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CAF123

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1) I know that the total energy and momentum operators don't commute, while the kinetic energy and momentum operators do. Why is this the case? (explanation rather than mathematically).

2) One form of the HUP says that we can't measure position and momentum of a particle simultaneously and when I evaluate the commuator , it gives a non zero operator. The other form of the HUP says that ## ΔEΔt ≥\frac{\hbar}{2}.##Is there a way to evaluate the commutator here - to similarly show that a non zero commutator between time and energy (if it exists) is in agreement with the HUP? (I.e do we define a time operator)?

Many thanks.