PlanetMath: 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:

$\displaystyle \Box \psi = \left(\frac{mc}{\hbar }\right)^2 \psi$

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.

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:

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

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

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

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

$\displaystyle \Box \psi = \left(\frac{mc}{\hbar }\right)^2 \psi$

"Klein-Gordon equation" is owned by invisiblerhino.

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Other names:

Klein Gordon equation, Klein-Gordon-Fock equation

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Cross-references: sides, equivalents, operator, square root, Dirac equation, expression, invariant, wave function, wave operator, equation
There are 3 references to this entry.

This is version 9 of Klein-Gordon equation, born on 2008-03-16, modified 2008-04-17.
Object id is 10412, canonical name is KleinGordonEquation.
Accessed 449 times total.

Classification:

AMS MSC:	35Q60 (Partial differential equations :: Equations of mathematical physics and other areas of application :: Equations of electromagnetic theory and optics)
	78A25 (Optics, electromagnetic theory :: General :: Electromagnetic theory, general)

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