Negative energy in bound states of a particle in a finite potential well Consider you have a particle in a finite potential well as depicted in the photo attached. Now we have three regions: 
$$V(x) =
\begin{cases}
0,  & \text{for } x<-a & (1)\\
-V_0, & \text{for }-a<x<a & (2)\\
0, & \text{for } x>a & (3)
\end{cases}$$
To describe the bound states of the system, we use $E$(Energy)$<0$ and this is somewhere I really am confused. A bit due to the fact that negative energy term is somewhat avoided generally. But the main concern is, in region (2), there is a negative potential, so we can have an overall negative energy but in regions (1) and (3), the potential is zero. What does then negative energy mean in those regions? What can we say about our physical system in regions (1) and (3) where potential is zero but energies are negative?

 A: The negative energy is no problem at all.
The wave functions $\psi(x)$ for energy $E$


*

*are oscillatory where $E > V(x)$, i.e. in region (2)

*and are decreasing exponentially where $E<V(x)$, i.e. in regions (1) and (3).
That means the particle penetrates a short distance into these regions.



(image taken from More 1D Problems)
Actually the situation in your negative potential well 
is qualitatively not very different from the hydrogen atom
(an electron in the negative Coulomb potential of the nucleus).

(image taken from this question)
A: It all depends on how you define the zero of eenrgy. In the problem at hand the zero energy has been defined as one in which the particle is at rest and away from the effects of the well. Any positive energy would mean that the particle has a kinetic energy in regions 1 and 3. A negative energy (kinetic plus potential) would imply that the particle cannot be in regions 1 and 3 because if it were there the kinetic eenrgy would have to be negative. It could only be in region 2. Quantum mechsanics however describes particles as waves and the wave does extend a certain small distance in region 1 and 3 for negative energy but not too far ( depends on the energy in the well). We say thet the negative enrgy particle has no chance of escape from the well according to your well.
A: See when you talk about the negative energy inside a well then you mean there is a positive kinetic energy present and motion of the particle is perceived physically but outside the well in a potential region it makes no sense as kinetic energy becomes negative which means velocity is imaginary.
