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量子力学中的角动量相加与矢量模型

1. 角动量相加与能级分析

在量子力学里，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，通过一系列运算：
[
\begin{align}
\frac{\kappa}{2}\hat{H}F |1 0\rangle&=\frac{1}{2}(\hat{\sigma}{1 +}\hat{\sigma}{2 -}+\hat{\sigma}{1 -}\hat{\sigma}{2 +})\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{1}{\sqrt{2}}\left((-1)\left|-\frac{1}{2},\frac{1}{2}\right\rang