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Lieb–Liniger Model
In physics, the Lieb–Liniger model describes a gas of particles moving in one dimension and satisfying Bose–Einstein statistics. More specifically, it describes a one dimensional Bose gas with Dirac delta interactions. It is named after Elliott H. Lieb and who introduced the model in 1963.Elliott H. Lieb and Werner Liniger, ''Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State'', Physical Review 130: 1605–1616, 1963 The model was developed to compare and test Nikolay Bogolyubov's theory of a weakly interaction Bose gas. Definition Given N bosons moving in one-dimension on the x -axis defined from ,L/math> with periodic boundary conditions, a state of the ''N''-body system must be described by a many-body wave function \psi(x_1, x_2, \dots, x_j, \dots,x_N). The Hamiltonian, of this model is introduced as : H = -\sum_^N \frac + 2c \sum_^N\sum_^N \delta(x_i-x_j)\ , where \delta is the Dirac delta function. The constant c denotes the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." It is one of the most fundamental scientific disciplines. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bose–Einstein Statistics
In quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting identical particles may occupy a set of available discrete energy states at thermodynamic equilibrium. The aggregation of particles in the same state, which is a characteristic of particles obeying Bose–Einstein statistics, accounts for the cohesive streaming of laser light and the frictionless creeping of superfluid helium. The theory of this behaviour was developed (1924–25) by Satyendra Nath Bose, who recognized that a collection of identical and indistinguishable particles could be distributed in this way. The idea was later adopted and extended by Albert Einstein in collaboration with Bose. Bose–Einstein statistics apply only to particles that do not follow the Pauli exclusion principle restrictions. Particles that follow Bose-Einstein statistics are called bosons, which have integer values of spin. In contrast, particles ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bose Gas
An ideal Bose gas is a quantum-mechanical phase of matter, analogous to a classical ideal gas. It is composed of bosons, which have an integer value of spin and abide by Bose–Einstein statistics. The statistical mechanics of bosons were developed by Satyendra Nath Bose for a photon gas and extended to massive particles by Albert Einstein, who realized that an ideal gas of bosons would form a condensate at a low enough temperature, unlike a classical ideal gas. This condensate is known as a Bose–Einstein condensate. Introduction and examples Bosons are quantum mechanical particles that follow Bose–Einstein statistics, or equivalently, that possess integer spin. These particles can be classified as elementary: these are the Higgs boson, the photon, the gluon, the W/Z and the hypothetical graviton; or composite like the atom of hydrogen, the atom of 16 O, the nucleus of deuterium, mesons etc. Additionally, some quasiparticles in more complex systems can also be co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Delta Function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Thus it can be Heuristic, represented heuristically as \delta (x) = \begin 0, & x \neq 0 \\ , & x = 0 \end such that \int_^ \delta(x) dx=1. Since there is no function having this property, modelling the delta "function" rigorously involves the use of limit (mathematics), limits or, as is common in mathematics, measure theory and the theory of distribution (mathematics), distributions. The delta function was introduced by physicist Paul Dirac, and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses. It is called the delta function because it is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete domain and takes values ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nikolay Bogolyubov
Nikolay Nikolayevich (Mykola Mykolayovych) Bogolyubov (; ; 21 August 1909 – 13 February 1992) was a Soviet, Ukrainian and Russian mathematician and theoretical physicist known for a significant contribution to quantum field theory, classical and quantum statistical mechanics, and the theory of dynamical systems; he was the recipient of the 1992 Dirac Medal for his works and studies. Biography Early life (1909–1921) Nikolay Bogolyubov was born on 21 August 1909 in Nizhny Novgorod, Russian Empire to Russian Orthodox Church priest and seminary teacher of theology, psychology and philosophy Nikolay Mikhaylovich Bogolyubov, and Olga Nikolayevna Bogolyubova, a teacher of music. Six months after Nicolay's birth, the family moved to Nizhyn, city of Chernihiv Oblast, where his father taught until 1913. From 1913 to 1918, the family lived in Kyiv. Nikolay received his initial education at home. His father taught him the basics of arithmetic, as well as German, French, and En ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-integer spin (1/2, 3/2, 5/2, ...). Every observed subatomic particle is either a boson or a fermion. Paul Dirac coined the name ''boson'' to commemorate the contribution of Satyendra Nath Bose, an Indian physicist. Some bosons are elementary particles occupying a special role in particle physics, distinct from the role of fermions (which are sometimes described as the constituents of "ordinary matter"). Certain elementary bosons (e.g. gluons) act as force carriers, which give rise to forces between other particles, while one (the Higgs boson) contributes to the phenomenon of mass. Other bosons, such as mesons, are composite particles made up of smaller constituents. Outside the realm of particle physics, multiple identical composite bosons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Boundary Conditions
Periodic boundary conditions (PBCs) are a set of boundary conditions which are often chosen for approximating a large (infinite) system by using a small part called a ''unit cell''. PBCs are often used in computer simulations and mathematical models. The topology of two-dimensional PBC is equal to that of a ''world map'' of some video games; the geometry of the unit cell satisfies perfect two-dimensional tiling, and when an object passes through one side of the unit cell, it re-appears on the opposite side with the same velocity. In topological terms, the space made by two-dimensional PBCs can be thought of as being mapped onto a torus (Compactification (mathematics), compactification). The large systems approximated by PBCs consist of an infinite number of unit cells. In computer simulations, one of these is the original simulation box, and others are copies called ''images''. During the simulation, only the properties of the original simulation box need to be recorded and propaga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Function
In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter), psi, respectively). Wave functions are complex number, complex-valued. For example, a wave function might assign a complex number to each point in a region of space. The Born rule provides the means to turn these complex probability amplitudes into actual probabilities. In one common form, it says that the squared modulus of a wave function that depends upon position is the probability density function, probability density of measurement in quantum mechanics, measuring a particle as being at a given place. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called ''normalization''. Since the wave function is complex-valued, only its relative phase and relative magnitud ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian (quantum Mechanics)
In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's ''energy spectrum'' or its set of ''energy eigenvalues'', is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory. The Hamiltonian is named after William Rowan Hamilton, who developed a revolutionary reformulation of Newtonian mechanics, known as Hamiltonian mechanics, which was historically important to the development of quantum physics. Similar to vector notation, it is typically denoted by \hat, where the hat indicates that it is an operator. It can also be written as H or \check. Introduction The Hamiltonian of a system represents the total energy of the system; that is, the sum of the kine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tonks–Girardeau Gas
In physics, a Tonks–Girardeau gas is a Bose gas in which the repulsive interactions between bosonic particles confined to one dimension dominate the system's physics. It is named after physicists Lewi Tonks, who developed a classical model in 1936, and Marvin D. Girardeau who generalized it to the quantum regime. It is not a Bose–Einstein condensate as it does not demonstrate any of the necessary characteristics, such as off-diagonal long-range order or a unitary two-body correlation function, even in a thermodynamic limit and as such cannot be described by a macroscopically occupied orbital (order parameter) in the Gross–Pitaevskii formulation. The Tonks–Girardeau gas is a particular case of the Lieb–Liniger model. Definition A row of bosons all confined to a one-dimensional line cannot pass each other and therefore cannot exchange places. The resulting motion has been compared to a traffic jam: the motion of each boson is strongly correlated with that of its two nei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
