Past exam of the theoretical physics II course of the University of Cambridge / 2012 / Paper 1 / 3 / a / Solution

The canonical commutation relation $[x^{\prime },p^{\prime }]=i\hslash I$ and hamiltonian $H^{\prime }=p^{\prime 2}/(2m)+m{\omega }^2{x}^{\prime 2}/2$ giveSubstitution in the Heisenberg equation of motion yields ${\dot{x}}^{\prime }=p^{\prime }/m$ and ${\dot{p}}^{\prime }=-m{\omega }^2{x}^{\prime }$. Differentiating once more gives the harmonic-oscillator equation for each operator. Its initial-value solution isDifferentiating these expressions at zero verifies both prescribed initial derivatives. They also preserve $[x^{\prime }(t),p^{\prime }(t)]=i\hslash I$.