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Born Reciprocity
In physics, Born reciprocity, also called reciprocal relativity or Born–Green reciprocity, is a principle set up by theoretical physicist Max Born that calls for a duality-symmetry among space and momentum. Born and his co-workers expanded his principle to a framework that is also known as reciprocity theory. Born noticed a symmetry among configuration space and momentum space representations of a free particle, in that its wave function description is invariant to a change of variables ''x'' → ''p'' and ''p'' → −''x''. (It can also be worded such as to include scale factors, e.g. invariance to ''x'' → ''ap'' and ''p'' → −''bx'' where ''a'', ''b'' are constants.) Born hypothesized that such symmetry should apply to the four-vectors of special relativity, that is, to the four-vector space coordinates : \mathbf = X^ := \left(X^0, X^1, X^2, X^3 \right) = \left(ct, x, y, z \right) and the four-vector momentum (four-mome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Born Rule
The Born rule (also called Born's rule) is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of finding a system in a given state, when measured, is proportional to the square of the amplitude of the system's wavefunction at that state. It was formulated by German physicist Max Born in 1926. Details The Born rule states that if an observable corresponding to a self-adjoint operator A with discrete spectrum is measured in a system with normalized wave function , \psi\rang (see Bra–ket notation), then: * the measured result will be one of the eigenvalues \lambda of A, and * the probability of measuring a given eigenvalue \lambda_i will equal \lang\psi, P_i, \psi\rang, where P_i is the projection onto the eigenspace of A corresponding to \lambda_i. : (In the case where the eigenspace of A corresponding to \lambda_i is one-dimensional and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Element
In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc length, is a function of the metric tensor and is denoted by ''ds''. Line elements are used in physics, especially in theories of gravitation (most notably general relativity) where spacetime is modelled as a curved Pseudo-Riemannian manifold with an appropriate metric tensor. General formulation Definition of the line element and arclength The coordinate-independent definition of the square of the line element ''ds'' in an ''n''-dimensional Riemannian or Pseudo Riemannian manifold (in physics usually a Lorentzian manifold) is the "square of the length" of an infinitesimal displacement d\mathbf (in pseudo Riemannian manifolds possibly negative) whose square root should be used for computing curve length: ds^2 = d\mathbf\cdot d\ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equations Of Physics
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in French an ''équation'' is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. ''Solving'' an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables. An equation is written as two expressions, connected by an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reciprocal Lattice
In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial function in real space known as the ''direct lattice''. While the direct lattice exists in real space and is commonly understood to be a physical lattice (such as the lattice of a crystal), the reciprocal lattice exists in the space of spatial frequencies known as reciprocal space or k space, where \mathbf refers to the wavevector. In quantum physics, reciprocal space is closely related to momentum space according to the proportionality \mathbf = \hbar \mathbf, where \mathbf is the momentum vector and \hbar is the Planck constant. The reciprocal lattice of a reciprocal lattice is equivalent to the original direct lattice, because the defining equations are symmetrical with respect to the vectors in real and reciprocal space. Mathematically, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Model (physics)
In mathematical physics, a lattice model is a mathematical model of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice. Currently, lattice models are quite popular in theoretical physics, for many reasons. Some models are exactly solvable, and thus offer insight into physics beyond what can be learned from perturbation theory. Lattice models are also ideal for study by the methods of computational physics, as the discretization of any continuum model automatically turns it into a lattice model. The exact solution to many of these models (when they are solvable) includes the presence of solitons. Techniques for solving these include the inverse scattering transform and the method of Lax pairs, the Yang–Baxter equation and quantum groups. The solution of these models has given i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crystal Structure
In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystal, crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns that repeat along the principal directions of Three-dimensional space (mathematics), three-dimensional space in matter. The smallest group of particles in the material that constitutes this repeating pattern is the unit cell of the structure. The unit cell completely reflects the symmetry and structure of the entire crystal, which is built up by repetitive Translation (geometry), translation of the unit cell along its principal axes. The translation vectors define the nodes of the Bravais lattice. The lengths of the principal axes, or edges, of the unit cell and the angles between them are the lattice constants, also called ''lattice parameters'' or ''cell parameters''. The symmetry properties of the crystal are described by the con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Geometry
