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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?

enter image description here

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    $\begingroup$ Remember that only energy differences are physically meaningful. You can adjust the overall potential function to be positive if you wish. It's totally arbitrary. But it's more convenient to have V be zero across most of the domain. $\endgroup$ – user1247 Jul 24 '19 at 21:22
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    $\begingroup$ negative energy term is somewhat avoided generally No, it isn’t. The energy of a hydrogen atom is negative, relative to the energy of a proton and an electron. The energy of the Earth-Sun system is negative, relative to the energy of the Earth and Sun separately. $\endgroup$ – G. Smith Jul 24 '19 at 21:23
  • $\begingroup$ The drawing shown is not adequate . It suggests that the wave function has the same amplitude outside the well as inside. It appears to be a mixup between an energy level diagram and wave function. $\endgroup$ – my2cts Jul 24 '19 at 22:46
  • $\begingroup$ @my2cts I haven't sketched the wave function in the drawing, just the potential and the energy $\endgroup$ – Harshdeep Singh Jul 25 '19 at 2:45
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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.

enter image description here
(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).

atomic energy levels
(image taken from this question)

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  • $\begingroup$ Yes. But at places where potential is equal to zero, then how can we explain negative energies. $\endgroup$ – Harshdeep Singh Jul 24 '19 at 21:55
  • $\begingroup$ These places are classically forbidden, but in quantum mechanics the wave function does not have to vanish there. In this case, it will decrease exponentially. If the wave function could not penetrate into classically forbidden regions, there would be no quantum tunneling. $\endgroup$ – G. Smith Jul 24 '19 at 22:04
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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.

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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.

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