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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 paradox is an unexpected consequence of relativity on the interaction of quantum particles with electrostatic potentials. The quantum mechanical problem of free particles striking an electrostatic step potential has two solutions when relativity is ignored.
The equation of motion for this theory is obtained by extremizing the action above. It takes the following form, ... This is the Klein–Gordon 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 ...
The scalar propagators are Green's functions for the Klein–Gordon equation. There are related singular functions which are important in quantum field theory. These functions are most simply defined in terms of the vacuum expectation value of products of field operators.
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]
The classical equations of motion of a field are typically identical in form to the (quantum) equations for the wave-function of one of its quanta. For example, the Klein–Gordon equation is the classical equation of motion for a free scalar field, but also the quantum equation for a scalar particle wave-function.
The photon field A satisfies the wave equation =. The Higgs field φ satisfies the Klein–Gordon equation. The weak interaction fields Z, W ± satisfy the Proca equation. These equations can be solved exactly.