Why hydrogen isn't enough — the central-field idea
Put two electrons in and the problem changes character. Each electron feels the nucleus and the other electron, so the potential no longer depends on a single distance r — it couples the electrons together and the Schrödinger equation stops separating. There is no closed-form solution for helium, let alone uranium. We need a physically-motivated approximation.
The key move is the central-field approximation: pretend each electron moves independently in an averaged, spherically-symmetric potential made of the bare nucleus plus the smeared-out cloud of all the other electrons. Because that averaged potential is still central, the angular part of each orbital is again a spherical harmonic, so the labels n, \ell, m_\ell survive. What changes is the energy: the inner electrons screen the nuclear charge, so an outer electron feels a reduced effective charge, and — crucially — the energy now depends on \ell as well as n.
Pauli, shells, and electron configuration
Electrons carry a fourth quantum number — spin — with m_s = \pm\tfrac12. They are also identical fermions, so their total wavefunction must be antisymmetric under exchange. The practical consequence is the Pauli exclusion principle: no two electrons can share all four quantum numbers (n,\ell,m_\ell,m_s). Each orbital holds at most two electrons, of opposite spin.
Filling orbitals from the bottom up, respecting Pauli, gives the ground-state electron configuration — for example carbon is 1s^2\,2s^2\,2p^2. Because a subshell \ell holds 2(2\ell+1) electrons, the shells close at 2, 10, 18, … electrons, and those closures are the noble gases. The periodic table's rows and blocks are simply Pauli plus screening made visible.
From configuration to terms: LS coupling
A configuration like carbon's 2p^2 is not one state but many: the two p-electrons can arrange their orbital and spin angular momenta in several ways, and the residual electron–electron repulsion (the part the central field averaged away) splits these into distinct energy terms. For light atoms the residual repulsion beats the spin–orbit force, so we first add up the orbital angular momenta into a total \mathbf{L}=\sum_i \boldsymbol{\ell}_i and the spins into a total \mathbf{S}=\sum_i \mathbf{s}_i. This is LS (Russell–Saunders) coupling, and it uses the addition of angular momenta you met in quantum mechanics.
A term is then labelled by a term symbol, which packages three numbers. Finally the weaker spin–orbit interaction couples \mathbf{L} and \mathbf{S} into a grand total \mathbf{J}=\mathbf{L}+\mathbf{S}, splitting each term into fine-structure levels distinguished by J (the subject of Guide 3).
The term symbol. The left superscript $2S+1 is the spin multiplicity (1 = singlet, 2 = doublet, 3 = triplet); L$ is written as a letter S, P, D, F for L=0,1,2,3; the right subscript J is the total angular momentum. Example: sodium's ground state is ^2S_{1/2}.
Hund's rules — worked example
Which term is the ground state? For that we use Hund's rules, an empirical but well-understood ordering: (1) maximize total spin S — parallel spins avoid each other, lowering repulsion; (2) for that S, maximize L; (3) then set J=|L-S| if the subshell is less than half full, and J=L+S if it is more than half full (a half-filled shell gives L=0, so J=S).