Thanks so much! I'll get rid of that extra $x$.

My attempt is available at this link. The final answers appear similar. physics.stackexchange.com/q/746519

Thank you very much. I am just posting another question where I am asking for solution verification of my work in momentum space. When I finish it in around 5 minutes would it be too much for me to ask for your reviewal?

You have clarified so much and probably just saved me hours of troubleshooting. To summarize, I can consider the inside of a cylindrical cavity and add a linear potential $V_0 x$ that discontinuously oscillates with a period of $T$ provided that I impose boundary conditions that will force the wave function to vanish. By the way, I came upon Lewis-Riesenfeld phases while I was searching on Google earlier today. Do you have suggestions for more reading material in this area? I'm clearly just a beginner in physics and am having difficulty prioritizing which books to read first. Thanks again

@Buzz Thanks for all of your advice on both of my questions. I just edited my problem to stipulate that $x\in[0,L]$ for some finite length $L$. Assuming the wave function vanishes on $(-\infty,0)$ and $(L,\infty)$, would I then be able to normalize the wave? Are you saying that there will be no ground state because I have not adequately defined boundary conditions? What if I also added that $\psi(0)=0$ and $\psi(L)=0$ to try mimicking the infinite well -- could I then define a ground state?

Yes, thank you so much. I neglected to state that I am considering a linear potential $V(x) = V_0 x$. (I have edited my question for clarity now.) I am imagining a cylinder of negligible radius that can be approximated as a one dimensional line. I want the electric field to remain constant throughout the space over a given time.

@Qmechanic♦, Thanks for the edits. My main confusion is about Claim 3. If we define the potential to be so large that the particle could reach a relativistic speed in a small time scale, would the Sudden Approximation still be valid? Or, would their be higher order nonlinearities that would ruin the model? Could I try solving the Dirac equation instead? Thanks again.