# PlanetPhysics/Klein Gordon Equation

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

${\displaystyle (\Box +\left({\frac {m}{\hbar c}}\right)^{2})\psi =0}$

Here the ${\displaystyle \Box }$ symbol refers to the wave operator, or D'Alembertian, and ${\displaystyle \psi }$ is the wavefunction 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:

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

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, ${\displaystyle E=i\hbar \partial _{t}}$ , ${\displaystyle p=-i\hbar \nabla }$ . This gives (in disembodied operator form)

${\displaystyle -\hbar ^{2}{\frac {\partial ^{2}}{\partial t^{2}}}=m^{2}c^{4}-\hbar ^{2}c^{2}\nabla ^{2}}$

Rearranging:

${\displaystyle \hbar ^{2}(c^{2}\nabla ^{2}-{\frac {\partial ^{2}}{\partial t^{2}}})+m^{2}c^{4}=0}$

Dividing both sides by ${\displaystyle \hbar ^{2}c^{2}}$ :

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

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

${\displaystyle (\Box +\left({\frac {m}{\hbar c}}\right)^{2})\psi =0}$