In order to understand the basics of the nucleon-nucleon interaction and the pertaining symmetries, we need to define the relevant quantum numbers and how we build up a single-particle state and a two-body state, and obviously our final holy grail, a many-boyd state.

• For the single-particle states, due to the fact that we have the spin-orbit force, the quantum numbers for the projection of orbital momentum \( l \), that is \( m_l \), and for spin \( s \), that is \( m_s \), are no longer so-called good quantum numbers. The total angular momentum \( j \) and its projection \( m_j \) are then so-called good quantum numbers.
• This means that the operator \( \hat{J}^2 \) does not commute with \( \hat{L}_z \) or \( \hat{S}_z \).
• We also start normally with single-particle state functions defined using say the harmonic oscillator. For these functions, we have no explicit dependence on \( j \). How can we introduce single-particle wave functions which have \( j \) and its projection \( m_j \) as quantum numbers?