Klein-Gordon equation

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

$$\mathrm{\square }\psi ={\left(\frac{mc}{\mathrm{\hslash }}\right)}^2\psi$$

Here the $\mathrm{\square }$ symbol refers to the wave operator, or D’Alembertian, ($\mathrm{\square }={\nabla }^2-\frac{1}{c^2}{\partial }_t^2$)and $\psi$ is the wave function of a particle.It is a Lorentz invariant expression.

0.1 Derivation

Like the Dirac equation, the Klein-Gordon equation is derived from the relativistic expression for total energy:

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

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, $E=i\mathrm{\hslash }{\partial }_t$, $p=-i\mathrm{\hslash }\nabla$. This gives (in disembodied operator form)

$$-{\mathrm{\hslash }}^2\frac{{\partial }^2}{\partial t^2}=m^2{c}^4-{\mathrm{\hslash }}^2{c}^2{\nabla }^2$$

Rearranging:

$${\mathrm{\hslash }}^2\left(c^2{\nabla }^2-\frac{{\partial }^2}{\partial t^2}\right)=m^2{c}^4$$

Dividing both sides by ${\mathrm{\hslash }}^2{c}^2$:

$$\left({\nabla }^2-\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2}\right)=\frac{m^2{c}^2}{{\mathrm{\hslash }}^2}$$

Identifying the expression in brackets as the D’Alembertian and right-multiplying the whole expression by $\psi$ , we obtain the Klein-Gordon equation:

$$\mathrm{\square }\psi ={\left(\frac{mc}{\mathrm{\hslash }}\right)}^2\psi$$