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 = p²c² + m²c⁴, where 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.