In a two-level unperturbed system with unperturbed Hamiltonian $H_0$ and corresponding eigenstates $|\phi_1\rangle$, $|\phi_2\rangle$ and eigenvalues $E_1$, $E_2$ of it, any state $|\psi(t)\rangle=a_1(t)|\phi_1\rangle+a_2(t)|\phi_1\rangle$, a stationary state.

Now if a perturbation W is included, such that $W_{11}=W_{22}=0$ (for simplicity) and $W_{12}=W^{*}_{21}\neq 0$, then $H=H_0+W$ and the corresponding eigenvectors and values of this hamiltonian would be $|\psi_{+}\rangle$ with $E_+$ and $|\psi_{-}\rangle$ with $E_-$. An arbitrary state $|\psi(t)\rangle=a_3(t)|\psi_{+}\rangle + a_4|\…

As I am progressing with my notes, I now am dealing with the angular momentum, its algebra, eigenvectors and values etc. In the notes we generalized by considering an angular momentum $\vec J$, that can represent the orbital angular momentum, the spin or any other type. The following claim is made:
$\vec J^2$ and $J_z$ build no complete set of commuting observables, therefore an additional index is needed to distinguish the different joint eigenvectors belonging to the same pair of eigenvalues $j(j+1)\hbar$ and $m\hbar$. An eigenstate can be written as $|k,j,m'rangle$.