Yesterday, I asked a question about solving the 1D Schrodinger equation in a time varying potential $f(t)x$ using a method solely in configuration space. Although this approach does not directly answer my original question — hence my asking it separately, I would be greatly appreciative if someone could verify my solution.
$$ \left(\frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + f(t)x\right) \Psi(x,t) = i\hbar\frac{\partial}{\partial t} \Psi(x,t) $$
Then, I used the cosmetic exchanges $x\mapsto i\hbar \frac{\partial}{\partial p}$ and $-i\hbar \frac{\partial}{\partial x} \mapsto p$ to produce the equation
$$ \left(\frac{p^2}{2m} + i\hbar f(t) \frac{\partial}{\partial p}\right) \Psi(x,t) = i\hbar\frac{\partial}{\partial t} \Psi(x,t) $$
Then, changing coordinates corresponds to a time-dependent boost in momentum space where $F(t)$ is antiderivative of $f(t)$
$$ \begin{align} p' &= p + F(t) \\ t' &= t \end{align} $$
producing the relationships
$$ \begin{align} \frac{\partial}{\partial p} &= \frac{\partial}{\partial p'}\\ \frac{\partial}{\partial t} &= \frac{\partial}{\partial t'} + f(t) \frac{\partial}{\partial p} \end{align} $$
and yielding the final equation
$$ -\frac{(p' - F(t))^2}{2m} \hat\Psi(p', t') = i\hbar \frac{\partial}{\partial t'} \hat\Psi(p', t') $$
The solution then becomes
$$ \hat \Psi(p', t') = C \exp\left(-\frac{i}{\hbar} \int_0^{t'} \frac{(p' - F(\tau))^2}{2m} d\tau\right) $$
or removing the primes by relabeling
$$\hat \Psi(p, t) = C \exp\left(-\frac{i}{\hbar} \int_0^{t} \frac{(p - F(\tau))^2}{2m} d\tau\right)$$
