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The first thing the book goes over (This is intro to QM by Griffiths) is a potential that has the form -A*deltafunction. Okay, that's just something he plucked for simplicity.

But then if the potential is lower than the particle's energy at that point, why does the math work out so that there's a

*higher*chance of finding it there? When we did problems with a modified infinite square well, where the bottom was V = 0 up to some point, then V = X afterwards, there was a higher probability of finding the particle where there is more potential energy (V = X), just like there would be classically.

I don't understand why the particle would "hover" at that point instead of leaving even faster.

The other thing is that this relates to scattering, right? I get it when you flip A and just get V = A*deltafunction, so that V is infinite, you have a barrier, and then you get scattering proper, i.e. the particle either hits the wall and bounces back or goes through.

How come the same thing applies when the well goes below the particles energy like in the first case? I would have thought it would just zoom past the potential well, since there's nothing stopping it.