Klein-Gordon Equation

Derivation

Start from

$$E^2=p^2c^2+m^2c^4.$$

Use the operator substitutions

$$E\to i\hbar\frac{\partial}{\partial t}, \qquad \mathbf p\to -i\hbar\nabla.$$

Then

$$\left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2} -\nabla^2 +\frac{m^2c^2}{\hbar^2}\right)\phi=0.$$

Interpretation

In modern language this is the field equation for a free scalar field, not a single-particle probability wave in the same sense as the Schrödinger equation.