# 泡利矩阵

## 维基百科，自由的百科全书

{\displaystyle {\begin{aligned}\sigma _{1}=\sigma _{x}&={\begin{pmatrix}0&1\\1&0\end{pmatrix))\\\sigma _{2}=\sigma _{y}&={\begin{pmatrix}0&-i\\i&0\end{pmatrix))\\\sigma _{3}=\sigma _{z}&={\begin{pmatrix}1&0\\0&-1\end{pmatrix))\end{aligned))}

## 数学性质

${\displaystyle \sigma _{a}={\begin{pmatrix}\delta _{a3}&\delta _{a1}-i\delta _{a2}\\\delta _{a1}+i\delta _{a2}&-\delta _{a3}\end{pmatrix))}$

### 本征值和本征向量

${\displaystyle \sigma _{1}^{2}=\sigma _{2}^{2}=\sigma _{3}^{2}=-i\sigma _{1}\sigma _{2}\sigma _{3}={\begin{pmatrix}1&0\\0&1\end{pmatrix))=I}$

{\displaystyle {\begin{aligned}\det(\sigma _{i})&=-1\\\operatorname {tr} (\sigma _{i})&=0\end{aligned))}

${\displaystyle {\begin{array}{lclc}\psi _{x+}=\displaystyle {\frac {1}{\sqrt {2))}\!\!\!\!\!&{\begin{pmatrix}{1}\\{1}\end{pmatrix))&\psi _{x-}=\displaystyle {\frac {1}{\sqrt {2))}\!\!\!\!\!&{\begin{pmatrix}{1}\\{-1}\end{pmatrix))\\\psi _{y+}=\displaystyle {\frac {1}{\sqrt {2))}\!\!\!\!\!&{\begin{pmatrix}{-i}\\{1}\end{pmatrix))&\psi _{y-}=\displaystyle {\frac {1}{\sqrt {2))}\!\!\!\!\!&{\begin{pmatrix}{1}\\{-i}\end{pmatrix))\\\psi _{z+}=&{\begin{pmatrix}{1}\\{0}\end{pmatrix))&\psi _{z-}=&{\begin{pmatrix}{0}\\{1}\end{pmatrix)).\end{array))}$

### 泡利向量

${\displaystyle {\vec {\sigma ))=\sigma _{1}{\hat {x))+\sigma _{2}{\hat {y))+\sigma _{3}{\hat {z))\,}$

{\displaystyle {\begin{aligned}{\vec {a))\cdot {\vec {\sigma ))&=(a_{i}{\hat {x))_{i})\cdot (\sigma _{j}{\hat {x))_{j})\\&=a_{i}\sigma _{j}{\hat {x))_{i}\cdot {\hat {x))_{j}\\&=a_{i}\sigma _{j}\delta _{ij}\\&=a_{i}\sigma _{i}\end{aligned))}

${\displaystyle \det {\vec {a))\cdot {\vec {\sigma ))=-{\vec {a))\cdot {\vec {a))=-|{\vec {a))|^{2))$

### 对易关系

${\displaystyle [\sigma _{a},\sigma _{b}]=2i\varepsilon _{abc}\,\sigma _{c}\,,}$

${\displaystyle \{\sigma _{a},\sigma _{b}\}=2\delta _{ab}\,I}$

### 和内积、外积的关系

{\displaystyle {\begin{aligned}\left[\sigma _{a},\sigma _{b}\right]+\{\sigma _{a},\sigma _{b}\}&=(\sigma _{a}\sigma _{b}-\sigma _{b}\sigma _{a})+(\sigma _{a}\sigma _{b}+\sigma _{b}\sigma _{a})\\2i\sum _{c}\varepsilon _{abc}\,\sigma _{c}+2\delta _{ab}I&=2\sigma _{a}\sigma _{b}\end{aligned))}

${\displaystyle \sigma _{a}\sigma _{b}=i\sum _{c}\varepsilon _{abc}\,\sigma _{c}+\delta _{ab}I}$

