# 克莱因-戈尔登方程本文重定向自 克莱因-戈尔登方程

## 陈述

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

${\displaystyle -\partial _{t}^{2}\psi +\nabla ^{2}\psi =m^{2}\psi }$

${\displaystyle \psi =e^{-i\omega t+ik\cdot x}=e^{ik_{\mu }x^{\mu }}}$

${\displaystyle -p_{\mu }p^{\mu }=E^{2}-P^{2}=\omega ^{2}-k^{2}=-k_{\mu }k^{\mu }=m^{2}\,}$

${\displaystyle \left[\nabla ^{2}-{\frac {m^{2}c^{2}}{\hbar ^{2}}}\right]\psi (\mathbf {r} )=0}$

## 相对论量子力学下的形式推导

${\displaystyle {\frac {\mathbf {p} ^{2}}{2m}}\psi =i\hbar {\frac {\partial }{\partial t}}\psi }$

${\displaystyle E={\sqrt {\mathbf {p} ^{2}c^{2}+m^{2}c^{4}}}}$

${\displaystyle (\Box ^{2}+\mu ^{2})\psi =0,}$

## 量子场论下的形式推导

${\displaystyle L={\frac {1}{2}}\partial _{\mu }\phi \partial ^{\mu }\phi -{\frac {1}{2}}m^{2}\phi ^{2}}$

## 自由粒子解

${\displaystyle \psi (\mathbf {r} ,t)=e^{i(\mathbf {k} \cdot \mathbf {r} -\omega t)}}$

${\displaystyle E=\pm {\sqrt {\mathbf {p} ^{2}c^{2}+m^{2}c^{4}}}}$

## 参考文献

• Sakurai, J. J. (1967). Advanced Quantum Mechanics. Addison Wesley. ISBN 0-201-06710-2.
• Greiner, W. (1990). Relativistic Quantum Mechanics. Springer-Verlag. ISBN 3-540-67457-8.