I took the course of quantum mechanics a while ago. I do not quite remember all the detail on how to derive the wave function for hydrogen but I still remember the general picture. I think the text always start the discussion with hydrogen because this atom contains 1 electron only. It is a simple 2-body model so we could solve the problem in the frame of center of mass. In other situation for atom has more than 1 electron but the outermost shell contains only 1 electron, we could reduce it as a hydrogen-like ion with 1 electron. Since the nucleus or the hydrogen-like ion is much heavier than the electron, we consider it a rest ion so the wave function we derive is mostly describing the motion of the electron? But in the book, they always mention the wave function of an atom. Is it the same thing to say the wave function for the atom or the electron (in the atom)?
As I still remember, if the wave function is used to describe the atom, the modulus square of the wave function is interpreted as the probability to find the atom in space; but if it is for electron, we should say it is interpreted as the probability of finding the electron in space. It is confusing.