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Double-well Potential
The so-called double-well potential is one of a number of Quartic function, quartic potentials of considerable interest in quantum mechanics, in quantum field theory and elsewhere for the exploration of various physical phenomena or mathematical properties since it permits in many cases explicit calculation without over-simplification. Thus the "symmetric double-well potential" served for many years as a model to illustrate the concept of instanton, instantons as a pseudo-classical configuration in a Euclideanised Quantum field theory, field theory. In the simpler quantum mechanical context this potential served as a model for the evaluation of Feynman path integral formulation, path integrals. or the solution of the Schrödinger equation by various methods for the purpose of obtaining explicitly the energy eigenvalues. The "inverted symmetric double-well potential", on the other hand, served as a nontrivial potential in the Schrödinger equation for the calculation of decay rates a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quartic Function
In algebra, a quartic function is a function of the form :f(x)=ax^4+bx^3+cx^2+dx+e, where ''a'' is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial. A '' quartic equation'', or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form :ax^4+bx^3+cx^2+dx+e=0 , where . The derivative of a quartic function is a cubic function. Sometimes the term biquadratic is used instead of ''quartic'', but, usually, biquadratic function refers to a quadratic function of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form :f(x)=ax^4+cx^2+e. Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If ''a'' is positive, then the function increases to positive infinity at both ends; and thus the function has a global minimum. Likewise, if ''a'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiclassical Physics
Semiclassical physics, or simply semiclassical refers to a theory in which one part of a system is described quantum mechanically whereas the other is treated classically. For example, external fields will be constant, or when changing will be classically described. In general, it incorporates a development in powers of Planck's constant, resulting in the classical physics of power 0, and the first nontrivial approximation to the power of (−1). In this case, there is a clear link between the quantum-mechanical system and the associated semi-classical and classical approximations, as it is similar in appearance to the transition from physical optics to geometric optics. Instances Some examples of a semiclassical approximation include: * WKB approximation: electrons in classical external electromagnetic fields. * semiclassical gravity: quantum field theory within a classical curved gravitational background (see general relativity). * quantum chaos; quantization of class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stirling Approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre. One way of stating the approximation involves the logarithm of the factorial: \ln(n!) = n\ln n - n +O(\ln n), where the big O notation means that, for all sufficiently large values of n, the difference between \ln(n!) and n\ln n-n will be at most proportional to the logarithm. In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to use instead the binary logarithm, giving the equivalent form \log_2 (n!) = n\log_2 n - n\log_2 e +O(\log_2 n). The error term in either base can be expressed more precisely as \tfrac12\log(2\pi n)+O(\tfrac1n), corresponding to an approximate formula for the factorial itself, n! \sim \sqrt\left(\frac\righ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Instantons
An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. More precisely, it is a solution to the equations of motion of the classical field theory on a Euclidean spacetime. In such quantum theories, solutions to the equations of motion may be thought of as critical points of the action. The critical points of the action may be local maxima of the action, local minima, or saddle points. Instantons are important in quantum field theory because: * they appear in the path integral as the leading quantum corrections to the classical behavior of a system, and * they can be used to study the tunneling behavior in various systems such as a Yang–Mills theory. Relevant to dynamics, families of instantons permit that instantons, i.e. different critical points of the equation of motion, be related t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lamé Equation , occasionally misspelled ''lamé''
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Lamé may refer to: *Lamé (fabric), a clothing fabric with metallic strands *Lamé (fencing), a jacket used for detecting hits * Lamé (crater) on the Moon * Ngeté-Herdé language, also known as Lamé, spoken in Chad *Peve language, also known as Lamé after its chief dialect, spoken in Chad and Cameroon *Lamé, a couple of the Masa languages of West Africa *Amy Lamé (born 1971), British radio presenter *Gabriel Lamé (1795–1870), French mathematician See also * Lamé curve, geometric figure *Lamé parameters * Lame (other) *Lame (kitchen tool) A lame () is a double-sided blade that is used to slash the tops of bread loaves in baking. A lame is used to ''score'' (also called ''slashing'' or ''docking'') bread just before the bread is placed in the oven. Often the blade's cutting edge wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Elliptic Function
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation \operatorname for \sin. The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by . Carl Friedrich Gauss had already studied special Jacobi elliptic functions in 1797, the lemniscate elliptic functions in particular, but his work was published much later. Overview There are twelve Jacobi ellipti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Instanton
