计谱原理（一）|作业10

10-1

假设 \(\mu\) 为常数，关于不同取值的S作 \(\alpha-S\) 图

10-2

\[ \begin{align*} P|m\rangle&=-i\sqrt\frac{\omega}{2}(a-a^\dagger)|m\rangle\\ &=-i\sqrt{\omega/2}(\sqrt m|m-1\rangle-\sqrt{m+1}|m+1\rangle)\\ &=-i\sqrt{ \frac{\omega}{2} }(\sqrt{ m }\ket{m-1} -\sqrt{ m+1 }\ket{m+1} ) \end{align*} \] \[ \begin{align*} [\hat H,Q]|m\rangle&=\sqrt{ \frac{1}{w} }([\hat{H},a^{\dagger}]\ket{m}+[\hat{H},a]\ket{m}) \\ &=\sqrt{ \frac{1}{2\omega} }(\omega a^{\dagger}\ket{m} -\omega a\ket{m} )\\ &=\sqrt{ \frac{2}{\omega} }(a^{\dagger}-a)\ket{m} \\ \end{align*} \] 得证： \[ \begin{align*} \hat{P}_{nm}=(i[\mathcal{H},\hat{Q}])_{nm} \end{align*} \]