# 泡利方程

## 方程

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

${\displaystyle \mathbf {p} }$动量算符p = −iħ∇，∇为梯度算符），
${\displaystyle {\vec {\sigma ))}$泡利矩阵
${\displaystyle |\psi \rangle :={\begin{pmatrix}|\psi _{+}\rangle \\|\psi _{-}\rangle \end{pmatrix))}$为泡利旋量

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

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

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

p = −iħ∇代入，可得到[1]

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

## 与施特恩-格拉赫实验的关系

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

${\displaystyle |\psi _{\pm }\rangle }$为泡利旋量，
${\displaystyle {\boldsymbol {\sigma ))=(\sigma _{x},\sigma _{y},\sigma _{z})}$泡利矩阵所构成的泡利向量，
B为外加磁场，与磁矢势A的关系为：${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$

${\displaystyle {\hat {1))}$为二阶单位矩阵${\displaystyle {\begin{pmatrix}1&0\\0&1\\\end{pmatrix))}$

## 与薛定谔方程、狄拉克方程的关系

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

## 参考文献

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

## 外部链接

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