Physical interpretation of negative total energy

Question

I am going through Griffith's text on Quantum Mechanics, in which he states $$\begin{cases} E < V(-\infty) \text{ and }V(\infty) \implies \text{bounded state}\\ E > V(-\infty) \text{ and }V(\infty) \implies \text{scattering state} \end{cases}$$ where $E$ is the total energy of the particle. This makes sense. However, he notes that most potentials tend to 0 as you approach infinity, and so the above simplifies to $$\begin{cases} E < 0 \implies \text{bounded state}\\ E > 0 \implies \text{scattering state} \end{cases}$$ This is where I am having some trouble with the physical interpretation. How can one have negative total energy? Can someone provide some intuition and maybe an example?

Answer 1

The total energy can be negative because, away from infinity, the potential can be negative. If the total energy is negative, then regions with zero potential are unaccessible for the particle, since it can't have negative kinetic energy. That's why Griffiths calls that situation "bounded": the cases with negative energy are constrained to the region of space in which the potential is negative enough for the total energy to be whatever value it is while the kinetic energy is still non-negative.

Particles with positive total energy, on the other hand, won't have those restrictions at infinity (for potentials vanishing at infinity). Hence, they can keep going on forever, without being restricted to a finite region. That's why Griffiths calls this case scattering.