When an electron falls to a proton

When a stone falls to the earth, it might bump several times, and rest on the surface of the earth. When an electron falls to a proton, will it rest on the surface of the proton?  It seems not. Compared with gravitational force, electric force is much stronger. The high energy electron will circle around the proton, much like the earth circle around the sun.

According to Maxwell’s theory of electromagnetism, a circling electron shall emit electromagnetic wave, lose energy and collapse into the proton. This is much like a satellite returning to the earth’s atmosphere, losing energy from friction and in the end, collapsing into the earth. However this is not what happens to a hydrogen atom, which is a combination of proton and an electron. A hydrogen atom is very stable. This means Maxwell’s theory is not applicable when the scale is very small. People invented quantum theory to explain phenomena in microscopic world.

Now suppose an electron is attached to a large ball and move around a big circle. The moving electron must emit electromagnetic wave, according to Maxwell’s theory. Imagine the moving circle becomes smaller and smaller. In the end, it becomes the size of a hydrogen atom. Somewhere, the applicable law changes from Maxwell’s theory to quantum theory. How it happens? This troubles me a lot.