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For $H=p^2/(2m)+V(x)$ and a time-independent operator $O$, differentiate its expectation value and use the Schrödinger equation and its adjoint. Moving $H$ across the inner product by integration by parts givesWork with wavefunctions in the displayed operators' domains; the vanishing at infinity removes the associated boundary terms. This is the Ehrenfest theorem derived from time evolution.

The canonical commutation relation follows directly from differentiating $x\psi$: $[p,x]\psi =-i\hslash \psi$. Thus $[p^2,x]=p[p,x]+[p,x]p=-2i\hslash p$, while $[V(x),x]=0$. It follows thatFor the momentum equation, apply both products to a test wavefunction:The kinetic term commutes with $p$, so $[H,p]=i\hslash V^{\prime }$. ThereforeThese are the classical equations from Ehrenfest theorem for position and momentum expectations; the force expectation need not equal the force evaluated at the expected position.

For a particle in the scalar potential energy $U(x)$,The canonical commutation relation $[x,p]=i\hslash$ givesThe Ehrenfest theorem with $O=x$ therefore givesSince $[p^2,p]=0$ and direct action on a wavefunction gives $[U(x),p]=i\hslash U^{\prime }(x)$,Finally $H$ has no explicit time dependence and $[H,H]=0$, so

The first two equations are the expectation-value counterparts of momentum equals mass times velocity and Newton's second law; the last is conservation of energy. They become the classical equations directly when the force varies negligibly across the wave packet, so that $⟨U^{\prime }(x)⟩\simeq U^{\prime }(⟨x⟩)$. These conclusions are summarized by the classical equations from Ehrenfest theorem.