than crossing, and which represents the close-mid back unrounded vowel . In certain nonstandard variations of

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8308-543: The Standard Model of particle physics , photons and other elementary particles are described as a necessary consequence of physical laws having a certain symmetry at every point in spacetime . The intrinsic properties of particles, such as charge , mass , and spin , are determined by gauge symmetry . The photon concept has led to momentous advances in experimental and theoretical physics, including lasers , Bose–Einstein condensation , quantum field theory , and

8442-713: The canonical commutation relation : Given a quantum state, the Born rule lets us compute expectation values for both X {\displaystyle X} and P {\displaystyle P} , and moreover for powers of them. Defining the uncertainty for an observable by a standard deviation , we have and likewise for the momentum: The uncertainty principle states that Either standard deviation can in principle be made arbitrarily small, but not both simultaneously. This inequality generalizes to arbitrary pairs of self-adjoint operators A {\displaystyle A} and B {\displaystyle B} . The commutator of these two operators

8576-496: The center of momentum frame , the colliding antiparticles have no net momentum, whereas a single photon always has momentum (determined 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. The energy of the two photons, or, equivalently, their frequency, may be determined from conservation of four-momentum . Seen another way,

8710-463: The degeneracy of the state i {\displaystyle i} and that of j {\displaystyle j} , respectively, E i {\displaystyle E_{i}} and E j {\displaystyle E_{j}} their energies, k {\displaystyle k} the Boltzmann constant and T {\displaystyle T}

8844-420: 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. In 1909 and 1916, Einstein showed that, if Planck's law regarding black-body radiation is accepted, the energy quanta must also carry momentum p = ⁠ h  / λ  ⁠ , making them full-fledged particles. This photon momentum

8978-423: The photoelectric effect . These early attempts to understand microscopic phenomena, now known as the " old quantum theory ", led to the full development of quantum mechanics in the mid-1920s by Niels Bohr , Erwin Schrödinger , Werner Heisenberg , Max Born , Paul Dirac and others. The modern theory is formulated in various specially developed mathematical formalisms . In one of them, a mathematical entity called

9112-411: The probabilistic interpretation of quantum mechanics. It has been applied to photochemistry , high-resolution microscopy , and measurements of molecular distances . Moreover, photons have been studied as elements of quantum computers , and for applications in optical imaging and optical communication such as quantum cryptography . The word quanta (singular quantum, Latin for how much )

9246-407: 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 {\displaystyle E=pc} , and may have extra polarization states; depending on the gauge used, virtual photons may have three or four polarization states, instead of

9380-562: The wave function provides information, in the form of probability amplitudes , about what measurements of a particle's energy, momentum, and other physical properties may yield. Quantum mechanics allows the calculation of properties and behaviour of physical systems. It is typically applied to microscopic systems: molecules, atoms and sub-atomic particles. It has been demonstrated to hold for complex molecules with thousands of atoms, but its application to human beings raises philosophical problems, such as Wigner's friend , and its application to

9514-414: The 1970s and 1980s by photon-correlation experiments. Hence, Einstein's hypothesis that quantization is a property of light itself is considered to be proven. Photons obey the laws of quantum mechanics, and so their behavior has both wave-like and particle-like aspects. When a photon is detected by a measuring instrument, it is registered as a single, particulate unit. However, the probability of detecting

9648-431: The Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors C 2 {\displaystyle \mathbb {C} ^{2}} with the usual inner product. Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are Hermitian (more precisely, self-adjoint ) linear operators acting on

9782-411: The Hilbert space of the combined system is the tensor product of the Hilbert spaces of the two components. For example, let A and B be two quantum systems, with Hilbert spaces H A {\displaystyle {\mathcal {H}}_{A}} and H B {\displaystyle {\mathcal {H}}_{B}} , respectively. The Hilbert space of the composite system

9916-432: The Hilbert space. A quantum state can be an eigenvector of an observable, in which case it is called an eigenstate , and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition . When an observable is measured, the result will be one of its eigenvalues with probability given by

