# 泡利矩陣

\begin{align} \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{align}

## 數學性質

$\sigma_a = \begin{pmatrix} \delta_{a3} & \delta_{a1} - i\delta_{a2}\\ \delta_{a1} + i\delta_{a2} & -\delta_{a3} \end{pmatrix}$

### 本徵質和本徵向量

$\sigma_1^2 = \sigma_2^2 = \sigma_3^2 = -i\sigma_1 \sigma_2 \sigma_3 = \begin{pmatrix} 1&0\\0&1\end{pmatrix} = I$

\begin{align} \det (\sigma_i) &= -1 \\ \operatorname{Tr} (\sigma_i) &= 0 \end{align}

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

### 包立向量

\begin{align} \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{align}

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

### 對易關係

$[\sigma_a, \sigma_b] = 2 i \varepsilon_{a b c}\,\sigma_c \, ,$

$\{\sigma_a, \sigma_b\} = 2 \delta_{a b}\,I.$

### 和內積、外積的關係

\begin{align} \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_{a b c}\,\sigma_c + 2 \delta_{a b}I &= 2\sigma_a \sigma_b \end{align}

$\sigma_a \sigma_b = i\sum_c\varepsilon_{a b c}\,\sigma_c + \delta_{a b}I$

\begin{align} 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{align}

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

### 包立向量的指數

$\vec{a} = a \hat{n}$ ，而且 $|\hat{n}|=1$ 對於偶數 n 可得： $(\hat{n} \cdot \vec{\sigma})^{2n} = I \,$

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

\begin{align} e^{i a(\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{align}

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

(2)

### 完備性關係

$\vec{\sigma}_{\alpha\beta}\cdot\vec{\sigma}_{\gamma\delta}\equiv \sum_{i=1}^3 \sigma^i_{\alpha\beta}\sigma^i_{\gamma\delta} = 2 \delta_{\alpha\delta} \delta_{\beta\gamma} - \delta_{\alpha\beta}\delta_{\gamma\delta}.\,$

$\sum_{i=0}^3 \sigma^i_{\alpha\beta}\sigma^i_{\gamma\delta} = 2 \delta_{\alpha\delta} \delta_{\beta\gamma}\,$.

### 和換位算符的關係

$P_{ij}|\sigma_i \sigma_j\rangle = |\sigma_j \sigma_i\rangle \,$

$P_{ij} = \tfrac{1}{2}(\vec{\sigma}_i\cdot\vec{\sigma}_j + 1)\,$

## 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）：

$1 \mapsto I, \quad i \mapsto - i \sigma_1, \quad j \mapsto - i \sigma_2, \quad k \mapsto - i \sigma_3.$

$1 \mapsto I, \quad i \mapsto i \sigma_3, \quad j \mapsto i \sigma_2, \quad k \mapsto i \sigma_1.$

## 參考文獻

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.