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Schrödinger–Newton Equation
The Schrödinger–Newton equation, sometimes referred to as the Newton–Schrödinger or Schrödinger–Poisson equation, is a nonlinear modification of the Schrödinger equation with a Newtonian gravitational potential, where the gravitational potential emerges from the treatment of the wave function as a mass density, including a term that represents interaction of a particle with its own gravitational field. The inclusion of a self-interaction term represents a fundamental alteration of quantum mechanics. It can be written either as a single integro-differential equation or as a coupled system of a Schrödinger and a Poisson equation. In the latter case it is also referred to in the plural form. The Schrödinger–Newton equation was first considered by Ruffini and Bonazzola in connection with self-gravitating boson stars. In this context of classical general relativity it appears as the non-relativistic limit of either the Klein–Gordon equation or the Dirac equation in a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schrödinger Equation
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, an Austrian physicist, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of the wave function, the quantum-mechanical characterization of an isolated physical system. The equation was postulated by Schrödinger based on a postulate of Louis de Broglie that all matter has an associated matter wave. The equati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vicinity of black holes or similar compact astrophysical objects, as well as in the early stages of the universe moments after the Big Bang. Three of the four fundamental forces of nature are described within the framework of quantum mechanics and quantum field theory: the Electromagnetism, electromagnetic interaction, the Strong interaction, strong force, and the Weak interaction, weak force; this leaves gravity as the only interaction that has not been fully accommodated. The current understanding of gravity is based on Albert Einstein's general theory of relativity, which incorporates his theory of special relativity and deeply modifies the understanding of concepts like time and space. Although general relativity is highly regarded for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavefunction Collapse
In various interpretations of quantum mechanics, wave function collapse, also called reduction of the state vector, occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an ''observation'' and is the essence of a measurement in quantum mechanics, which connects the wave function with classical observables such as position and momentum. Collapse is one of the two processes by which quantum systems evolve in time; the other is the continuous evolution governed by the Schrödinger equation. : In the Copenhagen interpretation, wave function collapse connects quantum to classical models, with a special role for the observer. By contrast, objective-collapse proposes an origin in physical processes. In the many-worlds interpretation, collapse does not exist; all wave function outcomes occur while quantum decoherence accounts for the appearanc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interferometry
Interferometry is a technique which uses the ''interference (wave propagation), interference'' of Superposition principle, superimposed waves to extract information. Interferometry typically uses electromagnetic waves and is an important investigative technique in the fields of astronomy, Optical fiber, fiber optics, engineering metrology, optical metrology, oceanography, seismology, spectroscopy (and its applications to chemistry), quantum mechanics, Nuclear physics, nuclear and particle physics, plasma physics, interactome, biomolecular interactions, surface profiling, microfluidics, mechanical stress/strain measurement, velocimetry, optometry, and making holograms. Interferometers are devices that extract information from interference. They are widely used in science and industry for the measurement of microscopic displacements, refractive index changes and surface irregularities. In the case with most interferometers, light from a single source is split into two beams that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac e^\,. The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma^2 is the variance. The standard deviation of the distribution is (sigma). A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Born–Oppenheimer Approximation
In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the assumption that the wave functions of atomic nuclei and electrons in a molecule can be treated separately, based on the fact that the nuclei are much heavier than the electrons. Due to the larger relative mass of a nucleus compared to an electron, the coordinates of the nuclei in a system are approximated as fixed, while the coordinates of the electrons are dynamic. The approach is named after Max Born and his 23-year-old graduate student J. Robert Oppenheimer, the latter of whom proposed it in 1927 during a period of intense ferment in the development of quantum mechanics. The approximation is widely used in quantum chemistry to speed up the computation of molecular wavefunctions and other properties for large molecules. There are cases where the assumption of separable motion no longer holds, which make the approximation lose validity (it is said to "break down"), but even then the ap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marginal Distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables in the subset without reference to the values of the other variables. This contrasts with a conditional distribution, which gives the probabilities contingent upon the values of the other variables. Marginal variables are those variables in the subset of variables being retained. These concepts are "marginal" because they can be found by summing values in a table along rows or columns, and writing the sum in the margins of the table. The distribution of the marginal variables (the marginal distribution) is obtained by marginalizing (that is, focusing on the sums in the margin) over the distribution of the variables being discarded, and the discarded variables are said to have been marginalized out. The context here is that the theoreti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiclassical Gravity
Semiclassical gravity is an approximation to the theory of quantum gravity in which one treats matter and energy fields as being quantum and the gravitational field as being classical. In semiclassical gravity, matter is represented by quantum matter fields that propagate according to the theory of quantum fields in curved spacetime. The spacetime in which the fields propagate is classical but dynamical. The dynamics of the theory is described by the ''semiclassical Einstein equations'', which relate the curvature of spacetime that is encoded by the Einstein tensor G_ to the expectation value of the energy–momentum tensor \hat T_ (a quantum field theory operator) of the matter fields, i.e. : G_ = \frac \left\langle \hat T_ \right\rangle_\psi, where ''G'' is the gravitational constant, and \psi indicates the quantum state of the matter fields. Energy–momentum tensor There is some ambiguity in regulating the energy–momentum tensor, and this depends upon the curvatu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galilei Invariance
Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames of reference. Galileo Galilei first described this principle in 1632 in his ''Dialogue Concerning the Two Chief World Systems'' using the example of a ship travelling at constant velocity, without rocking, on a smooth sea; any observer below the deck would not be able to tell whether the ship was moving or stationary. Formulation Specifically, the term ''Galilean invariance'' today usually refers to this principle as applied to Newtonian mechanics, that is, Newton's laws of motion hold in all frames related to one another by a Galilean transformation. In other words, all frames related to one another by such a transformation are inertial (meaning, Newton's equation of motion is valid in these frames). In this context it is sometimes called ''Newtonian relativity''. Among the axioms from Newton's theory are: #There exists an ''absolute space'', in which Newton's laws are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hartree Equation
In 1927, a year after the publication of the Schrödinger equation, Hartree formulated what are now known as the Hartree equations for atoms, using the concept of ''self-consistency'' that Lindsay had introduced in his study of many electron systems in the context of Bohr theory. Hartree assumed that the nucleus together with the electrons formed a spherically symmetric field. The charge distribution of each electron was the solution of the Schrödinger equation for an electron in a potential v(r) , derived from the field. Self-consistency required that the final field, computed from the solutions, was self-consistent with the initial field, and he thus called his method the self-consistent field method. History In order to solve the equation of an electron in a spherical potential, Hartree first introduced atomic units to eliminate physical constants. Then he converted the Laplacian from Cartesian to spherical coordinates to show that the solution was a product of a radi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear System
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gravitational Potential
In classical mechanics, the gravitational potential is a scalar potential associating with each point in space the work (energy transferred) per unit mass that would be needed to move an object to that point from a fixed reference point in the conservative gravitational field. It is analogous to the electric potential with mass playing the role of charge. The reference point, where the potential is zero, is by convention infinitely far away from any mass, resulting in a negative potential at any finite distance. Their similarity is correlated with both associated fields having conservative forces. Mathematically, the gravitational potential is also known as the Newtonian potential and is fundamental in the study of potential theory. It may also be used for solving the electrostatic and magnetostatic fields generated by uniformly charged or polarized ellipsoidal bodies. Potential energy The gravitational potential (''V'') at a location is the gravitational potential ener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
