Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / Section B / B7 / b / Solution

Orthogonality gives $⟨{\hat{a}}_{+}^2⟩=⟨{\hat{a}}_{-}^2⟩=0$, while the ladder coefficients give $⟨{\hat{a}}_{+}{\hat{a}}_{-}⟩=n$ and $⟨{\hat{a}}_{-}{\hat{a}}_{+}⟩=n+1$. Squaring the expressions for $\hat{x}$ and $\hat{p}$ therefore givesThusTheir equality is the harmonic-potential case of the quantum virial theorem, and their sum is the oscillator energy $E_n=\hslash \omega (n+1/2)$.