Two-Body Problem: Bound States

The simplest quantum systems consist of two interacting particles: the hydrogen atom, made up of a proton and an electron, interacting via the electrostatic force, on the atomic scale; and the deuteron nucleus, made up of a proton and a neutron, interacting via the nuclear force, on the nuclear scale. The interactions responsible for the stability of these two systems have similarities and differences to highlight for a better understanding of their properties. In the non-relativistic limit, the two interactions are instantaneous, time-independent and invariant under space translations and rotations, hence \(V(\vec {r}_1,t_1,\vec {r}_2,t_2) = V( |\vec {r}_{1}- \vec {r}_{2} |)\) , the positions of the two particles being \(\vec {r}_1\) and \(\vec {r}_2\) . Since the two interactions are attractive, bound states are expected both for the hydrogen atom and for the deuteron nucleus. In fact, both of them exist in nature. However, the size and energy scales of the two systems are quite different: the hydrogen is \(10^{5}\) times greater than the deuteron and its binding energy is \(10^{-6}\) times smaller. These significant differences must be traced back to the different nature of the two forces: weak and long-range the former, strong and short-range the latter. Although these properties have the same significance in both classical and quantum mechanics, their implications are quite different. Just to mention one, while the binding energy is negative in both cases, the classical values are continuous, whereas the quantum values are quantized. The same holds true for the angular momentum.