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Lippmann–Schwinger Equation
The Lippmann–Schwinger equation (named after Bernard Lippmann and Julian Schwinger) is one of the most used equations to describe particle collisions – or, more precisely, scattering – in quantum mechanics. It may be used in scattering of molecules, atoms, neutrons, photons or any other particles and is important mainly in atomic, molecular, and optical physics, nuclear physics and particle physics, but also for seismic scattering problems in geophysics. It relates the scattered wave function with the interaction that produces the scattering (the scattering potential) and therefore allows calculation of the relevant experimental parameters (scattering amplitude and cross sections). The most fundamental equation to describe any quantum phenomenon, including scattering, is the Schrödinger equation. In physical problems, this differential equation must be solved with the input of an additional set of initial and/or boundary conditions for the specific physical system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernard Lippmann
Bernard Abram Lippmann. (August 18, 1914 – February 12, 1988) was an American theoretical physicist. A former Professor of Physics at New York University, Lippmann is mainly known for the Lippmann-Schwinger equation, a widely used tool in non-relativistic scattering theory, which he formulated together with his doctoral supervisor Julian Schwinger Biography Bernard Lippmann was born in Brooklyn, New York City, in 1914. After initially attending the New York University Polytechnic School of Engineering, Polytechnic School of Brooklyn, where he attained a bachelor's degree in electrical engineering, he switched to physics and was admitted to the degree of Master of Science at the University of Michigan in 1935 Subsequently, Lippmann entered the industry, where he held various engineering roles until the entry of the United States into the World War II, Second World War when he joined Massachusetts Institute of Technology, MIT's Radiation Laboratory (MIT), Radiation Laboratory. Fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limiting Absorption Principle
In mathematics, the limiting absorption principle (LAP) is a concept from operator theory and scattering theory that consists of choosing the "correct" resolvent of a linear operator at the essential spectrum based on the behavior of the resolvent near the essential spectrum. The term is often used to indicate that the resolvent, when considered not in the original space (which is usually the L^2 space), but in certain weighted spaces (usually L^2_s, see below), has a limit as the spectral parameter approaches the essential spectrum. This concept developed from the idea of introducing complex parameter into the Helmholtz equation (\Delta+k^2)u(x)=-F(x) for selecting a particular solution. This idea is credited to Vladimir Ignatowski, who was considering the propagation and absorption of the electromagnetic waves in a wire. It is closely related to the Sommerfeld radiation condition and the limiting amplitude principle (1948). The terminology – both the limiting absorption pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Born Series
The Born series is the expansion of different scattering quantities in quantum scattering theory in the powers of the interaction potential V (more precisely in powers of G_0 V, where G_0 is the free particle Green's operator). It is closely related to Born approximation, which is the first order term of the Born series. The series can formally be understood as power series introducing the coupling constant by substitution V \to \lambda V . The speed of convergence and radius of convergence of the Born series are related to eigenvalues of the operator G_0 V . In general the first few terms of the Born series are good approximation to the expanded quantity for "weak" interaction V and large collision energy. Born series for scattering states The Born series for the scattering states reads : , \psi\rangle = , \phi \rangle + G_0(E) V , \phi\rangle + _0(E) V2 , \phi\rangle + _0(E) V3 , \phi\rangle + \dots It can be derived by iterating the Lippmann–Schwinger equation : , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Wave Analysis
Partial-wave analysis, in the context of quantum mechanics, refers to a technique for solving scattering problems by decomposing each wave into its constituent angular-momentum components and solving using boundary conditions. Preliminary scattering theory The following description follows the canonical way of introducing elementary scattering theory. A steady beam of particles scatters off a spherically symmetric potential V(r), which is short-ranged, so that for large distances r \to \infty, the particles behave like free particles. In principle, any particle should be described by a wave packet, but we instead describe the scattering of a plane wave \exp(ikz) traveling along the ''z'' axis, since wave packets can be expanded in terms of plane waves, and this is mathematically simpler. Because the beam is switched on for times long compared to the time of interaction of the particles with the scattering potential, a steady state is assumed. This means that the stationary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discretization
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable (creating a dichotomy for modeling purposes, as in binary classification). Discretization is also related to discrete mathematics, and is an important component of granular computing. In this context, ''discretization'' may also refer to modification of variable or category ''granularity'', as when multiple discrete variables are aggregated or multiple discrete categories fused. Whenever continuous data is discretized, there is always some amount of discretization error. The goal is to reduce the amount to a level conside ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis (linear Algebra)
