This means that the eigenvectors \( \phi_{n=0l=2m_l=2s=1/2m_s=-1/2} \) etc are not eigenvectors of \( \hat{j}^2 \). The above problems gives a \( 2\times2 \) matrix that mixes the vectors \( \psi_{n=0j=5/2m_j3/2;l=2m_ls=1/2m_s} \) and \( \psi_{n=0j=3/2m_j3/2;l=2m_ls=1/2m_s} \) with the states \( \phi_{n=0l=2m_l=2s=1/2m_s=-1/2} \) and \( \phi_{n=0l=2m_l=1s=1/2m_s=1/2} \). The unknown coefficients \( \alpha \) and \( \beta \) are the eigenvectors of this matrix. That is, inserting all values \( m_l,l,m_s,s \) we obtain the matrix $$ \left[ \begin{array} {cc} 19/4 & 2 \\ 2 & 31/4 \end{array} \right]$$ whose eigenvectors are the columns of $$ \left[ \begin{array} {cc} 2/\sqrt{5} &1/\sqrt{5} \\ 1/\sqrt{5} & -2/\sqrt{5} \end{array}\right]$$ These numbers define the so-called Clebsch-Gordan coupling coefficients (the overlaps between the two basis sets). We can thus write $$ \psi_{njm_j;ls}=\sum_{m_lm_s}\langle lm_lsm_s|jm_j\rangle\phi_{nlm_lsm_s}, $$ where the coefficients \( \langle lm_lsm_s|jm_j\rangle \) are the so-called Clebsch-Gordan coeffficients.