= Solution
Write $a=a_-$ and $a^\dagger=a_+$. Inverting the <annihilation operator> and <creation operator> definitions gives
$$
x=\sqrt{\frac{\hbar}{2m\omega}}(a+a^\dagger),\qquad
p=i\sqrt{\frac{m\hbar\omega}{2}}(a^\dagger-a).
$$
The ladder operators take $|n\rangle$ to orthogonal <quantum harmonic oscillator> <energy eigenstates> $|n-1\rangle$ and $|n+1\rangle$. Thus $\langle n|a|n\rangle=\langle n|a^\dagger|n\rangle=0$, and
$$
\boxed{\langle x\rangle_n=0,\qquad\langle p\rangle_n=0.}
$$
For $n=0$, the lowering term is simply zero. The conclusion concerns an individual <energy eigenstate>; interference between different eigenstates can produce nonzero <expectation values>.
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