10050-516: The IPA , the uppercase form is used. It is as a full-fledged majuscule and minuscule letter in the alphabets of some of languages of Africa such as Dagbani , Dinka , Kabye , and Ewe , and Berber languages using the Berber Latin alphabet . It is sometimes also used in the romanization of Pashto . The lowercase letter γ {\displaystyle \gamma } is used as

10184-547: 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. Although he accepted the validity of Maxwell's theory, Einstein pointed out that many anomalous experiments could be explained if

10318-467: The Nobel lectures of Wien , Planck and Millikan.) Instead, there was a widespread belief that energy quantization resulted from some unknown constraint on the matter that absorbed or emitted radiation. Attitudes changed over time. In part, the change can be traced to experiments such as those revealing Compton scattering , where it was much more difficult not to ascribe quantization to light itself to explain

10452-489: The Schrödinger equation are known for very few relatively simple model Hamiltonians including the quantum harmonic oscillator , the particle in a box , the dihydrogen cation , and the hydrogen atom . Even the helium atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in closed form . However, there are techniques for finding approximate solutions. One method, called perturbation theory , uses

10586-475: The Schrödinger equation for the particle in a box are or, from Euler's formula , Gamma Gamma ( / ˈ ɡ æ m ə / ; uppercase Γ , lowercase γ ; Greek : γάμμα , romanized :  gámma ) is the third letter of the Greek alphabet . In the system of Greek numerals it has a value of 3. In Ancient Greek , the letter gamma represented a voiced velar stop IPA: [ɡ] . In Modern Greek , this letter normally represents

10720-403: The analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy . Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion. One consequence of

10854-438: The atoms can emit and absorb that radiation. Thermal equilibrium requires that the energy density ρ ( ν ) {\displaystyle \rho (\nu )} of photons with frequency ν {\displaystyle \nu } (which is proportional to their number density ) is, on average, constant in time; hence, the rate at which photons of any particular frequency are emitted must equal

10988-526: The average across many interactions between matter and radiation. However, refined Compton experiments showed that the conservation laws hold for individual interactions. Accordingly, Bohr and his co-workers gave their model "as honorable a funeral as possible". Nevertheless, the failures of the BKS model inspired Werner Heisenberg in his development of matrix mechanics . A few physicists persisted in developing semiclassical models in which electromagnetic radiation

11122-606: The basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum. Both position and momentum are observables, meaning that they are represented by Hermitian operators . The position operator X ^ {\displaystyle {\hat {X}}} and momentum operator P ^ {\displaystyle {\hat {P}}} do not commute, but rather satisfy

11256-485: The coefficients A i j {\displaystyle A_{ij}} , B j i {\displaystyle B_{ji}} and B i j {\displaystyle B_{ij}} once physicists had obtained "mechanics and electrodynamics modified to accommodate the quantum hypothesis". Not long thereafter, in 1926, Paul Dirac derived the B i j {\displaystyle B_{ij}} rate constants by using

11390-404: The collection of probability amplitudes that pertain to another. One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between measurable quantities. The most famous form of this uncertainty principle says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for

11524-416: The concept in analyzing a thought experiment involving an electron and a high-energy photon . However, Heisenberg did not give precise mathematical definitions of what the "uncertainty" in these measurements meant. The precise mathematical statement of the position–momentum uncertainty principle is due to Kennard , Pauli , and Weyl . The uncertainty principle applies to situations where an experimenter has

11658-626: The continuous case, these formulas give instead the probability density . After the measurement, if result λ {\displaystyle \lambda } was obtained, the quantum state is postulated to collapse to λ → {\displaystyle {\vec {\lambda }}} , in the non-degenerate case, or to P λ ψ / ⟨ ψ , P λ ψ ⟩ {\textstyle P_{\lambda }\psi {\big /}\!{\sqrt {\langle \psi ,P_{\lambda }\psi \rangle }}} , in

11792-431: The dependence in position means that the momentum operator is equivalent (up to an i / ℏ {\displaystyle i/\hbar } factor) to taking the derivative according to the position, since in Fourier analysis differentiation corresponds to multiplication in the dual space . This is why in quantum equations in position space, the momentum p i {\displaystyle p_{i}}

