= Normalized two-state truncation of a displaced oscillator
{title2=$\langle x\rangle_{01}=x_0/(1+\eta^2)$}
Let $x_{\rm zp}=\sqrt{\hbar/(2m\omega)}$ and $\eta=x_0/(2x_{\rm zp})$. The normalized first-order truncation is $(|0\rangle+\eta|1\rangle)/\sqrt{1+\eta^2}$. Its mean displacement is $2\eta x_{\rm zp}/(1+\eta^2)$, agreeing with $x_0$ only at small $|\eta|$. The exact translated <ground state> is a <coherent state> whose expansion contains all <number states> and has mean occupation $\eta^2$. Correct linear response does not justify extrapolating a truncated state to large fields.
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