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to . The elements of a basis are called . Equivalently, a set is a basis if its elements are linearly independent and every element of is a linear combination of elements of . In other words, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the ''dimension'' of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Definition A basis of a vector space over a field (such as the real numbers or the complex numbers ) is a linearly independent subset of that spans . This me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Singularity
In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For example, the real function : f(x) = \frac has a singularity at x = 0, where the numerical value of the function approaches \pm\infty so the function is not defined. The absolute value function g(x) = , x, also has a singularity at x = 0, since it is not differentiable there. The algebraic curve defined by \left\ in the (x, y) coordinate system has a singularity (called a cusp) at (0, 0). For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory. Real analysis In real analysis, singularities are either discontinuities, or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). There are four kinds of discon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Element
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is a right inverse of . (An identity element is an element such that and for all and for which the left-hand sides are defined.) When the operation is associative, if an element has both a left inverse and a right inverse, then these two inverses are equal and unique; they are called the ''inverse element'' or simply the ''inverse''. Often an adjective is added for specifying the operation, such as in additive inverse, multiplicative inverse, and functional inverse. In this case (associative operation), an invertible element is an element that has an inverse. Inverses are commonly used in groupswhere every element is invertible, and ringswhere invertible elements are also called units. They are also commonly used for operations tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hellmann–Feynman Theorem
In quantum mechanics, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter, to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics. The theorem has been proven independently by many authors, including Paul Güttinger (1932), Wolfgang Pauli (1933), Hans Hellmann (1937) and Richard Feynman (1939). The theorem states where *\hat_ is a Hamiltonian operator depending upon a continuous parameter \lambda\,, *, \psi_\lambda\rangle, is an eigen-state (eigenfunction) of the Hamiltonian, depending implicitly upon \lambda, *E_\, is the energy (eigenvalue) of the state , \psi_\lambda\rangle, i.e. \hat_, \psi_\lambda\rangle = E_, \psi_\lambda\rangle. Proof This proof of the Hellmann–Feynma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenstate
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, while all other states are called mixed quantum states. A pure quantum state can be represented by a ray in a Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces. Pure states are also known as state vectors or wave functions, the latter term applying particularly when they are represented as functions of position or momentum. For example, when dealing with the energy spectrum of the electron in a hydrogen ato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian (quantum Mechanics)
Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian with two-electron nature ** Molecular Hamiltonian, the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule * Hamiltonian (control theory), a function used to solve a problem of optimal control for a dynamical system * Hamiltonian path, a path in a graph that visits each vertex exactly once * Hamiltonian group, a non-abelian group the subgroups of which are all normal * Hamiltonian economic program, the economic policies advocated by Alexander Hamilton, the first United States Secretary of the Treasury See also * Alexander Hamilton (1755 or 1757–1804), American statesman and one of the Founding Fathers of the US * Hamilton (other) Hamilton may refer to: People * Hamilton (name), a common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudopotential
In physics, a pseudopotential or effective potential is used as an approximation for the simplified description of complex systems. Applications include atomic physics and neutron scattering. The pseudopotential approximation was first introduced by Hans Hellmann in 1934. Atomic physics The pseudopotential is an attempt to replace the complicated effects of the motion of the core (i.e. non- valence) electrons of an atom and its nucleus with an effective potential, or pseudopotential, so that the Schrödinger equation contains a modified effective potential term instead of the Coulombic potential term for core electrons normally found in the Schrödinger equation. The pseudopotential is an effective potential constructed to replace the atomic all-electron potential (full-potential) such that core states are eliminated ''and'' the valence electrons are described by pseudo-wavefunctions with significantly fewer nodes. This allows the pseudo-wavefunctions to be described with far ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
