Klein-Gordon equation

The Klein-Gordon equationMathworldPlanetmath is an equation of mathematical physics that describes spin-0 particles. It is given by:


Here the symbol refers to the wave operator, or D’Alembertian, (=2-1c2t2) and ψ is the wave function of a particle. It is a Lorentz invariant expression.

0.1 Derivation

Like the Dirac equationMathworldPlanetmath, the Klein-Gordon equation is derived from the relativistic expression for total energy:


Instead of taking the square root (as Dirac did), we keep the equation in squared form and replace the momentum and energy with their operator equivalentsPlanetmathPlanetmath, E=it, p=-i. This gives (in disembodied operator form)




Dividing both sides by 2c2:


Identifying the expression in brackets as the D’Alembertian and right-multiplying the whole expression by ψ , we obtain the Klein-Gordon equation:

Title Klein-Gordon equation
Canonical name KleinGordonEquation
Date of creation 2013-03-22 17:55:11
Last modified on 2013-03-22 17:55:11
Owner invisiblerhino (19637)
Last modified by invisiblerhino (19637)
Numerical id 12
Author invisiblerhino (19637)
Entry type Definition
Classification msc 78A25
Classification msc 35Q60
Synonym Klein Gordon equation
Synonym Klein-Gordon-Fock equation
Related topic DiracEquation
Related topic SchrodingersWaveEquation