Past exam of the physics A course of the University of Cambridge / 2018 / Paper 1 / Section B / B9 / b / Solution

Expand the proposed product rule without interchanging any operator factors:Applying this commutator identity with $A=x$ and $B=C=p_x$ givesFor a differentiable multiplication potential, apply the other commutator to a test function:ThereforeThese operator equalities are understood on a common domain where the indicated products and derivatives are defined, with the boundary conditions appropriate to the state.