Klein–Gordon equation. The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic version of the Schrödinger equation. It is the equation of motion of a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. It cannot be straightforwardly interpreted as a Schrödinger equation for a quantum state, because it is second order in time and because it does not admit a positive definite conserved probability density. Still, with the appropriate interpretation, it does describe the quantum amplitude for finding a point particle in various places, the relativistic wavefunction, but the particle propagates both forwards and backwards in time. Any solution to the Dirac equation is automatically a solution to the Klein–Gordon equation, but the converse is not true.
Statement[edit] The Klein–Gordon equation is This is often abbreviated as where and is the d'Alembert operator, defined by The equation is most often written in natural units: is written. Schrödinger equation. In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of some physical system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger.[1] In classical mechanics, the equation of motion is Newton's second law, and equivalent formulations are the Euler–Lagrange equations and Hamilton's equations.
All of these formulations are used to solve for the motion of a mechanical system and mathematically predict what the system will do at any time beyond the initial settings and configuration of the system. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation for a quantum system (usually atoms, molecules, and subatomic particles whether free, bound, or localized). It is not a simple algebraic equation, but (in general) a linear partial differential equation. The concept of a state vector is a fundamental postulate of quantum mechanics. Airy function. This article is about the Airy special function. For the Airy stress function employed in solid mechanics, see Stress functions. In the physical sciences, the Airy function Ai(x) is a special function named after the British astronomer George Biddell Airy (1801–92).
The function Ai(x) and the related function Bi(x), which is also called the Airy function, but sometimes referred to as the Bairy function, are solutions to the differential equation known as the Airy equation or the Stokes equation. This is the simplest second-order linear differential equation with a turning point (a point where the character of the solutions changes from oscillatory to exponential). The Airy function is the solution to Schrödinger's equation for a particle confined within a triangular potential well and for a particle in a one-dimensional constant force field. Definitions[edit] Plot of Ai(x) in red and Bi(x) in blue y = Ai(x) satisfies the Airy equation This equation has two linearly independent solutions.
Euler–Bernoulli beam theory. This vibrating glass beam may be modeled as a cantilever beam with acceleration, variable linear density, variable section modulus, some kind of dissipation, springy end loading, and possibly a point mass at the free end. Additional analysis tools have been developed such as plate theory and finite element analysis, but the simplicity of beam theory makes it an important tool in the sciences, especially structural and mechanical engineering. History[edit] Schematic of cross-section of a bent beam showing the neutral axis. Prevailing consensus is that Galileo Galilei made the first attempts at developing a theory of beams, but recent studies argue that Leonardo da Vinci was the first to make the crucial observations. The Bernoulli beam is named after Jacob Bernoulli, who made the significant discoveries. Static beam equation[edit] Bending of an Euler–Bernoulli beam. The curve describes the deflection of the beam in the direction at some position , or other variables.
Note that where , and and . . . Dispersive partial differential equation. Dispersive partial differential equation From Wikipedia, the free encyclopedia Jump to: navigation , search In mathematics , a dispersive partial differential equation or dispersive PDE is a partial differential equation that is dispersive . In this context, dispersion means that waves of different wavelength propagate at different phase velocities .
Examples [ edit ] Linear equations [ edit ] Nonlinear equations [ edit ] See also [ edit ] External links [ edit ] The Dispersive PDE Wiki . Retrieved from " Categories : Navigation menu Personal tools Namespaces Variants Views Actions Navigation Interaction Toolbox Print/export Languages Edit links This page was last modified on 21 May 2012 at 20:15.