My confusion arose initially from the definition of binding energy being the lowest energy state (n=1) in the hydrogen atom. This, I assume, is simply because hydrogen only has one atom, and electrons don't exist in higher energy states stably. Then my next question was, why not? If these higher energy states exist, why can't electrons maintain those orbits? The above question seems to answer that question, but then I don't understand why bigger atoms CAN hold these higher energy eigenstates. Is is just because the lower electrons "prevent" the higher ones from decaying? I can see where the exclusion principle comes from, but I can't see how it would prevent decay, only how it would prevent more than two electrons from inhabiting the same orbit.
Here is a representation of the hydrogen atom energy levels.
It displays the availabe solutions of the Schrodiner equation for an atom composed of a proton in the nucleus and an electron existing in their mutual potential.
Systems stay in the minimum energy state, and for the single electron of hydrogen the minimum energy state is the n=1 state and the value of that energy is -13.6 eV.
It can happen that a photon of 10.2 eV scatters the electron to the n=2 state. This will be an unstable solution because there exists an empty lower energy state and the electron will radiate back to n=1. The same is true for the higher n states to which the electron can get scattered, and then can cascade down to the ground state. The radiation from these excitations is a spectrum measurable in the lab and it is how we know we have the correct quantum mechanical model of the hydrogen atom.
A second electron has no meaning in this solution of the hydrogen atom, which has zero charge as an atom. A second electron will not be attracted because there is no potential atom+ second electron .
Each atom has as many electrons as there are protons in the nucleus and there will be solutions that will give the energy levels those electrons can occupy.
For Z=2 and higher the Pauli exclusion principle does not allow two electrons in the same energy state. It is worth looking up Helium to get an idea of the complexity of the energy levels of multi electron atoms and the role of the Pep.