PlanetPhysics/Klein Gordon Equation

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

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

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

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

Rearranging:

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

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

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

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

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