Klein-Gordon equation

How to construct

Starting from the relativistic connection between energy and momentum:

$E^{2}={\mathbf {p}}^{2}c^{2}+m^{2}c^{4}$

Substituting $E\rightarrow i\hbar {\frac {\partial }{\partial t}}$ and ${\mathbf {p}}\rightarrow -i\hbar \nabla$, we get Klein-Gordon as follows:

$-\hbar ^{2}{\frac {\partial ^{2}\phi ({\mathbf {r}},t)}{\partial t^{2}}}=(-\hbar ^{2}c^{2}\nabla ^{2}+m^{2}c^{4})\phi ({\mathbf {r}},t)\qquad \qquad \qquad \qquad \qquad (9.2)$

Klein-Gordon can also be written as the following:

$(\square -K^{2})\phi ({\mathbf {r}},t)=0\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (9.3)$

where $\square =\nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}$ is d'Alembert operator and $K={\frac {mc}{\hbar }}$.

Equation (9.3) looks like a classical wave equation with an extra term $K^{2}$.