In theoretical physics, quantum geometry is the set of mathematical concepts generalizing the concepts of geometry whose understanding is necessary to describe the physical phenomena at distance scales comparable to the Planck length. At these distances, quantum mechanics has a profound effect on physical phenomena. Quantum gravity Each theory of quantum gravity uses the term "quantum geometry" in a slightly different fashion. String theory, a leading candidate for a quantum theory of gravity, uses the term quantum geometry to describe exotic phenomena such as T-duality and other geometric dualities, mirror symmetry, topology-changing transitions, minimal possible distance scale, and other effects that challenge intuition. More technically, quantum geometry refers to the shape of a spacetime manifold as experienced by D-branes which includes quantum corrections to the metric tensor, such as the worldsheet instantons. For example, the quantum volume of a cycle is computed from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
T-duality
In theoretical physics, T-duality (short for target-space duality) is an equivalence of two physical theories, which may be either quantum field theories or string theories. In the simplest example of this relationship, one of the theories describes strings propagating in a spacetime shaped like a circle of some radius R, while the other theory describes strings propagating on a spacetime shaped like a circle of radius proportional to 1/R. The idea of T-duality was first noted by Bala Sathiapalan in an obscure paper in 1987. The two T-dual theories are equivalent in the sense that all observable quantities in one description are identified with quantities in the dual description. For example, momentum in one description takes discrete values and is equal to the number of times the string winds around the circle in the dual description. The idea of T-duality can be extended to more complicated theories, including superstring theories. The existence of these dualities implies that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eduardo R
Eduardo is the Spanish and Portuguese form of the male given name Edward. Another version is Duarte. It may refer to: Association football * Eduardo Bonvallet, Chilean football player and sports commentator * Eduardo Carvalho, Portuguese footballer * Eduardo "Edu" Coimbra, Brazilian footballer * Eduardo Costa, Brazilian footballer * Eduardo da Conceição Maciel, Brazilian footballer * Eduardo da Silva, Brazilian-born Croatian footballer * Eduardo Adelino da Silva, Brazilian footballer * Eduardo Ribeiro dos Santos, Brazilian footballer * Eduardo Gómez (footballer), Chilean footballer * Eduardo Gonçalves de Oliveira, Brazilian footballer * Eduardo Jesus, Brazilian footballer * Eduardo Martini, Brazilian footballer * Eduardo Ferreira Abdo Pacheco, Brazilian footballer Music * Eduardo (rapper), Carlos Eduardo Taddeo, Brazilian rapper * Eduardo De Crescenzo, Italian singer, songwriter and multi-instrumentalist Politicians * Eduardo Año, Filipino politician and retired army genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eduard Prugovečki
Eduard Prugovečki (March 19, 1937 – October 13, 2003) was a Canadian physicist and mathematician of Croatian-Romanian descent. Prugovečki was born in Craiova, Romania to a Romanian mother, Helena (née Piatkowski), and Croatian father, Slavoljub. He completed the first four years of secondary education in Bucharest, before his family was forced to relocate to Zagreb in 1951, due to an anti- Yugoslav campaign by the communist authorities. He finished high school there and proceeded to study physics at the University of Zagreb, getting his diploma in 1959. He joined the Department of Theoretical Physics at the Institute Ruđer Bošković in Zagreb, where he worked as a research assistant until 1961. In 1961, as the best student of his generation in Zagreb, Prugovečki was sent to Princeton University, New Jersey, United States. He wrote his doctoral thesis under the direction of theoretical physicist Arthur Wightman, and earned his PhD from Princeton in 1964. In 1965, he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hideki Yukawa
was a Japanese theoretical physicist and the first Japanese Nobel laureate for his prediction of the pi meson, or pion. Biography He was born as Hideki Ogawa in Tokyo and grew up in Kyoto with two older brothers, two older sisters, and two younger brothers. He read the Confucian ''Doctrine of the Mean'', and later Lao-Tzu and Chuang-Tzu. His father, for a time, considered sending him to technical college rather than university since he was "not as outstanding a student as his older brothers". However, when his father broached the idea with his middle school principal, the principal praised his "high potential" in mathematics and offered to adopt Ogawa himself in order to keep him on a scholarly career. At that, his father relented. Ogawa decided against becoming a mathematician when in high school; his teacher marked his exam answer as incorrect when Ogawa proved a theorem but in a different manner than the teacher expected. He decided against a career in experimental physics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