11926-466: The direction of the photon's propagation, the magnitude of its momentum is The photon also carries spin angular momentum , which is related to photon polarization . (Beams of light also exhibit properties described as orbital angular momentum of light ). The angular momentum of the photon has two possible values, either +ħ or −ħ . These two possible values correspond to the two possible pure states of circular polarization . Collections of photons in

12060-483: 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. Current commonly accepted physical theories imply or assume

12194-450: The electromagnetic field. 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

12328-487: The electromagnetic wave, Δ N {\displaystyle \Delta N} , and the uncertainty in the phase of the wave, Δ ϕ {\displaystyle \Delta \phi } . However, this cannot be an uncertainty relation of the Kennard–Pauli–Weyl type, since unlike position and momentum, the phase ϕ {\displaystyle \phi } cannot be represented by

12462-438: 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 , 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. Since

12596-408: The energy stored within a material object should be regarded as composed of an integer number of discrete, equal-sized parts. To explain the photoelectric effect , Einstein introduced the idea that light itself is made of discrete units of energy. In 1926, Gilbert N. Lewis popularized the term photon for these energy units. Subsequently, many other experiments validated Einstein's approach. In

12730-501: 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 are provoked only by light of frequency higher than

12864-416: The galactic vector potential . Although the galactic vector potential is large because the galactic magnetic field exists on great length scales, only the magnetic field would be observable if the photon is massless. In the case that the photon has mass, the mass term ⁠ 1 / 2 ⁠ m A μ A would affect the galactic plasma. The fact that no such effects are seen implies an upper bound on

12998-415: The general case. The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr–Einstein debates , in which the two scientists attempted to clarify these fundamental principles by way of thought experiments . In the decades after the formulation of quantum mechanics,

13132-400: The influence of Isaac Newton . In the early 19th century, Thomas Young and August Fresnel clearly demonstrated the interference and diffraction of light, and by 1850 wave models were generally accepted. James Clerk Maxwell 's 1865 prediction that light was an electromagnetic wave – which was confirmed experimentally in 1888 by Heinrich Hertz 's detection of radio waves – seemed to be

13266-462: The interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave). However, such experiments demonstrate that particles do not form the interference pattern if one detects which slit they pass through. This behavior

13400-450: The light particle determined which of the two paths a single photon would take. Similarly, Einstein hoped for a more complete theory that would leave nothing to chance, beginning his separation from quantum mechanics. Ironically, Max Born 's probabilistic interpretation of the wave function was inspired by Einstein's later work searching for a more complete theory. In 1910, Peter Debye derived Planck's law of black-body radiation from

13534-430: The light passing through the slits is observed on a screen behind the plate. The wave nature of light causes the light waves passing through the two slits to interfere , producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles. However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves;

13668-483: The modes, while conserving energy and momentum overall. Dirac was able to derive Einstein's A i j {\displaystyle A_{ij}} and B i j {\displaystyle 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

13802-432: The momentum operator with momentum p = ℏ k {\displaystyle p=\hbar k} . The coefficients of the superposition are ψ ^ ( k , 0 ) {\displaystyle {\hat {\psi }}(k,0)} , which is the Fourier transform of the initial quantum state ψ ( x , 0 ) {\displaystyle \psi (x,0)} . It

13936-418: The number of atoms in state i {\displaystyle i} and those in state j {\displaystyle j} must, on average, be constant; hence, the rates R j i {\displaystyle R_{ji}} and R i j {\displaystyle R_{ij}} must be equal. Also, by arguments analogous to the derivation of Boltzmann statistics ,

14070-474: The observed results. Even after Compton's experiment, Niels Bohr , Hendrik Kramers and John Slater made one last attempt to preserve the Maxwellian continuous electromagnetic field model of light, the so-called BKS theory . An important feature of the BKS theory is how it treated the conservation of energy and the conservation of momentum . In the BKS theory, energy and momentum are only conserved on

14204-413: The oldest and most common is the " transformation theory " proposed by Paul Dirac , which unifies and generalizes the two earliest formulations of quantum mechanics – matrix mechanics (invented by Werner Heisenberg ) and wave mechanics (invented by Erwin Schrödinger ). An alternative formulation of quantum mechanics is Feynman 's path integral formulation , in which a quantum-mechanical amplitude

