For the atom, the Schrödinger Equation is known: write down $Z$, let the electrons interact by Coulomb, and out come the orbitals, the periodic table, the spectra — fixed by a law, nothing to fit. The atom is the solution to the equation.
For the nucleus, there is no such equation. You can write $H\Psi=E\Psi$, but the nuclear force has no closed form — only fitted potentials, each a different model, none canonical. There's no central potential either (the nucleus is self-bound), so no orbitals fall out — just a patchwork of effective models. And underneath it all isn't a Schrödinger problem at all, but QCD: nucleons and their "force" are merely emergent. For modern physicists asking for big money to a new accelerator to explore the nucleus, this is troublesome and root cause to the present crisis of particle physics.
- Atom: fundamental equation known, but hard to solve.
- Nucleus: no fundamental equation known, nothing canonical to even attempt to solve.
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