# 角动量

${\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} }$

${\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} =\mathbf {r} \times (m\mathbf {v} )=\mathbf {r} \times ({\boldsymbol {\omega }}\times (m\mathbf {r} ))=mr^{2}{\boldsymbol {\omega }}=I{\boldsymbol {\omega }}}$

## 角動量量子化

### 量子化角動量和不确定性原理

${\displaystyle [L_{i},L_{j}]=i\hbar \epsilon _{ijk}L_{k}}$
• ${\displaystyle \epsilon _{ijk}}$ 列維-奇維塔符號
• ${\displaystyle [A,B]=AB-BA}$ 交換子

${\displaystyle \left[L_{i},L^{2}\right]=0}$

${\displaystyle L^{2}=-{\frac {\hbar ^{2}}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial }{\partial \theta }}\right)-{\frac {\hbar ^{2}}{\sin ^{2}\theta }}{\frac {\partial ^{2}}{\partial \phi ^{2}}}}$

${\displaystyle L^{2}|l,m\rangle ={\hbar }^{2}l(l+1)|l,m\rangle }$
${\displaystyle L_{z}|l,m\rangle =\hbar m|l,m\rangle }$

${\displaystyle \langle \theta ,\phi |l,m\rangle =Y_{l,m}(\theta ,\phi )}$

${\displaystyle l}$ 是某非負整數。${\displaystyle -l\leq m\leq l}$ 是絕對值不大於${\displaystyle l}$ 的整數。

### 能量均分與角動量量子化

${\displaystyle E={\frac {L_{z}^{2}}{2I}}}$

${\displaystyle L_{z}}$ 是分子旋轉的角動量，${\displaystyle I}$ 轉動慣量和原子的距离平方成正比。從運用統計力學的配分函數

${\displaystyle Z=\int _{-\infty }^{\infty }dL_{z}e^{-\beta {\frac {L_{z}^{2}}{2I}}}={\sqrt {\frac {2\pi I}{\beta }}}}$

${\displaystyle \beta ={\frac {1}{k_{\mathrm {B} }T}}}$ 是温度${\displaystyle T}$ 的倒數）可以得到古典旋轉運動對平均能量的貢献：

${\displaystyle {\frac {\langle E\rangle }{N}}=-{\frac {\partial \log Z}{\partial \beta }}={\frac {1}{2\beta }}={\frac {k_{\mathrm {B} }T}{2}}}$

${\displaystyle Z=\sum _{n=-\infty }^{\infty }e^{-\beta {\frac {n^{2}\hbar ^{2}}{2I}}}}$

${\displaystyle Z\simeq 1+e^{-\beta {\frac {n^{2}\hbar ^{2}}{2I}}}+\cdots }$

${\displaystyle {\frac {\langle E\rangle }{N}}=-{\frac {\partial \log Z}{\partial \beta }}\simeq {\frac {n^{2}\hbar ^{2}}{2I}}}$

${\displaystyle T^{*}\approx {\frac {\hbar ^{2}}{2I}}}$

## 参考文献

1. ^ Roger G Newton. From Clockwork to Crapshoot: A History of Physics. Harvard University Press. 30 June 2009. ISBN 978-0-674-04149-3.
2. ^ Griffiths, David J., Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, 2004, ISBN 0-13-111892-7