Klein-Gordon equation

The Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is a relativistic version (describing scalar (or pseudoscalar) spinless particles) of the Schr�dinger equation.

The Schr�dinger equation for a free particle is

where is the momentum operator, using natural units where .

The Schr�dinger equation suffers from not being relativistically covariant, meaning it does not take into account Einstein's special theory of relativity.

It is natural to try and use the identity from special relativity

for the energy; then, just inserting the quantum mechanical momentum operator, yields the equation

This, however, is a cumbersome expression to work with because of the square root. Cumbersomeness, however, doesn't really count as an objection. But this equation, as it stands, is nonlocal.

Klein and Gordon instead worked with the square of this equation (the Klein-Gordon equation for a free particle), which in covariant notation reads

The Klein-Gordon equation was actually first found by Schr�dinger, before he made the discovery of the equation that now bears his name. He rejected it because he couldn't make it fit data (the equation doesn't take into account the spin of the electron); the way he found his equation was by making simplifications in the Klein-Gordon equation.

Fock in 1926 soon after the Schr�dinger equation was introduced wrote an article about its generalization for the case of magnetic fields, where forces were dependent on velocity and finally he independently wrote this equation. Both Klein and Fock used Kaluza and Klein's method. Fock also determined the gauge transformation for the wave equation.