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The Klein–Gordon equation (Klein–Fock–Gordon equation or sometimes Klein–Gordon–Fock equation) is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant .
The Klein–Gordon and Dirac equations for free particles [ edit ] Substituting the energy and momentum operators directly into the energy–momentum relation may at first sight seem appealing, to obtain the Klein–Gordon equation : [ 20 ]
The sine-Gordon equation is a second-order nonlinear partial differential equation for a function dependent on two variables typically denoted and , involving the wave operator and the sine of . It was originally introduced by Edmond Bour ( 1862 ) in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation ...
Although it is not a successful relativistic generalization of the Schrödinger equation, this equation is resurrected in the context of quantum field theory, where it is known as the Klein–Gordon equation, and describes a spinless particle field (e.g. pi meson or Higgs boson). Historically, Schrödinger himself arrived at this equation ...
The Klein–Gordon equation and the Dirac equation are two such equations. The Klein–Gordon equation, + =, was the first such equation to be obtained, even before the nonrelativistic one-particle Schrödinger equation, and applies to massive spinless particles. Historically, Dirac obtained the Dirac equation by seeking a differential equation ...
In 1927, Klein, Gordon and Fock also found it, but incorporated the electromagnetic interaction and proved that it was Lorentz invariant. De Broglie also arrived at the same equation in 1928. This relativistic wave equation is now most commonly known as the Klein–Gordon equation. [21]
Oskar Klein and Walter Gordon proposed the Klein–Gordon equation to describe quantum particles in the framework of relativity.Another important contribution by Gordon was to the theory of the Dirac equation, where he introduced the Gordon decomposition of the current into its center of mass and spin contributions, and so helped explain the = g-factor value in the electron's gyromagnetic ratio.
The Schrödinger equation is the low-velocity limiting case (v ≪ c) of the Klein–Gordon equation. When the relation is applied to a four-vector field ...