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This is commonly taken as a general solution to the free Klein–Gordon equation. Note that because the initial Fourier transformation contained Lorentz invariant quantities like p ⋅ x = p μ x μ {\displaystyle p\cdot x=p_{\mu }x^{\mu }} only, the last expression is also a Lorentz invariant solution to the Klein–Gordon equation.
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 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 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 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.
Without loss of generality, we will take λ to be non-negative. When λ is zero, the equation reduces to Poisson's equation.Therefore, when λ is very small, the solution approaches that of the unscreened Poisson equation, which, in dimension =, is a superposition of 1/r functions weighted by the source function f:
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 that would be first-order in both time and space, a desirable property for ...
which is an eigenvalue equation. Very often, only numerical solutions to the Schrödinger equation can be found for a given physical system and its associated potential energy. However, there exists a subset of physical systems for which the form of the eigenfunctions and their associated energies, or eigenvalues, can be found.