# 角动量算符对易关系

## 导引

${\displaystyle [L_{\alpha },L_{\beta }]=i\hbar \sum _{\gamma }\epsilon _{\alpha \beta \gamma }L_{\gamma },\quad \alpha ,\beta ,\gamma \in \{x,y,z\}}$

## 定义

${\displaystyle j_{x},j_{y},j_{z}}$

${\displaystyle [j_{\alpha },j_{\beta }]=i\hbar \sum _{\gamma }\epsilon _{\alpha \beta \gamma }j_{\gamma },\quad \alpha ,\beta ,\gamma \in \{x,y,z\}}$

${\displaystyle \mathbf {j} =(j_{x},j_{y},j_{z})}$

${\displaystyle \mathbf {j} \times \mathbf {j} =i\hbar \mathbf {j} }$

## 角动量平方算符

${\displaystyle \mathbf {j} ^{2}=\mathbf {j} \cdot \mathbf {j} =j_{x}^{2}+j_{y}^{2}+j_{z}^{2}}$

${\displaystyle [\mathbf {j} ^{2},j_{\alpha }]=0,\quad \alpha =x,y,z}$

## 升算符与降算符

${\displaystyle j_{+}=j_{x}+ij_{y},j_{-}=j_{x}-ij_{y}}$

${\displaystyle [\mathrm {j} ^{2},j_{\pm }]=0}$
${\displaystyle [j_{z},j_{\pm }]=\pm \hbar j_{\pm }}$
${\displaystyle j_{\pm }j_{\mp }=\mathbf {j} ^{2}-j_{z}^{2}\pm \hbar j_{z}}$
${\displaystyle [j_{+},j_{-}]=2\hbar j_{z}}$
${\displaystyle [j_{+},j_{-}]_{+}=2(\mathbf {j} ^{2}-j_{z}^{2})}$

## 本征函数

${\displaystyle |jm\rangle }$

${\displaystyle \mathbf {j} ^{2}|jm\rangle =\lambda \hbar ^{2}|jm\rangle ,\quad j_{z}|jm\rangle =m\hbar |jm\rangle }$

${\displaystyle \langle j'm'|jm\rangle =\delta _{jj'}\delta _{mm'}}$

${\displaystyle \langle j'm'|f|jm\rangle }$

${\displaystyle \langle j'm'|j_{\pm }|jm\rangle =\delta _{jj'}\delta _{m',m\pm 1}\langle jm\pm 1|j_{\pm }|jm\rangle }$

${\displaystyle 1=\sum _{j,m}|jm\rangle \langle jm|}$

${\displaystyle \langle jm\pm 1|j_{\pm }|jm\rangle \langle jm|j_{\mp }|jm\pm 1\rangle =\lambda -(m\pm 1)^{2}\pm (m\pm 1)}$

${\displaystyle j_{\pm }|jm\rangle =e^{i\delta }{\sqrt {\lambda -m(m\pm 1)}}|jm\pm 1\rangle }$

${\displaystyle \lambda =m_{\max(}m_{\max }+1)=m_{\min(}m_{\min }-1)}$

${\displaystyle m_{\max }-m_{\min }\in \mathbb {Z} }$

${\displaystyle \lambda =j(j+1),2j\in \mathbb {Z} _{0}^{+},m=-j,-j+1,\dots ,j-1,j}$

## 矩阵表示

j=0 时的矩阵表示是平凡的。

j=1/2 时的矩阵表示对应着泡利矩阵

${\displaystyle j_{z}={\begin{bmatrix}{\frac {1}{2}}&0\\0&-{\frac {1}{2}}\end{bmatrix}},j_{+}={\begin{bmatrix}0&0\\1&0\end{bmatrix}},j_{-}={\begin{bmatrix}0&1\\0&0\end{bmatrix}},j_{x}={\begin{bmatrix}0&{\frac {1}{2}}\\{\frac {1}{2}}&0\end{bmatrix}},j_{y}={\begin{bmatrix}0&{\frac {i}{2}}\\-{\frac {i}{2}}&0\end{bmatrix}}}$

j=1 时的矩阵表示，

${\displaystyle j_{z}={\begin{bmatrix}1&0&0\\0&0&0\\0&0&-1\end{bmatrix}},j_{+}={\begin{bmatrix}0&{\sqrt {2}}&0\\0&0&{\sqrt {2}}\\0&0&0\end{bmatrix}},j_{-}={\begin{bmatrix}0&0&0\\{\sqrt {2}}&0&0\\0&{\sqrt {2}}&0\end{bmatrix}},j_{x}={\begin{bmatrix}0&{\frac {\sqrt {2}}{2}}&0\\{\frac {\sqrt {2}}{2}}&0&{\frac {\sqrt {2}}{2}}\\0&{\frac {\sqrt {2}}{2}}&0\end{bmatrix}},j_{y}={\begin{bmatrix}0&-{\frac {\sqrt {2}}{2}}i&0\\{\frac {\sqrt {2}}{2}}i&0&-{\frac {\sqrt {2}}{2}}i\\0&{\frac {\sqrt {2}}{2}}i&0\end{bmatrix}},}$

## 注

1. ^ 参见角动量算符一文相关小节

## 参考文献

1. 曾谨言. 10. 量子力学卷 I （第四版）. 科学出版社. [2011]. ISBN 9787030181398.
2. ^ 徐光宪,黎乐民,王德民. 6. 量子化学:基本原理和从头计算法(第2版)(上册). 科学出版社. [2011]. ISBN 9787030192134.
3. ^ Lie algebra (PDF). [2014-09-08]. （原始内容存档 (PDF)于2021-05-06）.
• M. E. Rose. Elementary Theory of Angular Momentum. 2011.