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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. It is a differential equation version of the relativistic energy–momentum relation ${\displaystyle E^{2}=(pc)^{2}+\left(m_{0}c^{2}\right)^{2}\,}$.

Statement

The Klein–Gordon equation can be written in different ways. The equation itself usually refers to the position space form, where it can be written in terms of separated space and time components ${\displaystyle (t,\mathbf {x} )}$ or by combining them into a four-vector ${\displaystyle x^{\mu }=(ct,\mathbf {x} )}$. By Fourier transforming the field into momentum space, the solution is usually written in terms of a superposition of plane waves whose energy and momentum obey the energy-momentum dispersion relation from special relativity. Here, the Klein–Gordon equation is given for both of the two common metric signature conventions ${\displaystyle \eta _{\mu \nu }={\text{diag}}(\pm 1,\mp 1,\mp 1,\mp 1)}$.

Zdroj:https://en.wikipedia.org?pojem=Klein–Gordon_equation
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Klein–Gordon equation in normal units with metric signature ${\displaystyle \eta _{\mu \nu }={\text{diag}}(\pm 1,\mp 1,\mp 1,\mp 1)}$

	Position space ${\displaystyle x^{\mu }=(ct,\mathbf {x} )}$	Fourier transformation ${\displaystyle \omega =E/\hbar ,\quad \mathbf {k} =\mathbf {p} /\hbar }$	Momentum space ${\displaystyle p^{\mu }=(E/c,\mathbf {p} )}$
Separated time and space	${\displaystyle \left({\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}+{\frac {m^{2}c^{2}}{\hbar ^{2}}}\right)\psi (t,\mathbf {x} )=0}$	${\displaystyle \psi (t,\mathbf {x} )=\int {\frac {\mathrm {d} \omega }{2\pi \hbar }}\int {\frac {\mathrm {d} ^{3}k}{(2\pi \hbar )^{3}}}\,e^{\mp i(\omega t-\mathbf {k} \cdot \mathbf {x} )}\psi (\omega ,\mathbf {k} )}$