14338-412: The one-dimensional case in the x {\displaystyle x} direction, the time-independent Schrödinger equation may be written With the differential operator defined by with state ψ {\displaystyle \psi } in this case having energy E {\displaystyle E} coincident with the kinetic energy of the particle. The general solutions of

14472-472: The opposite direction; he derived Planck's law of black-body radiation by assuming B–E statistics). In Dirac's time, it was not yet known that all bosons, including photons, must obey Bose–Einstein statistics. 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 ,

14606-455: The original quantum system ceases to exist as an independent entity (see Measurement in quantum mechanics ). The time evolution of a quantum state is described by the Schrödinger equation: Here H {\displaystyle H} denotes the Hamiltonian , the observable corresponding to the total energy of the system, and ℏ {\displaystyle \hbar }

14740-403: The overall uncertainty as far as quantum mechanics allows. Quantum optics makes use of coherent states for modes of the electromagnetic field. There is a tradeoff, reminiscent of the position–momentum uncertainty relation, between measurements of an electromagnetic wave's amplitude and its phase. This is sometimes informally expressed in terms of the uncertainty in the number of photons present in

14874-399: The photon can be considered as its own antiparticle (thus an "antiphoton" is simply a normal photon with opposite momentum, equal polarization, and 180° out of phase). The reverse process, pair production , is the dominant mechanism by which high-energy photons such as gamma rays lose energy while passing through matter. That process is the reverse of "annihilation to one photon" allowed in

15008-583: The photon mass is generated via the Higgs mechanism then the upper limit of m ≲ 10 eV/ c from the test of Coulomb's law is valid. In most theories up to the eighteenth century, light was pictured as being made 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), Robert Hooke (1665), and Christiaan Huygens (1678); however, particle models remained dominant, chiefly due to

15142-536: The photon mass of m < 3 × 10 eV/ c . The galactic vector potential can also be probed directly by measuring the torque exerted on a magnetized ring. Such methods were used to obtain the sharper upper limit of 1.07 × 10 eV/ c (the equivalent of 10  daltons ) given by the Particle Data Group . These sharp limits from the non-observation of the effects caused by the galactic vector potential have been shown to be model-dependent. If

15276-464: The photon probably derives from gamma rays , which were discovered in 1900 by Paul Villard , named by Ernest Rutherford in 1903, and shown to be a form of electromagnetic radiation in 1914 by Rutherford and Edward Andrade . In chemistry and optical engineering , photons are usually symbolized by hν , which is the photon energy , where h is the Planck constant and the Greek letter ν ( nu )

15410-472: The photon to be strictly massless. If photons were not purely massless, their speeds would vary with frequency, with lower-energy (redder) photons moving slightly slower than higher-energy photons. Relativity would be unaffected by this; the so-called speed of light, c , would then not be the actual speed at which light moves, but a constant of nature which is the upper bound on speed that any object could theoretically attain in spacetime. Thus, it would still be

15544-428: The position becomes more and more uncertain. The uncertainty in momentum, however, stays constant. The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere inside a certain region, and therefore infinite potential energy everywhere outside that region. For

15678-400: The question of what constitutes a "measurement" has been extensively studied. Newer interpretations of quantum mechanics have been formulated that do away with the concept of " wave function collapse " (see, for example, the many-worlds interpretation ). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled so that

15812-443: The rate at which they are absorbed . Einstein began by postulating simple proportionality relations for the different reaction rates involved. In his model, the rate R j i {\displaystyle R_{ji}} for a system to absorb a photon of frequency ν {\displaystyle \nu } and transition from a lower energy E j {\displaystyle E_{j}} to

15946-524: The ratio of N i {\displaystyle N_{i}} and N j {\displaystyle N_{j}} is g i / g j exp ⁡ ( E j − E i ) / ( k T ) , {\displaystyle g_{i}/g_{j}\exp {(E_{j}-E_{i})/(kT)},} where g i {\displaystyle g_{i}} and g j {\displaystyle g_{j}} are

