The total two-nucleon state function has to be anti-symmetric. The total function contains a spatial part, a spin part and an isospin part. If isospin is conserved, this leads to in case we have an \( s \)-wave with spin \( S=0 \) to an isospin two-body state with \( T=1 \) since the spatial part is symmetric and the spin part is anti-symmetric.

Since the projections for \( T \) are \( T_z=-1,0,1 \), we can have a \( pp \), an \( nn \) and a \( pn \) state.

For \( l=0 \) and \( S=1 \), a so-called triplet state, \( ^3S_1 \), we must have \( T=0 \), meaning that we have only one state, a \( pn \) state. For other partial waves, the following table lists states up to \( f \) waves. We can systemize this in a table as follows, recalling that \( |\mathbf{l}-\mathbf{S}| \le |\mathbf{J}| \le |\mathbf{l}+\mathbf{S}| \),