In the shell model, each nucleon moves in an average mean field generated by all others. A strong spin-orbit term splits single-particle levels and explains the magic numbers $2, 8, 20, 28, 50, 82, 126$. The mean-field picture is the starting point for more advanced configuration-interaction approaches.

Nucleon pairing produces superfluid-like behavior in nuclei, especially in even-even systems. In a BCS-like treatment, the pairing gap $\Delta$ reduces odd-even mass staggering and alters low-lying spectra. Quasiparticles are the natural excitations of the paired mean field.

When nuclei are deformed, rotational bands emerge with energies approximately following $$E_J \approx \frac{\hbar^2}{2\mathcal{I}}J(J+1)$$ where $\mathcal{I}$ is the moment of inertia. Such spectra reveal intrinsic quadrupole deformation and broken rotational symmetry at the mean-field level.