Coulomb potential

The Coulomb potential is the energy stored in the electric attraction between two charges, and in the atom it is what holds the negatively charged electron to the positively charged nucleus. Its defining feature is that it grows weaker in proportion to one over the distance: pull the electron twice as far out and the binding energy drops to half. This gentle, long-reaching 1/r shape is the well that the electron lives in.

When you place this potential into the Schrödinger equation for hydrogen, its precise mathematical form is what makes an exact solution possible and dictates the famous pattern of energy levels. The same 1/r law that Newton wrote for gravity governs electric attraction too, which is why hydrogen and the orbits of planets share a deep mathematical kinship even though one is quantum and the other is not.

It helps to remember what the potential is and is not. It is not a force you feel directly; it is a stored energy that becomes more negative as the electron settles closer to the nucleus, just as a ball has lower energy at the bottom of a valley. The electron does not crash into the proton, despite the ever-deepening well, because quantum mechanics forbids it from being pinned down to a single point — confining it that tightly would cost more energy than the well can offer.