Past exam of the theoretical physics II course of the University of Cambridge / 2018 / 4 / a

Using $[x,p]=i\hslash$, the Heisenberg picture equations for the quantum harmonic oscillator areSolving these coupled linear equations givesThe oscillator annihilation operator and creation operator areThey obey $[a,a^{†}]=1$ and $H_0=\hslash \omega (a^{†}a+1/2)$. Substitution of the position and momentum solutions, or the corresponding commutators with $H_0$, yields