= Solution
Inverting the definitions of the <creation operator> and <annihilation operator> gives $x=\sqrt{\hbar/(2m\omega)}(a+a^\dagger)$. Their actions and the <orthogonality> of the oscillator <energy eigenstates> give
$$
\langle n_2|x|n_1\rangle=\sqrt{\frac{\hbar}{2m\omega}}\left[\sqrt{n_1}\,\delta_{n_2,n_1-1}+\sqrt{n_1+1}\,\delta_{n_2,n_1+1}\right].
$$
Thus the <electric dipole transition> <selection rule> is
$$
\boxed{n_2=n_1\pm1,}
$$
with the lower alternative absent for $n_1=0$. The field exchanges one oscillator quantum in first-order dipole coupling. For a long weak monochromatic drive, its <frequency> must also be resonant with the level separation, here $\omega$, to give a substantial transition <probability>. Higher-order multiphoton processes are beyond the stated single-position-<matrix element> approximation.
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