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Einstein–Brillouin–Keller Method
The Einstein–Brillouin–Keller method (EBK) is a semiclassical method (named after Albert Einstein, Léon Brillouin, and Joseph B. Keller) used to compute eigenvalues in quantum-mechanical systems. EBK quantization is an improvement from Bohr-Sommerfeld quantization which did not consider the caustic phase jumps at classical turning points. This procedure is able to reproduce exactly the spectrum of the 3D harmonic oscillator, particle in a box, and even the relativistic fine structure of the hydrogen atom. In 1976–1977, Michael Berry and M. Tabor derived an extension to Gutzwiller trace formula for the density of states of an integrable system starting from EBK quantization. There have been a number of recent results on computational issues related to this topic, for example, the work of Eric J. Heller and Emmanuel David Tannenbaum using a partial differential equation gradient descent approach. Procedure Given a separable classical system defined by coordinates (q_i, ... [...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 classical c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Of States
In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states in the system of volume V whose energies lie in the range from E to E+\delta E. It is mathematically represented as a distribution by a probability density function, and it is generally an average over the space and time domains of the various states occupied by the system. The density of states is directly related to the dispersion relations of the properties of the system. High DOS at a specific energy level means that many states are available for occupation. Generally, the density of states of matter is continuous. In isolated systems however, such as atoms or molecules in the gas phase, the density distribution is discrete, like a spectral density. Local variations, most often due to distortions of the original system, are often referr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Langer Correction
The Langer correction, named after the mathematician Rudolf Ernest Langer, is a correction to the WKB approximation for problems with radial symmetry. Description In 3D systems When applying WKB approximation method to the radial Schrödinger equation, : -\frac \frac + -V_\textrm(r)R(r) = 0 , where the effective potential is given by :V_\textrm(r)=V(r)-\frac ( \ell the azimuthal quantum number related to the angular momentum operator), the eigenenergies and the wave function behaviour obtained are different from the real solution. In 1937, Rudolf E. Langer suggested a correction :\ell(\ell+1) \rightarrow \left(\ell+\frac\right)^2 which is known as Langer correction or Langer replacement. This manipulation is equivalent to inserting a 1/4 constant factor whenever \ell(\ell+1) appears. Heuristically, it is said that this factor arises because the range of the radial Schrödinger equation is restricted from 0 to infinity, as opposed to the entire real line. By such a changing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angular Momentum Operator
In quantum mechanics, the angular momentum operator is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum problems involving rotational symmetry. Such an operator is applied to a mathematical representation of the physical state of a system and yields an angular momentum value if the state has a definite value for it. In both classical and quantum mechanical systems, angular momentum (together with linear momentum and energy) is one of the three fundamental properties of motion.Introductory Quantum Mechanics, Richard L. Liboff, 2nd Edition, There are several angular momentum operators: total angular momentum (usually denoted J), orbital angular momentum (usually denoted L), and spin angular momentum (spin for short, usually denoted S). The term ''angular momentum operator'' can (confusingly) refer to either the total or the orbital angular momen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neumann Boundary Condition
In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann. When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative applied at the boundary of the domain. It is possible to describe the problem using other boundary conditions: a Dirichlet boundary condition specifies the values of the solution itself (as opposed to its derivative) on the boundary, whereas the Cauchy boundary condition, mixed boundary condition and Robin boundary condition are all different types of combinations of the Neumann and Dirichlet boundary conditions. Examples ODE For an ordinary differential equation, for instance, :y'' + y = 0, the Neumann boundary conditions on the interval take the form :y'(a)= \alpha, \quad y'(b) = \beta, where and are given numbers. PDE For a partial differential equation, for instance, :\nabla^2 y + y = 0, where denotes the Laplace operator, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Boundary Condition
In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential equation, it specifies the values that a solution needs to take along the boundary of the domain. In finite element method (FEM) analysis, ''essential'' or Dirichlet boundary condition is defined by weighted-integral form of a differential equation. The dependent unknown ''u in the same form as the weight function w'' appearing in the boundary expression is termed a ''primary variable'', and its specification constitutes the ''essential'' or Dirichlet boundary condition. The question of finding solutions to such equations is known as the Dirichlet problem. In applied sciences, a Dirichlet boundary condition may also be referred to as a fixed boundary condition. Examples ODE For an ordinary differential equation, for instance, y'' + y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maslov Index
