Solution
= Solution
Since $[X,H]=iP$ and $\langle P\rangle=\sqrt{2\omega}\Im\alpha$, the bound is
$\Delta X\Delta H\ge\sqrt{\omega/2}|\Im\alpha|$. The left side is $\sqrt{\omega/2}|\alpha|$, so equality holds exactly when $\Re\alpha=0$.
= Solution
Since $[X,H]=iP$ and $\langle P\rangle=\sqrt{2\omega}\Im\alpha$, the bound is
$\Delta X\Delta H\ge\sqrt{\omega/2}|\Im\alpha|$. The left side is $\sqrt{\omega/2}|\alpha|$, so equality holds exactly when $\Re\alpha=0$.