AI摘要：Summary: This text discusses the energy of diatomic molecules, including total energy, translational, rotational, and vibrational energy, as well as quantum numbers for rotation and vibration.

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已知：

质心$\vec{r_c}=\frac{1}{2}\left( \vec{r_1}+\vec{r_2} \right)$$\vec{v_c}=\frac{1}{2}\left( \vec{v_1}+\vec{v_2} \right)$
O1$\vec{r_1}$$\vec{v_1}$
O2$\vec{r_2}$$\vec{v_2}$

总能量：

$$ E_{total}=\frac{1}{2}m(\vec{v_1}^2+\vec{v_2}^2) $$

平动能：

$$ E_{tra}=\frac{1}{2} \cdot 2m \cdot \vec{v_c}^2 $$

转动能：

角动量：

$$ \vec{L}=m \left[ \left( \vec{r_1}-\vec{r_c} \right) \times \left( \vec{v_1}-\vec{r_c} \right) + \left( \vec{r_2}-\vec{r_c} \right) \times \left( \vec{v_2}-\vec{r_c} \right) \right] $$

角速度：

$$ \vec{\omega} = \frac{\left( \vec{r_1}-\vec{r_c} \right) \times \left(\vec{v_1}-\vec{r_c} \right) }{\left( \vec{r_1}-\vec{r_c} \right)^2} $$

$$ E_{rot}=\frac{1}{2} \vec{\omega} \cdot \vec{L} $$

振动能

$$ E_{vib}=E_{total}-E_{tra}-E_{rot} $$

转动量子数

$$ E_{rot}=\frac{h^2}{8 \pi^2 I}l(l+1) \Rightarrow l $$

振动量子数

把分子近似考虑为谐振子：

$$ E_{vib}=h \nu \left(v+\frac{1}{2} \right) $$

零点能？？？？？