Past exam of the physics A course of the University of Cambridge / 2017 / Paper 1 / Section A / 2 / Solution

Use the canonical commutation relation $[\hat{x},\hat{p}]=i\hslash I$ on a common invariant domain of the operator products. Write $A=\hat{p}\hat{x}$, so $\hat{x}\hat{p}=A+i\hslash I$. Associativity then giveswhereasThereforeAs a direct position-representation check, the momentum operator is $\hat{p}=-i\hslash d/dx$. Acting on a smooth function $f$ givesThis is the equality of two quadratic position-momentum orderings; it does not make arbitrary rearrangements of noncommuting linear operators interchangeable.