What equation accurately describes the behavior of the electron in a hydrogen atom?

That’s kind of an abstract question

Hydrogen usually comes in 2 different forms - elemental hydrogen and hydronium ions

An equation with a hydronium ion would be the dissociation of an Acid in water

Ex, HCl —> H+ + Cl-

Hydrogen’s electron is now orbiting around Chlorine, thus making it simply a Neutron and Proton.

In elemental form, hydrogen is diatomic it can only be found by itself if bonded to another H atom

Ex (unbalanced), H2 + O2 —> H2O

Hope this helps