Klein-Gordon equation

The Klein-Gordon equation is an equation of mathematical physics that describes spin-0 particles. It is given by:

Here the symbol refers to the wave operator, or D'Alembertian, ( ) and $\psi$ is the wave function of a particle. It is a Lorentz invariant expression.

Derivation

Like the Dirac equation, the Klein-Gordon equation is derived from the relativistic expression for total energy:$$ E^2 = m^2c^4 + p^2c^2$$ Instead of taking the square root (as Dirac did), we keep the equation in squared form and replace the momentum and energy with their operator equivalents, $E = i \hbar \partial_t$ , $p = -i \hbar \nabla$ . This gives (in disembodied operator form)$$ -\hbar^2 \frac{\partial^2}{\partial t^2} = m^2 c^4 - \hbar^2 c^2 \nabla^2$$ Rearranging:$$ \hbar^2\left(c^2 \nabla^2 -\frac{\partial^2}{\partial t^2} \right) = m^2 c^4$$ Dividing both sides by $\hbar^2 c^2$ :$$ \left( \nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} \right) = \frac{m^2 c^2}{\hbar^2}$$ Identifying the expression in brackets as the D'Alembertian and right-multiplying the whole expression by $\psi$ , we obtain the Klein-Gordon equation: