# 包立方程式

## 方程式

 包立方程式 （廣義形式） ${\displaystyle \left[{\frac {1}{2m}}({\boldsymbol {\sigma }}\cdot (\mathbf {p} -q\mathbf {A} ))^{2}+q\phi \right]|\psi \rangle =i\hbar {\frac {\partial }{\partial t}}|\psi \rangle }$

${\displaystyle \mathbf {p} }$ 動量算符p = −iħ∇，∇為梯度算符），
${\displaystyle {\vec {\sigma }}}$ 包立矩陣
${\displaystyle |\psi \rangle :={\begin{pmatrix}|\psi _{+}\rangle \\|\psi _{-}\rangle \end{pmatrix}}}$ 為包立旋量

${\displaystyle {\hat {H}}|\psi \rangle =i\hbar {\frac {\partial }{\partial t}}|\psi \rangle }$

${\displaystyle {\hat {H}}={\frac {1}{2m}}({\boldsymbol {\sigma }}\cdot (\mathbf {p} -q\mathbf {A} ))^{2}+q\phi }$

${\displaystyle ({\boldsymbol {\sigma }}\cdot \mathbf {a} )({\boldsymbol {\sigma }}\cdot \mathbf {b} )=\mathbf {a} \cdot \mathbf {b} +i{\boldsymbol {\sigma }}\cdot \left(\mathbf {a} \times \mathbf {b} \right)}$

p = −iħ∇代入，可得到[1]

${\displaystyle {\hat {H}}={\frac {1}{2m}}\left[\left(\mathbf {p} -q\mathbf {A} \right)^{2}-q\hbar {\boldsymbol {\sigma }}\cdot \mathbf {B} \right]+q\phi }$

## 與斯特恩-革拉赫實驗的關係

 包立方程式 （磁場B） ${\displaystyle \underbrace {i\hbar {\frac {\partial }{\partial t}}|\psi _{\pm }\rangle =\left({\frac {(\mathbf {p} -q\mathbf {A} )^{2}}{2m}}+q\phi \right){\hat {1}}|\psi _{\pm }\rangle } _{\mathrm {Schr{\ddot {o}}dinger~equation} }-\underbrace {{\frac {q\hbar }{2m}}{\boldsymbol {\sigma }}\cdot \mathbf {B} |\psi _{\pm }\rangle } _{\text{Stern-Gerlach term}}}$

${\displaystyle |\psi _{\pm }\rangle }$ 為包立旋量，
${\displaystyle {\boldsymbol {\sigma }}=(\sigma _{x},\sigma _{y},\sigma _{z})}$ 包立矩陣所構成的包立向量，
B為外加磁場，與磁向量勢A的關係為：${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$

${\displaystyle {\hat {1}}}$ 為二階單位矩陣${\displaystyle {\begin{pmatrix}1&0\\0&1\\\end{pmatrix}}}$

## 與薛丁格方程式、狄拉克方程式的關係

${\displaystyle \left[{\frac {\mathbf {p} ^{2}}{2m}}+q\phi \right]{\begin{pmatrix}\psi _{+}\\\psi _{-}\end{pmatrix}}=i\hbar {\frac {\partial }{\partial t}}{\begin{pmatrix}\psi _{+}\\\psi _{-}\end{pmatrix}}}$

## 參考文獻

1. ^ Bransden, BH; Joachain, CJ. Physics of Atoms and Molecules 1st. Prentice Hall. 1983: 638-638. ISBN 0-582-44401-2.