Ehrenfest’s theorem makes expectation values obey Hamiltonian commutator equations analogous to classical motion.
For a possibly time-dependent observable ,
Differentiating and using and its adjoint gives
For , the canonical commutation relation gives
These are the expectation-value analogues of Newton's second law and conservation of energy.
The correspondence principle states that quantum predictions approach classical mechanics in the appropriate large-scale or semiclassical regime.

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