I was wondering about the following:
If you have the time-dependent Schrödinger equation such that $i \hbar \partial_t \psi(x,t) = - \frac{\hbar^2}{2m} \partial_x^2\psi(x,t) + V(x,t) \psi(x,t),$ $$i \hbar \frac{\partial\psi(x,t)}{\partial t} = - \frac{\hbar^2}{2m} \frac{\partial^2\psi(x,t)}{\partial x^2} + V(x,t) \psi(x,t),$$
where also the potential is also time dependent. What is the general strategy to solve this one? Separation of Variables or are there better techniques available?-Especially Especially if $V(x,t) = V_1(t)V_2(x)$. For example if you know the solution to $E_n = - \frac{\hbar^2}{2m} \partial_x^2\psi(x) + V_2(x) \psi(x)$, does $$E_n = - \frac{\hbar^2}{2m} \frac{\partial^2\psi(x,t)}{\partial x^2} + V_2(x) \psi(x),$$ does this help to find the general solution?
