# Klein-Gordon Equation

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## Klein-Gordon equation

[′klīn ′gȯrd·ən i‚kwā·zhən]## Klein-Gordon Equation

a relativistic (that is satisfying the requirements of the theory of relativity) quantum equation for particles with zero spin.

Historically, the Klein-Gordon equation was the first relativistic equation in quantum mechanics for the wave function ψ of a particle. It was proposed in 1926 by E. Schrödinger as a relativistic generalization of the Schrödinger equation. It was also proposed, independently, by the Swedish physicist O. Klein, the Soviet physicist V. A. Fok, and the German physicist W. Gordon.

For a free particle, the Klein-Gordon equation is written

This equation is associated with the relativistic relationship between the energy £ and the momentum *p* of a particle £ ^{2} = *p ^{2}c^{2}* +

*m*where

^{2}c^{4},*m*is the mass of the particle and

*c*is the speed of light.

The solution to the equation is the function ψ *(x, y, z, t),* which is a function only of the coordinates *(x, y, z)* and the time *(t).* Consequently, the particles described by this function have no other internal degrees of freedom—that is, they are spinless (the π-meson and the K-meson are of this type). However, analysis of the equation indicated that its solution ψ differed fundamentally in physical meaning from the ordinary wave function, which is considered as the probability amplitude of detecting a particle at a given point in space and a given moment in time: ψ*(x, y, z, f)* is not determined uniquely by the value of ψ at the initial moment (such an unambiguous relationship is postulated in nonrelativistic quantum mechanics).

Furthermore, the expression for the probability of a given state can take on not only positive values but also negative values, which are devoid of physical meaning. Therefore, the Klein-Gordon equation was at first rejected. However, in 1934, W. Pauli and W. Weisskopf discovered a suitable interpretation for the equation within the scope of quantum field theory; treating it like a field equation analogous to Maxwell’s equations for an electromagnetic field, they quantized it, so that ψ became an operator.

M. A. LIBERMAN