To see this, we consider the following example and fix
$$
j=\frac{3}{2}=l-s\hspace{1cm} m_j=\frac{3}{2}.
$$
and
$$
j=\frac{5}{2}=l+s\hspace{1cm} m_j=\frac{3}{2}.
$$
It means we can have, with \( l=2 \) and \( s=1/2 \) being fixed, in order to have \( m_j=3/2 \) either \( m_l=1 \) and \( m_s=1/2 \) or
\( m_l=2 \) and \( m_s=-1/2 \). The two states
$$
\psi_{n=0j=5/2m_j=3/2;l=2s=1/2}
$$
and
$$
\psi_{n=0j=3/2m_j=3/2;l=2s=1/2}
$$
will have admixtures from \( \phi_{n=0l=2m_l=2s=1/2m_s=-1/2} \) and \( \phi_{n=0l=2m_l=1s=1/2m_s=1/2} \).
How do we find these admixtures? Note that we don't specify the values of \( m_l \) and \( m_s \)
in the functions \( \psi \) since
\( \hat{j}^2 \) does not commute with \( \hat{L}_z \) and \( \hat{S}_z \).