转动算符A：$(\vec{r}\cdot\vec{s})=(A\vec{r}\cdot A\vec{s})$保持变换前后内积不变。要求A是实正交矩阵$A^T A=1$,$det A=\pm1$
固有转动与非固有转东:是否改变坐标系的手性。若转动后，再做空间反演，则改变坐标系的手征性
幺模矩阵：行列式为 1 的矩阵 detA=1。
$SO(3)$：三维幺模实正交矩阵$\mathrm{R}(\hat{\mathfrak{n}},\omega)$描写绕三维空间方向$\hat{n}$转动ω角的变换，按照矩阵的乘积规则，它的集合构成群。
$O(3)$：三维实正交矩阵群。$SO(3)$$\otimes$空间反演群。SO(3)群是O(3)群的不变子群
$\mathrm{R}(\vec{\mathbf{e}}_3,\omega)= \begin{pmatrix} \cos\omega & -\sin\omega & 0 \\ \sin\omega & \cos\omega & 0 \\ 0 & 0 & 1 \end{pmatrix}$
该矩阵可用泡利矩阵写成指数矩阵形式
$\text{泡利矩阵: }\quad\sigma_1= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},\sigma_2= \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix},\sigma_3= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$
$\begin{aligned} \text{三个矩阵之间的关系:} & & \mathcal{E}{abc}= \begin{cases} 1 & & abc:123,231,312 \\ -1 & & abc:321,213,132 \\ 0 & & others & & \end{cases} \\ \sigma_a^2=1,\sigma_1\sigma_2=i\sigma_3,Tr\sigma_a=0,Tr(\sigma_a\sigma_b)=2\delta{ab},etc. \end{aligned}$
$S(\phi,\theta)$:
$\begin{aligned} & \mathrm{S}(\varphi,\theta)=\mathrm{R}(\vec{\mathsf{e}}{3},\varphi)\mathrm{R}(\vec{\mathsf{e}}{2},\theta) \\ & = \begin{pmatrix} \cos\varphi & -\sin\varphi & 0 \\ \sin\varphi & \cos\varphi & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\Theta & 0 & \cos\Theta \end{pmatrix} \\ & \left.=\left( \begin{array} {ccc}{\cos{\varphi}\cos{\theta}} & {-\sin{\varphi}} & {\cos{\varphi}\sin{\theta}} \\ {\sin{\varphi}\cos{\theta}} & {\cos{\varphi}} & {\sin{\varphi}\sin{\theta}} \\ {-\sin{\theta}} & {0} & {\cos{\theta}} \end{array}\right.\right) \end{aligned}$