= Solution
Complete the square in the <potential energy>. With $x_F=F/(M\omega^2)$,
$$
H=\frac{p^2}{2M}+\frac{M\omega^2}{2}(x-x_F)^2-\frac{F^2}{2M\omega^2}.
$$
The <quantum harmonic oscillator> is translated without changing its level spacing. If $\varphi_n(x)$ is its normalized unforced <energy eigenstate>, then
$$
\boxed{\psi_n^{(F)}(x)=\varphi_n(x-x_F),\qquad
E_n^{(F)}=\hbar\omega\left(n+\frac12\right)-\frac{F^2}{2M\omega^2},\quad n=0,1,2,\ldots.}
$$
Equivalently, $|n;F\rangle=e^{-ix_Fp/\hbar}|n\rangle$. The <ground state> <position> is centered at $x_F$, and its <position uncertainty> remains $\sqrt{\hbar/(2M\omega)}$.
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