The one atom solved exactly: its levels, shells, spectrum and fine structure.
With the force pinned, the electron's radial equation in 4D is our own, with one change: its angular momentum is shifted by a half, L = l + ½. So 4D hydrogen can be solved in closed form, like ours. Its levels are
3D: E = −Z² / 2n² 4D: E = −Z² / 2(n + ½)² (in hartrees, 27.2 eV)
The ground state is bound by 6.05 eV, not 13.6, and the atom is twice the size (its average radius is 3 bohr to our 1.5). As in 3D, every level's energy depends only on n, not l. That's the same hidden symmetry that closes planets' orbits, and it survives in any dimension as long as the force is 1/r. But the shells are bigger, because a 4D electron has more ways to point: (l + 1)² orbitals for each l, where we have 2l + 1.
The light 4D hydrogen gives out. Our Balmer lines (falling to n = 2) run red to violet. Its own fall to n = 2 crowds into the red and orange, ending in yellow at 570 nm, and its Lyman series is out in the ultraviolet.
The (l + 1)² orbitals of each l are the harmonic polynomials of degree l in four variables. They are the pair of spins (l/2, l/2) under 4D turning's two halves. An electron's spin joins the left half only (motion and spin). So the coupling between spin and orbit splits each level in two, as j = l ± ½ does ours: one part with spin added to the left half's l/2 and one with it taken away. The counts:
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