Past exam of the mathematics course of the University of Cambridge / 2001 / ib / Paper 2 / 9F / Solution

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.