  We seek to introduce a mathematical method to derive the Klein-Gordon
equation and a set of relevant laws strictly, which combines the relativistic
wave functions in two inertial frames of reference. If we define the stationary
state wave functions as special solutions like
$\Psi(\mathbf{r},t)=\psi(\mathbf{r})e^{-iEt/\hbar}$, and define $m=E/c^2$,
which is called the mass of the system, then the Klein-Gordon equation can
clearly be expressed in a better form when compared with the non-relativistic
limit, which not only allows us to transplant the solving approach of the
Schr\"{o}dinger equation into the relativistic wave equations, but also proves
that the stationary solutions of the Klein-Gordon equation in a potential field
have the probability significance. For comparison, we have also discussed the
Dirac equation. By introducing the concept of system mass into the Klein-Gordon
equation with the scalar and vector potentials, we prove that if the Schr\"{o
dinger equation in a certain potential field can be solved exactly, then under
the condition that the scalar and vector potentials are equal, the Klein-Gordon
equation in the same potential field can also be solved exactly by using the
same method.
