The Klein-Gordon equation is a relativistic version (describing spinless particles) of the Schrödinger equation.
The Schrödinger equation for a free particle is
where is the momentum operator, using natural units where .
The Schrödinger 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 Schrödinger, 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.