PlanetPhysics: Klein-Gordon equation

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Klein-Gordon equation

(Definition)

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

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

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:

$\displaystyle E^2 = m^2c^4 + p^2c^2$

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)

$\displaystyle -\hbar^2 \frac{\partial^2}{\partial t^2} = m^2 c^4 - \hbar^2 c^2 \nabla^2$

Rearranging:

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

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

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

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

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

"Klein-Gordon equation" is owned by invisiblerhino.

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