Past exam of the chemistry A course of the University of Cambridge / 2024 / Paper 1 / A2 / e

Write $\hat{N}={\hat{a}}^{†}\hat{a}$. The creation and annihilation operators obeyConsequently,The resulting function is not zero: its squared norm isThus ${\hat{a}}^{†}{\Psi }_0$ is the normalized first excited state, up to phase. More generally the creation operator raises the energy by one quantum $\hslash \omega$: ${\hat{a}}^{†}|n⟩=\sqrt{n+1}|n+1⟩$. The factor is required because applying it to a higher state does not preserve normalization.