The hydrogen atom and atomic structure

Hydrogen — one proton, one electron — is the one atom whose quantum mechanics can be solved exactly, and it is the template for everything after it. Solving it means finding the wavefunctions of an electron in the proton's Coulomb attraction, and just as with a particle in a box, only certain solutions fit. The allowed energies come out as −13.6 eV/n², reproducing the spectral lines of hydrogen that had been measured for decades with no explanation.

Notice the shape of that ladder. Because of the 1/n², the levels are not evenly spaced: they crowd closer and closer together as n rises, converging on zero, which is the energy of an electron just barely unbound. The first jump, n = 1 to 2, costs 10.2 eV; the jump from n = 5 to 6 costs about 0.17 eV. This is why hydrogen's spectral series look the way they do, bunching up towards a series limit.

The solutions are labelled by three quantum numbers — n for the energy and overall size, l for the shape of the angular distribution, m for its orientation — and these are what generalise to every other element. What they describe are orbitals, and an orbital is not a path. There is no trajectory, no little planet circling; there is a standing wave, and |ψ|² gives the probability of finding the electron in each region. The familiar lobed p-orbital pictures are surfaces enclosing most of that probability, not routes. Add spin and the exclusion principle and these same orbitals, filled in order, produce the whole structure of the periodic table.