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I am confused, hydrogen just has one electron in the 1s orbit. but why can we plot all kind of orbitals (higher energy eigenstates for that atom)? My assumption: So physically spoken these orbits around the hydrogen nucleus exist and they go to infinity, but they are just empty?

Additional question: When we want to plot orbitals of other atoms e.g. silicon, we need to use numerical methods. So instead of doing complex numerical methods, do we use simply the hydrogen solutions as approximation then?

Is there a big difference between let's say the shape of a numerically solved silicon p-orbital and an analytically solved hydrogen p-orbital?

http://www.webelements.com/shop/shopimages/products/extras/POS0007-A2-orbitron-2010-800.jpg

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Your assumption is correct. The higher energy orbitals exist for the hydrogen atom, but with only one electron, the ground state is occupied when the system is on its lowest energy state. Excitations to higher levels can be obtained with the absortion of photons by the atom.

For the second question, the approximation of hydrogenic orbitals is not very good unless the atom is an hydrong-like atom. So, the exact shapes of the real orbitals and their energies are not so close to the hydrogen atom, in general. The study of this and the methods used is a field called electronic structure.

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If one heat up hydrogen the electrons didn't leave the atom (or the H2 molecule) at once. The electrons are able to store energy on a lot of discrete levels. These levels depend on the nucleus and its energy level. For hydrogen these levels are well studied and there are solvable equations. If the nucleus is composed from a lot of protons and neutrons it was not found an equation without iteration, that considered the loss of energy from the excited electrons to the nucleus. And perhaps such an equation could not exist, but about this I'm not sure.

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