We have introduced a single-particle Hamiltonian $$ H_0=\sum_{i=1}^A \hat{h}_0(x_i) = \sum_{i=1}^A\left(\hat{t}(x_i) + \hat{u}_{\mathrm{ext}}(x_i)\right), $$ with an external and central symmetric potential \( u_{\mathrm{ext}}(x_i) \), which is often approximated by a harmonic oscillator potential or a Woods-Saxon potential. Being central symmetric leads to a degeneracy in energy which is not observed experimentally. We see this from for example our discussion of separation energies and magic numbers. There are, in addition to the assumed magic numbers from a harmonic oscillator basis of \( 2,8,20,40,70\dots \) magic numbers like \( 28 \), \( 50 \), \( 82 \) and \( 126 \).

To produce these additional numbers, we need to add a phenomenological spin-orbit force which lifts the degeneracy, that is $$ \hat{h}(x_i) = \hat{t}(x_i) + \hat{u}_{\mathrm{ext}}(x_i) +\xi(\boldsymbol{r})\boldsymbol{ls}=\hat{h}_0(x_i)+\xi(\boldsymbol{r})\boldsymbol{ls}. $$