|
Simon Problems
In mathematics, the Simon problems (or Simon's problems) are a series of fifteen questions posed in the year 2000 by Barry Simon, an American mathematical physicist. Inspired by other collections of mathematical problems and open conjectures, such as the famous list by David Hilbert, the Simon problems concern quantum operators. Eight of the problems pertain to anomalous spectral behavior of Schrödinger operators, and five concern operators that incorporate the Coulomb potential. In 2014, Artur Avila won a Fields Medal for work including the solution of three Simon problems. Among these was the problem of proving that the set of energy levels of one particular abstract quantum system was in fact the Cantor set, a challenge known as the "Ten Martini Problem" after the reward that Mark Kac offered for solving it. The 2000 list was a refinement of a similar set of problems that Simon had posed in 1984. Context Background definitions for the "Coulomb energies" problems (N non ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simon's Problem
In computational complexity theory and quantum computing, Simon's problem is a computational problem that is proven to be solved exponentially faster on a quantum computer than on a classical (that is, traditional) computer. The quantum algorithm solving Simon's problem, usually called Simon's algorithm, served as the inspiration for Shor's algorithm. Both problems are special cases of the abelian hidden subgroup problem, which is now known to have efficient quantum algorithms. The problem is set in the model of decision tree complexity or query complexity and was conceived by Daniel Simon in 1994. Simon exhibited a quantum algorithm that solves Simon's problem exponentially faster and with exponentially fewer queries than the best probabilistic (or deterministic) classical algorithm. In particular, Simon's algorithm uses a linear number of queries and any classical probabilistic algorithm must use an exponential number of queries. This problem yields an oracle separation b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ground State
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. In quantum field theory, the ground state is usually called the vacuum state or the vacuum. If more than one ground state exists, they are said to be degenerate. Many systems have degenerate ground states. Degeneracy occurs whenever there exists a unitary operator that acts non-trivially on a ground state and commutes with the Hamiltonian of the system. According to the third law of thermodynamics, a system at absolute zero temperature exists in its ground state; thus, its entropy is determined by the degeneracy of the ground state. Many systems, such as a perfect crystal lattice, have a unique ground state and therefore have zero entropy at absolute zero. It is also possible for the highest excited state to have absolute zero temper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cosmic Censorship
The weak and the strong cosmic censorship hypotheses are two mathematical conjectures about the structure of gravitational singularities arising in general relativity. Singularities that arise in the solutions of Einstein's equations are typically hidden within event horizons, and therefore cannot be observed from the rest of spacetime. Singularities that are not so hidden are called ''naked''. The weak cosmic censorship hypothesis was conceived by Roger Penrose in 1969 and posits that no naked singularities exist in the universe. Basics Since the physical behavior of singularities is unknown, if singularities can be observed from the rest of spacetime, causality may break down, and physics may lose its predictive power. The issue cannot be avoided, since according to the Penrose–Hawking singularity theorems, singularities are inevitable in physically reasonable situations. Still, in the absence of naked singularities, the universe, as described by the general theory of relativi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved. QED mathematically describes all phenomena involving electrically charged particles interacting by means of exchange of photons and represents the quantum counterpart of classical electromagnetism giving a complete account of matter and light interaction. In technical terms, QED can be described as a perturbation theory of the electromagnetic quantum vacuum. Richard Feynman called it "the jewel of physics" for its extremely accurate predictions of quantities like the anomalous magnetic moment of the electron and the Lamb shift of the energy levels of hydrogen. History The first formulation of a quantum theory describing radiation and matter interaction is attributed to British scientist Paul Dirac, who ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Field Theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. QFT treats particles as excited states (also called Quantum, quanta) of their underlying quantum field (physics), fields, which are more fundamental than the particles. The equation of motion of the particle is determined by minimization of the Lagrangian, a functional of fields associated with the particle. Interactions between particles are described by interaction terms in the Lagrangian (field theory), Lagrangian involving their corresponding quantum fields. Each interaction can be visually represented by Feynman diagrams according to perturbation theory (quantum mechanics), perturbation theory in quantum mechanics. History Quantum field theory emerged from the wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Chromodynamics
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of quantum field theory called a non-abelian gauge theory, with symmetry group SU(3). The QCD analog of electric charge is a property called ''color''. Gluons are the force carriers of the theory, just as photons are for the electromagnetic force in quantum electrodynamics. The theory is an important part of the Standard Model of particle physics. A large body of experimental evidence for QCD has been gathered over the years. QCD exhibits three salient properties: * Color confinement. Due to the force between two color charges remaining constant as they are separated, the energy grows until a quark–antiquark pair is spontaneously produced, turning the initial hadron into a pair of hadrons instead of isolating a color charge. Although ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-avoiding Walk
