Different statements of the Heisenberg uncertainty principle

Question

I know that from the Heisenberg uncertainty principle, ∆x∆p=ℏ/2 . And I know that this equation can be rewritten as ∆t∆E=ℏ/2.

From QED I also know that the equation ∆t∆E=ℏ/2 claims that some energy E can exist for some short time t then disappear. Is that true that the equation ∆x∆p=ℏ/2 also claims that some momentum p can exist for some short distance x and then disappear?

I wonder if momentum can be not conserved for some short displacement as energy can do so for some short time.

Answer 1

$\Delta E \Delta t\ge \frac{\hbar}{2}$ is not a “proper” uncertainty since time is not an observable in QM: there is no such thing as the commutator $[\hat H,\hat t]=i\hbar\hat{1}$. Indeed there are several examples where the inequality is violated. Thus any argument based on this should be scrutinized for a precise meaning of $\Delta t$.