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In the Wikipedia derivation of this theorem, there is a step which I do not follow. This uses the Heisenberg picture to get Ehrenfest's theorem. A is an operator at t=0 and A(t) is the evolved operator.

[tex]\frac{d}{dt}A(t) = \frac{i}{\hbar}e^{iHt/\hbar}(HA-AH)e^{-iHt/\hbar} +e^{iHt/\hbar}(\frac{\partial A}{\partial t})e^{-iHt/\hbar} = \frac{i}{\hbar}(HA-AH)+\frac{\partial A}{\partial t}[/tex]

The last step is said to be true because [itex]e^{iHt/\hbar}[/itex] commutes with H. But I don't see how that leads to the equality. Can someone explain?

Secondly, when the Ehrenfest theorem is applied, we get classical laws. For instance, Newton's second law with the original classical variables replaced by expectation values of the quantum operator i.e. [itex]x[/itex] becomes [itex]<x>[/itex] and so on. But in general, is it true that any classical law can be written in terms of QM through Ehrenfest's theorem (provided we can define the relevant operators)? Is Ehrenfest's theorem the thing that actually connects QM with classical mechanics?

Thank you!

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# Ehrenfest theorem

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