Why do non-hydrogen atomic orbitals have the same degeneracy structure as hydrogen orbitals? The solutions of the Schrödinger equation for hydrogen are the "electronic orbitals", shown in this picture:
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They have the following degeneracy structure:
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It is often said that atoms in other elements simply have more electrons "filling up" the orbitals in increasing order of energy, as required by the Pauli exclusion principle.  This is somewhat confirmed via xray spectroscopy:
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But why would non-hydrogen atoms have the same orbital degeneracy structure as hydrogen, if the actual Hamiltonian is different from one atom to another?
 A: Although John Rennie is right to remind us that when talking about orbitals for a multi-electron atom, it is only an approximation, I would like to point out that the source of the exact degeneracy of the (many-electron) atomic states, it is the isotropy of the atom in the absence of external polarizing fields. This makes the (total) angular momentum J and its projection m, exact quantum numbers, applicable to classification of atomic terms.
A: Polyelectronic atoms don't have atomic orbitals - though they are a very useful approximation for describing the properties of polyelectronic atoms.
The 1s, 2s, etc orbitals are solutions for a central potential, and for any smooth monotonic central potential we'll get solutions of this form. The radial part of the orbitals will be different for different central potentials but the angular part is dictated by the spherical symmetry and is the same for all (smooth monotonic) central potentials.
But for any atom with more than two electrons the potential is not centrally symmetric because it includes terms like $1/r_{ij}$ for the interaction between the $i$th and $j$th electrons. This means the hydrogenic orbitals are not solutions.
However because the electrons are delocalised over the whole atom the potential is approximately central. By this I mean that if we take a time average potential the $1/r_{ij}$ terms tend to average out to a central force. In that case we do get hydrogenic type orbital as solutions, but we have to bear in mind that they are approximate solutions. They should be regarded as a useful way of building up the complete electronic structure but they are not themselves real. For example in a lithium atom there is not actually two electrons in a 1s orbital and one in a 2s orbital. There is a single three electron wavefunction that can be approximately decomposed into hydrogenic 1s and 2s orbitals for convenience.
Having said this, for most purposes the hydrogenic orbitals are very good approximations. For example when we are considering atomic spectra we usually describe them as transitions between the hydrogenic orbitals and this works pretty well.