In mathematics, a self-avoiding walk (SAW) is a sequence of moves on a lattice (a lattice path) that does not visit the same point more than once. This is a special case of the graph theoretical notion of a path. A self-avoiding polygon (SAP) is a closed self-avoiding walk on a lattice. Very little is known rigorously about the self-avoiding walk from a mathematical perspective, although physicists have provided numerous conjectures that are believed to be true and are strongly supported by numerical simulations. In computational physics, a self-avoiding walk is a chain-like path in or with a certain number of nodes, typically a fixed step length and has the property that it doesn't cross itself or another walk. A system of SAWs satisfies the so-called excluded volume condition. In higher dimensions, the SAW is believed to behave much like the ordinary random walk. SAWs and SAPs play a central role in the modeling of the topological and knot-theoretic behavior of thread- a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roth Number
Roth may refer to: Places Germany * Roth (district), in Bavaria, Germany ** Roth, Bavaria, capital of that district ** Roth (electoral district), a federal electoral district * Rhineland-Palatinate, Germany: ** Roth an der Our, in the district Bitburg-Prüm ** Roth bei Prüm, in the district Bitburg-Prüm ** Roth, Altenkirchen, in the district of Altenkirchen ** Roth, Bad Kreuznach, in the district of Bad Kreuznach ** Roth, Rhein-Hunsrück, in the district Rhein-Hunsrück ** Roth, Rhein-Lahn, in the district Rhein-Lahn-Kreis France * Roth, Moselle, a village in the commune of Hambach, Moselle United States * Roth, Illinois, a community * Roth, North Dakota, a community * Roth, Virginia, a community Rivers * Roth (Danube), a river of Bavaria, Germany, tributary of the Danube * Roth (Rednitz), a river of Bavaria, Germany, tributary of the Rednitz * Roth (Zusam), a river of Bavaria, Germany, tributary of the Zusam * Rot (Apfelstädt), a river also called Roth, of Thuringia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville Number
In number theory, a Liouville number is a real number ''x'' with the property that, for every positive integer ''n'', there exists a pair of integers (''p, q'') with ''q'' > 1 such that :0 1 + \log_2(d) ~) no pair of integers ~(\,p,\,q\,)~ exists that simultaneously satisfies the pair of bracketing inequalities :0 0 ~, then, since c\,q - d\,p is an integer, we can assert the sharper inequality \left, c\,q - d\,p \ \ge 1 ~. From this it follows that :\left, x - \frac\= \frac \ge \frac Now for any integer ~n > 1 + \log_2(d)~, the last inequality above implies :\left, x - \frac \ \ge \frac > \frac \ge \frac ~. Therefore, in the case ~ \left, c\,q - d\,p \ > 0 ~ such pair of integers ~(\,p,\,q\,)~ would violate the ''second'' inequality in the definition of a Liouville number, for some positive integer . We conclude that there is no pair of integers ~(\,p,\,q\,)~, with ~ q > 1 ~, that would qualify such an ~ x = c / d ~, as a Liouville number. Hence a Liouville number, if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Renormalization Group
In theoretical physics, the term renormalization group (RG) refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying force laws (codified in a quantum field theory) as the energy scale at which physical processes occur varies, energy/momentum and resolution distance scales being effectively conjugate under the uncertainty principle. A change in scale is called a scale transformation. The renormalization group is intimately related to ''scale invariance'' and ''conformal invariance'', symmetries in which a system appears the same at all scales (so-called self-similarity). As the scale varies, it is as if one is changing the magnifying power of a notional microscope viewing the system. In so-called renormalizable theories, the system at one scale will generally be seen to consist of self-similar copies of itself when viewed at a smaller sca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ferromagnetism
Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials are the familiar metals noticeably attracted to a magnet, a consequence of their large magnetic permeability. Magnetic permeability describes the induced magnetization of a material due to the presence of an ''external'' magnetic field, and it is this temporarily induced magnetization inside a steel plate, for instance, which accounts for its attraction to the permanent magnet. Whether or not that steel plate acquires a permanent magnetization itself, depends not only on the strength of the applied field, but on the so-called coercivity of that material, which varies greatly among ferromagnetic materials. In physics, several different types of material magnetism are distinguished. Ferromagnetism (along with the similar effect ferrimagnetis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thermodynamic Beta
In statistical thermodynamics, thermodynamic beta, also known as coldness, is the reciprocal of the thermodynamic temperature of a system:\beta = \frac (where is the temperature and is Boltzmann constant).J. Meixner (1975) "Coldness and Temperature", ''Archive for Rational Mechanics and Analysis'' 57:3, 281-29abstract It was originally introduced in 1971 (as "coldness function") by , one of the proponents of the rational thermodynamics school of thought, based on earlier proposals for a "reciprocal temperature" function.Day, W.A. and Gurtin, Morton E. (1969) "On the symmetry of the conductivity tensor and other restrictions in the nonlinear theory of heat conduction", ''Archive for Rational Mechanics and Analysis'' 33:1, 26-32 (Springer-Verlag)abstractJ. Castle, W. Emmenish, R. Henkes, R. Miller, and J. Rayne (1965) Science by Degrees: ''Temperature from Zero to Zero'' (Westinghouse Search Book Series, Walker and Company, New York). Thermodynamic beta has units reciprocal to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
