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Any solution of the free Dirac equation is, for each of its four components, a solution of the free Klein–Gordon equation. Despite historically it was invented as a single particle equation the Klein–Gordon equation cannot form the basis of a consistent quantum relativistic one-particle theory, any relativistic theory implies creation and ...
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.
A classical free scalar field satisfies the Klein–Gordon equation. If a scalar field is denoted φ {\displaystyle \varphi } , a quartic interaction is represented by adding a potential energy term ( λ / 4 !
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 equations for stationary configurations of the FK model reduce to those of the standard map or Chirikov–Taylor map of stochastic theory. [1] In the continuum-limit approximation the FK model reduces to the exactly integrable sine-Gordon (SG) equation, which allows for soliton solutions.
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.
Pokhozhaev's identity is an integral relation satisfied by stationary localized solutions to a nonlinear Schrödinger equation or nonlinear Klein–Gordon equation. It was obtained by S.I. Pokhozhaev [1] and is similar to the virial theorem. This relation is also known as G.H. Derrick's theorem. Similar identities can be derived for other ...
There is a close algebraic analogy between the Helmholtz equation (governing scalar optics) and the Klein–Gordon equation; and between the matrix form of the Maxwell's equations (governing vector optics) and the Dirac equation. So it is natural to use the powerful machinery of standard quantum mechanics (particularly, the Foldy–Wouthuysen ...