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Analogy with Classical Mechanics
In classical mechanics, the total time derivative of a physical quantity $$A$$ is given as :

\frac{dA}{dt} = \frac{\partial A}{\partial t} + \{A, H\} $$

This bears striking resemblance to the Ehrenfest Theorem. It implies that a physical quantity $$A$$ is conserved if its Poisson Bracket with the Hamiltonian is zero and it does not depend on time explicitly. This condition in classical mechanics is very similar to the condition in quantum mechanics for the conservation of an observable (as implied by Ehrenfest Theorem:Poisson bracket is replaced by commutator)