I'm a chemist, not a physicist, but am taking a quantum chemistry course right now and I'm having difficulty grappling with the following:
For the particle in a finite potential well, so long as no forces act on it inside the well its potential will be constant and we set it to $0$. That makes sense, what I don't understand is why the particles potential energy must be $U_0$ outside the well. I understand it obviously cant be less than $U_0$, and it makes sense that it could be exactly $U_0$, but why cant it be greater than $U_0$?
If its greater than $U_0$ wouldn't it still be found outside the walls of the box?
I've tried looking for answers but I cant find a satisfactory one, which makes me think I'm thinking about the problem incorrectly. Does it have something to do with the energy being less than the potential? Any help on rectifying this?