Driven Quantum Harmonic Oscillator Consider the Hamiltonian
$$
H = \frac{p^2}{2} + \frac{ x^2}{2} - F(t) x.
$$
This is essentially a time dependent shifted harmonic oscillator, which can be represented as
$$
H' = \frac{p^2}{2} + \frac{1}{2} (x - F(t))^2- \frac{F^2(t)}{2}.
$$
Will the ground state eigenfunction for this driven oscillator be just $\psi(x,t)  = \exp\left(-(x- F(t))^{2}\right)$, a function of parameter $t$, such that at instant $t_{0}$ this acts as the instantaneous ground state of the Hamiltonian $H(t_{0})$ with energy eigenvalue (using $\hbar = 1$)
$$
E(t_{0}) = \frac{1}{2} - \frac{F^2(t_{0})}{2}?
$$
Given a coherent state of the driven oscillator at some instant of time $t_{0}$ of the Hamiltonian $H(t_{0})$, will it remain a coherent state of $H(t_{0})$ under time evolution to some time $t_{1} > t_{0}$ under $H(t)$?
PS: This is not a homework question, I am simply curious about this system.
 A: +1 for non-dimensionalizing $\hbar=1$ and the rest superfluous parameters.
Define $K\equiv \frac{1 }{2}(p^2 + x^2)$. Then you see that the canonical transformation $x\mapsto x-F(t), \qquad p\mapsto p$
preserves the commutation relations; and hence shifts the spectrum of K, to that of H,
$$
e^{-ipF} K e^{ipF}=H+ F^2/2,
$$
so the spectrum of H is $n+(1 -F^2)/2$.
It is then evident, as you surmised (but for an exponent factor of 1/2), that the unnormalized "ground" state of
H is a plain Gaussian,
$$
\tfrac{1}{2} ( -\partial^2+x^2 -2Fx) e^{-(x-F)^2/2}= {(1-F^2)\over 2} e^{-(x-F)^2/2},
$$
etc, for the excited states, mutatis mutandis....
When you diagonalize to Dirac creation and annihilation operators, $\sqrt{2} a(t)\equiv x-F(t)+ip$, you hardly see substantial traces of the shift, beyond the ferocious dependence of all former "constants" on time, $H(t)=a^\dagger (t) a(t) + (1-F(t)^2)/2$.
You may then address your problem, the TDSE,
$$
\Bigl (i\partial_t - n-(1-F(t)^2)/2\Bigr )~~ \psi_n(x,t)=0,
$$
... it is not trivial to solve.$^\natural$

$^\natural$For instance, it is evidently impossible to solve for facile  Ansätze such as $e^{-(x-F(t))^2+ G(t)}$, etc. You might have to devise very special Fs, but perhaps I have missed your desideratum. 
A: Seek a solution $U(t)$  to the driven harmonic oscillator evolution equation
$$
i\partial_t U = \{\Omega a^\dagger a +f(t) a^\dagger +f^*(t) a\}U
$$
in the form
$$
U= e^{i\theta} e^{-{\textstyle \frac 12}|z|^2}e^{za^\dagger} e^{-z^*a} e^{-i\omega a^\dagger a},\nonumber\\
U^{-1} = e^{-i\theta} e^{+{\textstyle \frac 12}|z|^2}e^{i\omega a^\dagger a} e^{z^*a} e^{-za^\dagger}, \nonumber
$$
Now, using
$$
e^{-i\varphi a^\dagger a}\left[\matrix{a\cr a^\dagger}\right] e^{i\varphi a^\dagger a}=  \left[\matrix{ae^{+i\varphi}\cr a^\dagger  e^{-i\varphi}}\right]\nonumber\\
e^{-\lambda a^\dagger} \,a\, e^{+\lambda a^\dagger}= a+\lambda\nonumber\\
 e^{+\lambda^*  a} a^\dagger e^{-\lambda^* a}= a^\dagger+\lambda^*\nonumber
$$
we find
$$
U^{-1}\partial_t U= i e^{i\omega a^\dagger a}\left[ \frac 12 (-\dot z z^*-  \dot z^* z) +\dot z (a^\dagger+z^*)- \dot z a^\dagger -i\dot \omega a^\dagger a+i\dot \theta\right] e^{-i\omega a^\dagger a},
$$
while
$$
U^{-1} \{\Omega(t) a^\dagger a +f(t) a^\dagger +f^*(t) a\}U= e^{i\omega a^\dagger a}[\Omega(t)(a^\dagger +z^*)(a+z)+f(t)(a^\dagger+z^*)+f^*(t)(a+z)]e^{-i\omega a^\dagger a}
$$
Thus, comparing coefficients of $a^\dagger a$, $a^\dagger$, $a$ and $1$ we read off
that
$$
\dot \omega = \Omega\nonumber\\
i\dot z=\Omega z+f\nonumber\\
-i\dot z^* =\Omega z^* +f^*\nonumber\\
\dot \theta = -{\textstyle \frac 12}(fz^*+f^*z)\nonumber.
$$
Now the displacement operator
$$
D(z)= e^{za ^\dagger - z a}= e^{-{\textstyle \frac12}|z|^2} e^{za^\dagger}e^{-z^*a} 
$$
acs on the ground state as
$$ 
|{z}\rangle=  e^{za^\dagger -z^* a}|z\rangle =e^{-{\textstyle \frac 12}|z|^2}e^{za^\dagger} |z\rangle
$$
so  under our  time evolution we have
$$
|x\rangle \mapsto e^{i\theta(t)} |z(t)\rangle  
$$
where
$$
 \theta =\int_0^t \left\{\Omega|z|^2 -\frac{i}{2} (\dot z z^*-\dot z^*z)\right\}dt .
$$
is the classical action associated with the evolution.
