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Thomas–Fermi Equation
In mathematics, the Thomas–Fermi equation for the neutral atom is a second order non-linear ordinary differential equation, named after Llewellyn Thomas and Enrico Fermi, which can be derived by applying the Thomas–Fermi model to atoms. The equation reads :\frac = \frac y^ subject to the boundary conditions :y(0)=1 \quad ; \quad y(+\infty)=0 If y approaches zero as x becomes large, this equation models the charge distribution of a neutral atom as a function of radius x. Solutions where y becomes zero at finite x model positive ions. For solutions where y becomes large and positive as x becomes large, it can be interpreted as a model of a compressed atom, where the charge is squeezed into a smaller space. In this case the atom ends at the value of x for which dy/dx=y/x. Transformations Introducing the transformation z=y/x converts the equation to :\frac\frac\left(x^2\frac\right) - z^=0 This equation is similar to Lane–Emden equation with polytropic index 3/2 except the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Llewellyn Thomas
Llewellyn Hilleth Thomas (21 October 1903 – 20 April 1992) was a British physicist and applied mathematician. He is best known for his contributions to atomic and molecular physics and solid-state physics. His key achievements include calculating relativistic effects on the spin-orbit interaction in a hydrogen atom (Thomas precession), creating an approximate theory of N-body quantum systems ( Thomas-Fermi theory), and devising an efficient method for solving tridiagonal system of linear equations (Thomas algorithm). Life and education Born in London, he studied at Cambridge University, receiving his BA, PhD, and MA degrees in 1924, 1927 and 1928 respectively. While on a Traveling Fellowship for the academic year 1925–1926 at Bohr's Institute in Copenhagen, he proposed Thomas precession in 1926, to explain the difference between predictions made by spin-orbit coupling theory and experimental observations. In 1929 he obtained a job as a professor of physics at the Ohio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enrico Fermi
Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and the "architect of the atomic bomb". He was one of very few physicists to excel in both theoretical physics and experimental physics. Fermi was awarded the 1938 Nobel Prize in Physics for his work on induced radioactivity by neutron bombardment and for the discovery of transuranium elements. With his colleagues, Fermi filed several patents related to the use of nuclear power, all of which were taken over by the US government. He made significant contributions to the development of statistical mechanics, Quantum mechanics, quantum theory, and nuclear physics, nuclear and particle physics. Fermi's first major contribution involved the field of statistical mechanics. After Wolfgang Pauli formulated his Pauli exclusion principle, exclusion pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas–Fermi Model
The Thomas–Fermi (TF) model, named after Llewellyn Thomas and Enrico Fermi, is a quantum mechanical theory for the electronic structure of many-body systems developed semiclassically shortly after the introduction of the Schrödinger equation. It stands separate from wave function theory as being formulated in terms of the electronic density alone and as such is viewed as a precursor to modern density functional theory. The Thomas–Fermi model is correct only in the limit of an infinite nuclear charge. Using the approximation for realistic systems yields poor quantitative predictions, even failing to reproduce some general features of the density such as shell structure in atoms and Friedel oscillations in solids. It has, however, found modern applications in many fields through the ability to extract qualitative trends analytically and with the ease at which the model can be solved. The kinetic energy expression of Thomas–Fermi theory is also used as a component in more ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lane–Emden Equation
In astrophysics, the Lane–Emden equation is a dimensionless form of Poisson's equation for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid. It is named after astrophysicists Jonathan Homer Lane and Robert Emden. The equation reads : \frac \frac \left(\right) + \theta^n = 0, where \xi is a dimensionless radius and \theta is related to the density, and thus the pressure, by \rho=\rho_c\theta^n for central density \rho_c. The index n is the polytropic index that appears in the polytropic equation of state, : P = K \rho^\, where P and \rho are the pressure and density, respectively, and K is a constant of proportionality. The standard boundary conditions are \theta(0)=1 and \theta'(0)=0. Solutions thus describe the run of pressure and density with radius and are known as ''polytropes'' of index n. If an isothermal fluid (polytropic index tends to infinity) is used instead of a polytropic fluid, one obtains the Emden–Chan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretical physics. He served as doctoral supervisor for many Nobel Prize winners in physics and chemistry (only J. J. Thomson's record of mentorship is comparable to his). He introduced the second quantum number (azimuthal quantum number) and the third quantum number (magnetic quantum number). He also introduced the fine-structure constant and pioneered X-ray wave theory. Early life and education Sommerfeld was born in 1868 to a family with deep ancestral roots in Prussia. His mother Cäcilie Matthias (1839–1902) was the daughter of a Potsdam builder. His father Franz Sommerfeld (1820–1906) was a physician from a leading family in Königsberg, where Arnold's grandfather had resettled from the hinterland in 1822 for a career as Court Postal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ettore Majorana
Ettore Majorana (,, uploaded 19 April 2013, retrieved 14 December 2019 ; born on 5 August 1906 – possibly dying after 1959) was an Italian theoretical physicist who worked on neutrino masses. On 25 March 1938, he disappeared under mysterious circumstances after purchasing a ticket to travel by ship from Palermo to Naples. The Majorana equation and Majorana fermions are named after him. In 2006, the Majorana Prize was established in his memory. Life and work In 1938, Enrico Fermi was quoted as saying about Majorana: "There are several categories of scientists in the world; those of second or third rank do their best but never get very far. Then there is the first rank, those who make important discoveries, fundamental to scientific progress. But then there are the geniuses, like Galilei and Newton. Majorana was one of these." Gifted in mathematics Majorana was born in Catania, Sicily. Mathematically gifted, he was very young when he joined Enrico Fermi's team in Rome a ... [...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] |
