Unless we have observables which depend on the magnetic quantum numbers, the degeneracy given by these quantum numbers is not seen experimentally. The typical situation when we perform shell-model calculations is that the results depend on the magnetic quantum numbers. The reason for this is that it is easy to implement the Pauli principle for many particles when we work in what we dubbed for \( m \)-scheme.

A resulting state in a shell-model calculations will thus depend on the total value of \( M \) defined as $$ M=\sum_{i=1}^{A}m_{j_i}. $$ A shell model many-body state is given by a linear combination of Slater determinants \( \vert \Phi_i\rangle \). That is, for some conserved quantum numbers \( \lambda \) we have $$ \vert Psi_{\lambda}\rangle = \sum_{i} C_i \vert \Phi_i\rangle, $$ where the coefficients are the overlaps between the many-body basis sets \( \Psi \) and \( \Phi \) and are the resulting eigenvectors from the shell-model calculations.