I Uncertainty principle##\Delta H\Delta Q##, where Q is time-independent

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Why is ##\Delta H\Delta Q \ge |\frac{ħ}{2}##d<Q>/dt##|##
Let Q be a time-independent operator.
##[H,Q] = iħ[\frac{d}{dt},Q]##
Since Q is time-independent, ##[H,Q]=0##

And from the uncertainty principle :
##\Delta H\Delta Q \ge |<\Psi|\frac{1}{2i}[H,Q]|\Psi>|##
From ##[H,Q] = 0##, I concluded that ##\Delta H\Delta Q \ge 0##

But by evaluating d<Q>/dt, it can be found that ##\Delta H\Delta Q \ge |\frac{ħ}{2}##d<Q>/dt##|##

I know that the right answer is the latter, but I just want to know why ##\Delta H\Delta Q \ge 0## is wrong.
 
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