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Shell Model of the Nucleus

Describes the arrangement of nucleons within the nucleus, analogous to electron shells in atoms, explaining energy levels and stability.
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The statement of the theorem

The nuclear shell model describes the energy levels En,l,jE_{n, l, j} of nucleons (protons and neutrons) within the mean field potential V(r)V(r) generated by all other nucleons. The single-particle Hamiltonian is h^=p⃗22m+V(r)\hat{h} = \frac{\vec{p}^2}{2m} + V(r). The solutions to the Schrödinger equation, h^ψ=Eψ\hat{h} \psi = E \psi, yield quantized energy levels. The effective potential V(r)V(r) is often approximated by a Woods-Saxon potential. The total energy of the nucleus is determined by filling these single-particle states according to the Pauli principle, leading to the total energy Etotal=∑i=1Z+NEiE_{total} = \sum_{i=1}^{Z+N} E_i, where EiE_i are the occupied single-particle energy eigenvalues.