Past exam of the physics A course of the University of Cambridge / 2019 / Paper 1 / Section B / B9 / a

Let $c=\sqrt{m\omega /(2\hslash )}$ and $d=1/\sqrt{2m\hslash \omega }$, so $a=cx+idp$ and $a^{†}=cx-idp$. The canonical commutation relation $[x,p]=i\hslash$ givesHence $\boxed{H=\hslash \omega (a^{†}a+1/2)}$. Similarly,Using the product rule for a commutator, $[AB,C]=A[B,C]+[A,C]B$, givesThus $a^{†}a$ is the number operator, while $a$ and $a^{†}$ are the annihilation operator and creation operator.