量子力学中的角动量相加、自旋与矢量模型
1. 角动量相加与能级分析
在量子体系里，角动量相加是一个关键概念。以特定的角动量态 $|1 0\rangle$ 为例，对其进行相关算符操作后：
$$
\begin{align}
\frac{2}{\kappa}\hat{H}F |1 0\rangle&=\frac{1}{2} \left( \hat{\sigma}{1+} \hat{\sigma}{2-} + \hat{\sigma}{1-} \hat{\sigma}{2+} \right) \frac{1}{\sqrt{2}} \left( \left| -\frac{1}{2}, \frac{1}{2} \right\rangle + \left| \frac{1}{2}, -\frac{1}{2} \right\rangle \right) + \hat{\sigma}{1z} \hat{\sigma}_{2z} \frac{1}{\sqrt{2}} \left( \left| -\frac{1}{2}, \frac{1}{2} \right\rangle + \left| \frac{1}{2}, -\frac{1}{2} \right\rangle \right)\
&=\frac{1}{2} \frac{1}{\sqrt{2}} \left( 4 \left| \frac{1}{2}, -\frac{1}{2} \right\rangle + 4 \left| -\frac{1}{2}, \frac{1}{2} \right\rangle \right) + \frac{
