= Solution
Orthogonality gives $\langle\hat a_+^2\rangle=\langle\hat a_-^2\rangle=0$, while the ladder coefficients give $\langle\hat a_+\hat a_-\rangle=n$ and $\langle\hat a_-\hat a_+\rangle=n+1$. Squaring the expressions for $\hat x$ and $\hat p$ therefore gives
$$
\langle x^2\rangle=\frac\hbar{m\omega}(n+\tfrac12),\qquad\langle p^2\rangle=m\hbar\omega(n+\tfrac12).
$$
Thus
$$
\boxed{\langle V\rangle=\frac12m\omega^2\langle x^2\rangle=\frac12\hbar\omega(n+\tfrac12),\qquad\langle K\rangle=\frac{\langle p^2\rangle}{2m}=\frac12\hbar\omega(n+\tfrac12).}
$$
Their equality is the harmonic-potential case of the <quantum virial theorem>, and their sum is the oscillator energy $E_n=\hbar\omega(n+1/2)$.
Back to article page
