= First-order selection rule for a linearly driven oscillator
{title2=$\langle n|x|0\rangle=\sqrt{\hbar/(2m\omega)}\,\delta_{n1}$}
The ladder identity $x=\sqrt{\hbar/(2m\omega)}(a+a^\dagger)$ proves the matrix element in the title. In <time-dependent perturbation theory>, a linear force excites only $n=1$ directly from $n=0$. <Parity> alone rules out even excited states; the ladder identity also rules out odd $n>1$. Higher-order evolution can reach other levels, so the driven oscillator is not an exact <two-level system>.
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