Time-dependent SE for 'discrete' time steps

[DougD720]

Hi guys,

I'm trying to solve the S.E. For a potential that's time dependent but the time variable is not continuous. Essentially the potential is a finite square well and it shifts over time but not continuously.

I.e. At time 0<t<t' it's a finite square well centered about some x'

At time t'<t<t" it's a finite square well centered about x"

And so on. All the t's and x's are known numbers. Also it's not exactly a finite square well, here's an example at some time:

0<t<t' => V(x) = {∞ ; x<0 , V0 ; 0<x<P or P'<x<P" , 0 ; otherwise}

Where P, P', P" and V0 are known constants.

I thought I might be able to solve each situation time-independently and then just multiply by the time in each well over the total time but I don't think I can do that. If someone could let me know how to go about tackling this problem I'd appreciate it! I've taken 2 semesters of QM but am a little rusty. Just having trouble figuring out the plan of attack here (it's not for HW just something I'm working on where I'm modeling a process as a particle in the types of potentials shown).

Thanks!

 

[DougD720]

A picture's worth a thousand words... here's the scenario if there were only two sets of times, hence the potential function only changes once: