Why does negative energy imply that a system is bounded?

Question

I wanted to know why "negative energy" of a two particle system implies that it is bounded. That is what happens in the case of a hydrogen atom; my textbooks say so, but they do not give any reason for that and simply state it. I tried looking on Wikipedia but it offers the "principle of minimum potential energy" as a reason. Is there any other reason for this?

Answer 1

The notion of negative energies is more like a convention, since potential energy is defined up to an arbitrary constant. In the case of the Coulomb potential it is assumed as a convention that the potential generated by a charge is zero at infinity, where there would be no interaction with other charges. Energy levels which are then lower than this one, I.e. zero in this convention, represent energetically more favourable configurations and therefore lead to bound states. In many physical situation this argument is enough to reach this conclusion, but technically there are a few other conditions that need to be checked in order to be sure that one actually has a bound state. Cosmically one shot check that the sub manifold of phase space given by the constraint $H=E<0$, where $H$is the Hamiltonian, is bounded. In a quantum theory one needs to check that the energy level is a discrete eigenvalue of the Hamiltonian.