The special-relativistic expression for the kinetic energy may be used to form the classical free particle Hamiltonian
The analogous quantum mechanical expression may be constructed
by replacing the classical momentum,
p, with its quantum mechanical operator, 
which yields the free particle wave equation
This equation, however, does not satisfy some of the conditions required by special relativity. The wave equation is not invariant to a Lorentz transform, and the square root term introduces ambiguity. The Klein-Gordon equation rectifies both of these problems simply by taking the square of the original energy expression and extending the result to a quantum mechanical wave equation:
The resulting wave equation is Lorentz covariant and well defined, but
suffers from other problems. Negative energy solutions to this equation are
possible, which do not have a readily obvious explanation, and the
probability density, 
, fluctuates with time as does its
integral over all space.
The foibles of the Klein-Gordon equation may make it a poor equation for the electron, but those weaknesses helped to point Dirac in the right direction and to develop a single particle equation which successfully surmounted all these problems.