16080-413: The result can be the creation of quantum entanglement : their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "... the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought". Quantum entanglement enables quantum computing and

16214-566: The results of a Bell test will be constrained in a particular, quantifiable way. Many Bell tests have been performed and they have shown results incompatible with the constraints imposed by local hidden variables. It is not possible to present these concepts in more than a superficial way without introducing the mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also linear algebra , differential equations , group theory , and other more advanced subjects. Accordingly, this article will present

16348-463: The same physical system. In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space . The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex square-integrable functions L 2 ( C ) {\displaystyle L^{2}(\mathbb {C} )} , while

16482-461: The speed of light. If Coulomb's law is not exactly valid, then that would allow the presence of an electric field to exist within a hollow conductor when it is subjected to an external electric field. This provides a means for precision tests of Coulomb's law . A null result of such an experiment has set a limit of m ≲ 10 eV/ c . Sharper upper limits on the mass of light have been obtained in experiments designed to detect effects caused by

16616-412: The speed of spacetime ripples ( gravitational waves and gravitons ), but it would not be the speed of photons. If a photon did have non-zero mass, there would be other effects as well. Coulomb's law would be modified and the electromagnetic field would have an extra physical degree of freedom . These effects yield more sensitive experimental probes of the photon mass than the frequency dependence of

16750-499: The summation as well; for example, two photons may interact indirectly through virtual electron – positron pairs . Such photon–photon scattering (see two-photon physics ), as well as electron–photon scattering, is meant to be one of the modes of operations of the planned particle accelerator, the International Linear Collider . In modern physics notation, the quantum state of the electromagnetic field

16884-625: The superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if ψ A {\displaystyle \psi _{A}} and ϕ A {\displaystyle \phi _{A}} are both possible states for system A {\displaystyle A} , and likewise ψ B {\displaystyle \psi _{B}} and ϕ B {\displaystyle \phi _{B}} are both possible states for system B {\displaystyle B} , then

17018-586: The system's temperature . From this, it is readily derived that g i B i j = g j B j i {\displaystyle g_{i}B_{ij}=g_{j}B_{ji}} and The A i j {\displaystyle A_{ij}} and B i j {\displaystyle B_{ij}} are collectively known as the Einstein coefficients . Einstein could not fully justify his rate equations, but claimed that it should be possible to calculate

17152-575: The term in a letter to Nature on 18 December 1926. The same name was used earlier but was never widely adopted before Lewis: in 1916 by the American physicist and psychologist Leonard T. Troland , in 1921 by the Irish physicist John Joly , in 1924 by the French physiologist René Wurmser (1890–1993), and in 1926 by the French physicist Frithiof Wolfers (1891–1971). The name was suggested initially as

17286-441: The theory is that it usually cannot predict with certainty what will happen, but only give probabilities. Mathematically, a probability is found by taking the square of the absolute value of a complex number , known as a probability amplitude. This is known as the Born rule , named after physicist Max Born . For example, a quantum particle like an electron can be described by a wave function, which associates to each point in space

17420-421: 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

17554-437: The universe as a whole remains speculative. Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy . For example, the refinement of quantum mechanics for the interaction of light and matter, known as quantum electrodynamics (QED), has been shown to agree with experiment to within 1 part in 10 when predicting the magnetic properties of an electron. A fundamental feature of

17688-526: The value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle ). Quantum mechanics arose gradually from theories to explain observations that could not be reconciled with classical physics, such as Max Planck 's solution in 1900 to the black-body radiation problem, and the correspondence between energy and frequency in Albert Einstein 's 1905 paper , which explained

17822-497: Was observed experimentally by Arthur Compton , for which he received the Nobel Prize in 1927. The pivotal question then, was 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 . (See § Quantum field theory and § As

17956-536: Was used before 1900 to mean particles or amounts of different quantities , including electricity . In 1900, the German physicist Max Planck was studying black-body radiation , and he suggested that the experimental observations, specifically at shorter wavelengths , would be explained if the energy stored within a molecule was a "discrete quantity composed of an integral number of finite equal parts", which he called "energy elements". In 1905, Albert Einstein published

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