In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space ''V''. Its dimension is ''n''(''n'' + 1) (where the dimension of ''V'' is ''2n''). It may be identified with the homogeneous space :, where is the unitary group and the orthogonal group. Following Vladimir Arnold it is denoted by Λ(''n''). The Lagrangian Grassmannian is a submanifold of the ordinary Grassmannian of V. A complex Lagrangian Grassmannian is the complex homogeneous manifold of Lagrangian subspaces of a complex symplectic vector space ''V'' of dimension 2''n''. It may be identified with the homogeneous space of complex dimension ''n''(''n'' + 1) :, where is the compact symplectic group. Topology The stable topology of the Lagrangian Grassmannian and complex Lagrangian Grassmannian is completely understood, as these spaces appear in the Bott periodicity theorem: \Omega(\mathrm/\mathrm U) \simeq \mathrm U/\mathrm O, and \Omega(\math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Action-angle Coordinates
In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequency, frequencies of oscillatory or rotational motion without solving the equations of motion. Action-angle coordinates are chiefly used when the Hamilton–Jacobi equations are completely separable. (Hence, the Hamiltonian (quantum mechanics), Hamiltonian does not depend explicitly on time, i.e., the conservation of energy, energy is conserved.) Action-angle variables define an invariant torus, so called because holding the action constant defines the surface of a torus, while the angle variables parameterize the coordinates on the torus. The Bohr–Sommerfeld quantization conditions, used to develop quantum mechanics before the advent of Schrödinger equation#Particles as waves, wave mechanics, state that the action must be an integral multiple of Planck's constant; similarly, Albert Einstein, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamilton–Jacobi Equation
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely. The Hamilton–Jacobi equation is also the only formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, it fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the eighteenth century) of finding an analogy between the propagation of light and the motion of a particle. The wave equation followed by mechanical systems is similar to, but not identical with, Schrödinger's equation, as described below; for this reason, the Ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emmanuel David Tannenbaum
Emmanuel David Tannenbaum (June 28, 1978 – May 28, 2012) was an Israeli/American biophysicist and applied mathematician. He worked as a professor and researcher in the Department of Chemistry at the Ben-Gurion University of the Negev and the Department of Biology at the Georgia Institute of Technology, specializing in the fields of mathematical biology, systems biology, and quantum physics. Tannenbaum's initial work was in quantum chemistry as part of his Harvard University doctoral thesis where he developed a novel partial differential equation approach to the EBK quantization of nearly separable Hamiltonians in the quasi-integrable regime. Emmanuel Tannenbaum subsequently devoted his research to studying various problems in evolutionary dynamics using quasispecies models. His seminal work centered on the key question of the evolutionary advantages of sexual reproduction. Tannenbaum demonstrated a strong selective advantage for sexual reproduction with fewer and much less ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eric J
Eric J Dubowsky also known as Eric J, is a Grammy, Emmy, ARIA, and APRA award-winning mixer, songwriter and record producer. Eric grew up in the New York City suburb of Tenafly, New Jersey and graduated from Tenafly High School in 1993. After attending Syracuse University he worked at Greene St. Recording in New York City, the home of early hip-hop artists Run-DMC, and Public Enemy. It was here Eric assisted engineer/producer, Rod Hui. That led to a job working with Atlantic Records producer Arif Mardin in 1998. Eric has worked with artists such as Flume, Weezer, ODESZA, Twenty One Pilots, Chet Faker, Brandy, Jeff Bhasker, Andy Grammer, Tove Lo, St. Vincent, The Chemical Brothers, Alessia Cara, Dua Lipa, Demi Lovato, Angus & Julia Stone, Freeform Five, Ruel, Kimbra, Mansionair, Panama, Kylie Minogue, Meg Mac, Yuka Honda, Flight Facilities, Joss Stone, The Rubens, Marc Kinchen, Lisa Mitchell, Carolina Liar, and actress Emmy Rossum. Eric received the 2014 ARIA award ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrable System
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; that is, its evolution is restricted to a submanifold within its phase space. Three features are often referred to as characterizing integrable systems: * the existence of a ''maximal'' set of conserved quantities (the usual defining property of complete integrability) * the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability) * the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability) Integrable systems may be seen as very different in qualitative character from mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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