# How should a *good* nuclear theory explains spin-orbit coupling?

Under the framework of the independent-particle model in nuclear physics, the general form for the Hamiltonian is given by $$$$\label{1} H=\sum_{k=1}^{A}[T(k)+U(k)]+\left[\sum_{1=k

It is a sum of the independent-particle hamiltonian and a residual interaction in the form of a perturbation. The general idea was to reduce the problem of $$A$$-nucleon to $$A$$ problems of independent-nucleon. At first, the experienced potential $$U(k)$$ is chosen to be the harmonic oscillator, but this was not enough to produce all magic numbers, hence the introduction of the spin-orbit coupling $$U_{\mathrm{S} \mathrm{O}}(r)=f(r) l \cdot s$$ which could be defined as the interaction of orbital motion and intrinsic spin of each nucleon inside the nucleus; each nucleon is assumed to orbit around the center of the nucleus under the effect of other nucleons' potential.

From the book of Brussard "Shell Model Applications in Nuclear Spectroscopy", it is stated, "In spite of the strong empirical evidence for the spin-orbit term in the independent-particle model, satisfactory theoretical explanation of the presence of spin-orbit term has not yet been provided." My question is, could someone explain precisely how far this satisfactory theory goes beyond SM and supposed to be described? Because each theory has its limitations why the shell model is described as an empirical approach or not a satisfactory theory despite being an approximation to the nuclear interaction and the effective interactions used to calculate energies?