{\displaystyle {\begin{aligned}a_{p}b_{q}\sigma _{p}\sigma _{q}&=a_{p}b_{q}\left(i\sum _{r}\varepsilon _{pqr}\,\sigma _{r}+\delta _{pq}I\right)\\a_{p}\sigma _{p}b_{q}\sigma _{q}&=i\sum _{r}\varepsilon _{pqr}\,a_{p}b_{q}\sigma _{r}+a_{p}b_{q}\delta _{pq}I\end{aligned))}

${\displaystyle ({\vec {a))\cdot {\vec {\sigma )))({\vec {b))\cdot {\vec {\sigma )))=({\vec {a))\cdot {\vec {b)))\,I+i({\vec {a))\times {\vec {b)))\cdot {\vec {\sigma ))}$

### 泡利向量的指数

${\displaystyle {\vec {a))=a{\hat {n))}$，而且${\displaystyle |{\hat {n))|=1}$对于偶数n可得：

${\displaystyle ({\hat {n))\cdot {\vec {\sigma )))^{2n}=I\,}$

${\displaystyle ({\hat {n))\cdot {\vec {\sigma )))^{2n+1}={\hat {n))\cdot {\vec {\sigma ))\,}$

{\displaystyle {\begin{aligned}e^{ia({\hat {n))\cdot {\vec {\sigma )))}&=\sum _{n=0}^{\infty }{\frac {i^{n}\left[a({\hat {n))\cdot {\vec {\sigma )))\right]^{n)){n!))\\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(a{\hat {n))\cdot {\vec {\sigma )))^{2n)){(2n)!))+i\sum _{n=0}^{\infty }{\frac {(-1)^{n}(a{\hat {n))\cdot {\vec {\sigma )))^{2n+1)){(2n+1)!))\\&=I\sum _{n=0}^{\infty }{\frac {(-1)^{n}a^{2n)){(2n)!))+i{\hat {n))\cdot {\vec {\sigma ))\left(\sum _{n=0}^{\infty }{\frac {(-1)^{n}a^{2n+1)){(2n+1)!))\right)\\\end{aligned))}

 ${\displaystyle e^{ia({\hat {n))\cdot {\vec {\sigma )))}=I\cos {a}+i({\hat {n))\cdot {\vec {\sigma )))\sin {a}\,}$

(2)

### 完备性关系

${\displaystyle {\vec {\sigma ))_{\alpha \beta }\cdot {\vec {\sigma ))_{\gamma \delta }\equiv \sum _{i=1}^{3}\sigma _{\alpha \beta }^{i}\sigma _{\gamma \delta }^{i}=2\delta _{\alpha \delta }\delta _{\beta \gamma }-\delta _{\alpha \beta }\delta _{\gamma \delta }.\,}$

${\displaystyle \sum _{i=0}^{3}\sigma _{\alpha \beta }^{i}\sigma _{\gamma \delta }^{i}=2\delta _{\alpha \delta }\delta _{\beta \gamma }\,}$.

### 和换位算符的关系

${\displaystyle P_{ij}|\sigma _{i}\sigma _{j}\rangle =|\sigma _{j}\sigma _{i}\rangle \,}$

${\displaystyle P_{ij}={\tfrac {1}{2))({\vec {\sigma ))_{i}\cdot {\vec {\sigma ))_{j}+1)\,}$

## SU (2)

### 四元数与泡利矩阵

{I, 1, 2, 3}的实数张成与四元数的实代数同构，可透过下列映射得到对应关系（注意到泡利矩阵的负号）：

${\displaystyle 1\mapsto I,\quad \mathbf {i} \mapsto -\sigma _{2}\sigma _{3}=-i\sigma _{1},\quad \mathbf {j} \mapsto -\sigma _{3}\sigma _{1}=-i\sigma _{2},\quad \mathbf {k} \mapsto -\sigma _{1}\sigma _{2}=-i\sigma _{3}.}$

${\displaystyle 1\mapsto I,\quad \mathbf {i} \mapsto i\sigma _{3},\quad \mathbf {j} \mapsto i\sigma _{2},\quad \mathbf {k} \mapsto i\sigma _{1}.}$

## 参考文献

1. ^ Pauli matrices. Planetmath website. 28 March 2008 [28 May 2013].
2. ^ Nakahara, Mikio. Geometry, topology, and physics 2nd. CRC Press. 2003. ISBN 978-0-7503-0606-5., pp. xxii.