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: $$ E^2 = m^2c^4 + p^2c^2 $$ 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: $$ \hbar^2(c^2 \nabla^2 - \frac{\partial^2}{\partial t^2}) + m^2 c^4 = 0 $$ 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: $$ (\Box + \left(\frac{m}{\hbar c}\right)^2) \psi = 0 $$