# 泡利矩陣

（重定向自泡利矩阵

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

## 數學性質

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

### 本徵值和本徵向量

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

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

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

### 包立向量

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

{\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}}}

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

### 對易關係

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

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

### 和內積、外積的關係

{\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}}}

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

### 包立向量的指數

${\displaystyle {\vec {a}}=a{\hat {n}}}$ ，而且${\displaystyle |{\hat {n}}|=1}$ 對於偶數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)

### 完備性關係

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

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

### 和換位算符的關係

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

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

## SU (2)

### 四元數與包立矩陣

{I, 1, 2, 3}的實數張成與四元數的實代數同構，可透過下列映射得到對應關係（注意到包立矩陣的負號）：

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

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

## 參考文獻

1. ^ Pauli matrices. Planetmath website. 28 March 2008 [28 May 2013]. （原始内容存档于2017-09-26）.
2. ^ Nakahara, Mikio. Geometry, topology, and physics 2nd. CRC Press. 2003. ISBN 978-0-7503-0606-5., pp. xxii页面存档备份，存于互联网档案馆）.