Klein-Gordon equationKleinGordonEquation2008-03-16 14:54:182008-04-17 06:19:54Definition% this is the default PlanetMath preamble. as your knowledge
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The Klein-Gordon equation is an equation of mathematical physics that describes spin-0 particles. It is given by:
\[
\Box \psi = \left(\frac{mc}{\hbar }\right)^2 \psi
\]
Here the $\Box$ symbol refers to the wave operator, or D'Alembertian, ($\Box = \nabla^2 - \frac{1}{c^2} \partial^2_t$)
and $\psi$ is the wave function of a particle.
It is a Lorentz invariant expression.
\subsection{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:
\[
\hbar^2\left(c^2 \nabla^2 -\frac{\partial^2}{\partial t^2} \right) = m^2 c^4
\]
Dividing both sides by $\hbar^2 c^2$:
\[
\left( \nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} \right) = \frac{m^2 c^2}{\hbar^2}
\]
Identifying the expression in brackets as the D'Alembertian and right-multiplying the whole expression by $\psi$ , we obtain the Klein-Gordon equation:
\[
\Box \psi = \left(\frac{mc}{\hbar }\right)^2 \psi
\]