Periodicity or periodic may refer to: Mathematics * Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups * Periodic function, a function whose output contains values that repeat periodically * Periodic mapping Physical sciences * Periodic table of chemical elements * Periodic trends, relative characteristics of chemical elements observed * Redshift periodicity, astronomical term for redshift quantization Other uses * Fokker periodicity blocks, which mathematically relate musical intervals * Periodic acid, a compound of iodine * Principle of periodicity, a concept in generally accepted accounting principles * Quasiperiodicity, property of a system that displays irregular periodicity See also * Aperiodic (other) * Cycle (other) * Frequency (other) * Period (other) * Periodical * Seasonality In time series data, seasonality is the presence of variations that occur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oblate Spheroidal Wave Functions
In applied mathematics, oblate spheroidal wave functions (like also prolate spheroidal wave functions and other related functions) are involved in the solution of the Helmholtz equation in oblate spheroidal coordinates. When solving this equation, \Delta \Phi + k^2 \Phi=0, by the method of separation of variables, (\xi,\eta,\varphi), with: :\ z=(d/2) \xi \eta, :\ x=(d/2) \sqrt \cos \varphi, :\ y=(d/2) \sqrt \sin \varphi, :\ \xi \ge 0 \text , \eta, \le 1. the solution \Phi(\xi,\eta,\varphi) can be written as the product of a radial spheroidal wave function R_(-i c,i \xi) and an angular spheroidal wave function S_(-i c,\eta) by e^. Here c=kd/2, with d being the interfocal length of the elliptical cross section of the oblate spheroid. The radial wave function R_(-i c,i \xi) satisfies the linear 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prolate Spheroidal Wave Functions
The prolate spheroidal wave functions are eigenfunctions of the Laplacian in prolate spheroidal coordinates, adapted to boundary conditions on certain ellipsoids of revolution (an ellipse rotated around its long axis, “cigar shape“). Related are the oblate spheroidal wave functions (“pancake shaped” ellipsoid). Solutions to the wave equation Solve the Helmholtz equation, \nabla^2 \Phi + k^2 \Phi=0, by the method of separation of variables in prolate spheroidal coordinates, (\xi,\eta,\varphi), with: :\ x=a \sqrt \cos \varphi, :\ y=a \sqrt \sin \varphi, :\ z=a \, \xi \, \eta, and \xi \ge 1, , \eta, \le 1 , and 0 \le \varphi \le 2\pi. Here, 2a > 0 is the interfocal distance of the elliptical cross section of the prolate spheroid. Setting c=ka, the solution \Phi(\xi,\eta,\varphi) can be written as the product of e^, a radial spheroidal wave function R_(c,\xi) and an angular spheroidal wave function S_(c,\eta). The radial wave function R_(c,\xi) satisfies the line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathieu Functions
In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation : \frac + (a - 2q\cos(2x))y = 0, where a and q are parameters. They were first introduced by Émile Léonard Mathieu, who encountered them while studying vibrating elliptical drumheads.Morse and Feshbach (1953).Brimacombe, Corless and Zamir (2021) They have applications in many fields of the physical sciences, such as optics, quantum mechanics, and general relativity. They tend to occur in problems involving periodic motion, or in the analysis of partial differential equation boundary value problems possessing elliptic symmetry.Gutiérrez-Vega (2015). Definition Mathieu functions In some usages, ''Mathieu function'' refers to solutions of the Mathieu differential equation for arbitrary values of a and q. When no confusion can arise, other authors use the term to refer specifically to \pi- or 2\pi-periodic solutions, which exist only for special v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lamé Function
In mathematics, a Lamé function, or ellipsoidal harmonic function, is a solution of Lamé's equation, a second-order ordinary differential equation. It was introduced in the paper . Lamé's equation appears in the method of separation of variables applied to the Laplace equation in elliptic coordinates. In some special cases solutions can be expressed in terms of polynomials called Lamé polynomials. The Lamé equation Lamé's equation is :\frac + (A+B\weierp(x))y = 0, where ''A'' and ''B'' are constants, and \wp is the Weierstrass elliptic function. The most important case is when B\weierp(x) = - \kappa^2 \operatorname^2x , where \operatorname is the elliptic sine function, and \kappa^2 = n(n+1)k^2 for an integer ''n'' and k the elliptic modulus, in which case the solutions extend to meromorphic functions defined on the whole complex plane. For other values of ''B'' the solutions have branch points. By changing the independent variable to t with t=\operatorname x, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Instantons
Periodic instantons are finite energy solutions of Euclidean-time field equations which communicate (in the sense of quantum tunneling) between two turning points in the barrier of a potential and are therefore also known as bounces. Vacuum instantons, normally simply called instantons, are the corresponding zero energy configurations in the limit of infinite Euclidean time. For completeness we add that ``sphalerons´´ are the field configurations at the very top of a potential barrier. Vacuum instantons carry a winding (or topological) number, the other configurations do not. Periodic instantons werde discovered with the explicit solution of Euclidean-time field equations for double-well potentials and the cosine potential with non-vanishing energy and are explicitly expressible in terms of Jacobian elliptic functions (the generalization of trigonometrical functions). Periodic instantons describe the oscillations between two endpoints of a potential barrier between two potential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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