# Klein-Gordon equation

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

$$\mathrm{\square}\psi ={\left(\frac{mc}{\mathrm{\hslash}}\right)}^{2}\psi $$ |

Here the $\mathrm{\square}$ symbol refers to the wave operator, or D’Alembertian, ($\mathrm{\square}={\nabla}^{2}-\frac{1}{{c}^{2}}{\partial}_{t}^{2}$) 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^{}, $E=i\mathrm{\hslash}{\partial}_{t}$, $p=-i\mathrm{\hslash}\nabla $. This gives (in disembodied operator form)

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

Rearranging:

$${\mathrm{\hslash}}^{2}\left({c}^{2}{\nabla}^{2}-\frac{{\partial}^{2}}{\partial {t}^{2}}\right)={m}^{2}{c}^{4}$$ |

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

$$\left({\nabla}^{2}-\frac{1}{{c}^{2}}\frac{{\partial}^{2}}{\partial {t}^{2}}\right)=\frac{{m}^{2}{c}^{2}}{{\mathrm{\hslash}}^{2}}$$ |

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

$$\mathrm{\square}\psi ={\left(\frac{mc}{\mathrm{\hslash}}\right)}^{2}\psi $$ |

Title | Klein-Gordon equation |
---|---|

Canonical name | KleinGordonEquation |

Date of creation | 2013-03-22 17:55:11 |

Last modified on | 2013-03-22 17:55:11 |

Owner | invisiblerhino (19637) |

Last modified by | invisiblerhino (19637) |

Numerical id | 12 |

Author | invisiblerhino (19637) |

Entry type | Definition |

Classification | msc 78A25 |

Classification | msc 35Q60 |

Synonym | Klein Gordon equation |

Synonym | Klein-Gordon-Fock equation |

Related topic | DiracEquation |

Related topic | SchrodingersWaveEquation |