# 魏尔方程式

（重定向自外爾方程式

## 方程式

${\displaystyle \sigma ^{\mu }\partial _{\mu }\psi =0}$

SI單位中可寫為：

${\displaystyle I_{2}{\frac {1}{c}}{\frac {\partial \psi }{\partial t}}+\sigma _{x}{\frac {\partial \psi }{\partial x}}+\sigma _{y}{\frac {\partial \psi }{\partial y}}+\sigma _{z}{\frac {\partial \psi }{\partial z}}=0}$

${\displaystyle \sigma _{\mu }=(\sigma _{0},\sigma _{1},\sigma _{2},\sigma _{3})=(I_{2},\sigma _{x},\sigma _{y},\sigma _{z})}$

### 魏尔旋量

${\displaystyle \psi ={\begin{pmatrix}\psi _{1}\\\psi _{2}\\\end{pmatrix}}=\chi e^{-i(\mathbf {k} \cdot \mathbf {r} -\omega t)}=\chi e^{-i(\mathbf {p} \cdot \mathbf {r} -Et)/\hbar }}$

${\displaystyle \chi ={\begin{pmatrix}\chi _{1}\\\chi _{2}\\\end{pmatrix}}}$

${\displaystyle |\mathbf {p} |=\hbar |\mathbf {k} |=\hbar \omega /c\,\rightarrow \,|\mathbf {k} |=\omega /c}$

{\displaystyle {\begin{aligned}&\sigma ^{\mu }\partial _{\mu }\psi _{R}=0\\&{\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{L}=0\end{aligned}}}

## 推演

${\displaystyle {\mathcal {L}}=i\psi _{R}^{\dagger }\sigma ^{\mu }\partial _{\mu }\psi _{R}}$
${\displaystyle {\mathcal {L}}=i\psi _{L}^{\dagger }{\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{L}}$

## 參考資料

1. ^ Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0
2. ^ The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.