PlanetPhysics/Klein Gordon Equation

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

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

Here the $\Box$ symbol refers to the wave operator, or D'Alembertian, and $\psi$ 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:

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

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, $E=i\hbar \partial _{t}$, $p=-i\hbar \nabla$. 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:

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

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

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

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

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