Editing Azimuthal quantum number (section)

=== Total angular momentum of an electron in the atom ===
[[File:LS coupling.svg|thumb|"Vector cones" of total angular momentum '''J''' (purple), orbital '''L''' (blue), and spin '''S''' (green). The cones arise due to [[quantum uncertainty]] between measuring angular momentum component.]]

Due to the [[spin–orbit interaction]] in an atom, the orbital angular momentum no longer [[commutator|commutes]] with the [[Hamiltonian (quantum mechanics)|Hamiltonian]], nor does the [[Spin (physics)|spin]]. These therefore change over time. However the [[total angular momentum]] {{math|'''J'''}} does commute with the one-electron Hamiltonian and so is constant. {{math|'''J'''}} is defined as
<math display="block">\mathbf{J} = \mathbf{L} + \mathbf{S}</math>
{{math|'''L'''}} being the [[angular momentum operator|orbital angular momentum]] and {{math|'''S'''}} the spin. The total angular momentum satisfies the same [[Angular momentum operator#Commutation relations|commutation relations as orbital angular momentum]], namely
<math display="block">[J_i, J_j ] = i \hbar \varepsilon_{ijk} J_k</math>
from which it follows that
<math display="block">\left[J_i, J^2 \right] = 0</math>
where {{math|''J''<sub>''i''</sub>}} stand for {{math|''J''<sub>''x''</sub>}}, {{math|''J''<sub>''y''</sub>}}, and {{math|''J''<sub>''z''</sub>}}.

The quantum numbers describing the system, which are constant over time, are now {{math|''j''}} and {{math|''m''<sub>''j''</sub>}}, defined through the action of {{math|'''J'''}} on the wavefunction <math>\Psi</math>
<math display="block">\begin{align}
\mathbf{J}^2\Psi &= \hbar^2 j(j+1) \Psi \\[1ex]
\mathbf{J}_z\Psi &= \hbar m_j\Psi
\end{align}</math>

So that {{math|''j''}} is related to the norm of the total angular momentum and {{math|''m''<sub>''j''</sub>}} to its projection along a specified axis. The ''j'' number has a particular importance for [[relativistic quantum chemistry]], often featuring in subscript in for deeper states near to the core for which spin-orbit coupling is important.

As with any angular momentum in quantum mechanics, the projection of {{math|'''J'''}} along other axes cannot be co-defined with {{math|''J''<sub>z</sub>}}, because they do not commute.
The [[eigenvector]]s of {{math|''j''}}, {{math|''s''}}, {{math|''m''<sub>''j''</sub>}} and parity, which are also eigenvectors of the Hamiltonian, are linear combinations of the eigenvectors of {{math|''ℓ''}}, {{math|''s''}}, {{math|''m''<sub>''ℓ''</sub>}} and {{math|''m''<sub>''s''</sub>}}.