There are several questions asking by the meaning of Time-Energy uncertainly relation and also by its derivation.
My problem is about the second. In all documents I have seen, the same is supposed alwas: Let A an observable which does not depend explictly on t. (from John Baez's web page to Griffiths (p.113) and also in the original paper, formula 4). Then
ΔTAΔH≥ℏ2.
My question is: What happens if we allow ⟨∂A/∂t⟩≠0?.
EDIT.
@ZoltanZimboras demands me to write the definition of ΔTA. The truth is I'm not sure how to answer him. Usually, ΔTA is defined as
ΔTA=⟨A⟩d⟨A⟩dt.
If we set the same definition, then the Heisenberg's inequality comes
ΔTAΔH≥ℏ2|1−⟨∂A/∂t⟩d⟨A⟩/dt|.
The advantage of this definition is physics understand very well the meaning of the quantity ΔTA, but then the above equation comes more difficult. On the other hand, if we set
ΔTA=⟨A⟩d⟨A⟩dt−⟨∂A∂t⟩,
Then the Heisenberg's principle takes its usual form, but we have to reinterpret the quantity ΔTA.
Answer
Following the argument in Introduction to Quantum Mechanics by Griffiths, ΔHΔQ=12i⟨[H,Q]⟩=ℏ2|d⟨Q⟩dt−⟨∂Q∂t⟩|
Then, defining ΔE=ΔH, |d⟨Q⟩dt−⟨∂Q∂t⟩|Δt≡ΔQ
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