# Klein-Gordon equation

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.

## 0.1 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:

 $\hbar^{2}\left(c^{2}\nabla^{2}-\frac{\partial^{2}}{\partial t^{2}}\right)=m^{2% }c^{4}$

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:

 $\Box\psi=\left(\frac{mc}{\hbar}\right)^{2}\psi$
Title Klein-Gordon equation KleinGordonEquation 2013-03-22 17:55:11 2013-03-22 17:55:11 invisiblerhino (19637) invisiblerhino (19637) 12 invisiblerhino (19637) Definition msc 78A25 msc 35Q60 Klein Gordon equation Klein-Gordon-Fock equation DiracEquation SchrodingersWaveEquation