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Chemistry LibreTexts

8.9: The Allowed Values of J - the Total Angular Momentum Quantum Number

The total angular momentum quantum number parameterizes the total angular momentum of a given particle, by combining its orbital angular momentum and its intrinsic angular momentum (i.e., its spin).  Due to the spin-orbit interaction in the atom, the orbital angular momentum no longer commutes with the Hamiltonian, nor does the spin. These therefore change over time. However the total angular momentum \(J\) does commute with the Hamiltonian and so is constant. J is defined through

Figure \(\PageIndex{1}\): "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 components (see vector model of the atom. Image is used with permission (Public Domain; Maschen).

\[\begin{align} & |\bf{L}| = \hbar \sqrt{\ell(\ell+1)}, \quad L_z = m_\ell \hbar, \\& |\bf{S}| = \hbar \sqrt{s(s+1)}, \quad S_z = m_s \hbar, \\& |\bf{J}| = \hbar \sqrt{j(j+1)}, \quad J_z = m_j \hbar, \\\end{align} \]

where

  •  \(l\), is the azimuthal quantum number,
  • \(s\), is the spin quantum number intrinsic to the type of particle,
  • \(j\), is the  total angular momentum quantum number,

which respectively take the values:

\[\begin{align} & m_\ell \in \{ -\ell, -(\ell-1) \cdots \ell-1, \ell \} , \quad \ell \in \{ 0,1 \cdots n-1 \} \\& m_s \in \{ -s, -(s-1) \cdots s-1, s \} , \\& m_j \in \{ -j, -(j-1) \cdots j-1, j  \} , \\& m_j=m_\ell+m_s, \quad j=|\ell+s|\\\end{align} \]

and the magnitudes are:

 \[\begin{align} & |\bf{J}| = \hbar\sqrt{j(j+1)} \\& |\bf{J}_1| = \hbar\sqrt{j_1(j_1+1)} \\& |\bf{J}_2| = \hbar\sqrt{j_2(j_2+1)} \\\end{align} \]

in which

\[ j \in \{ |j_1 - j_2|, |j_1 - j_2| - 1 \cdots j_1 + j_2 - 1, j_1 + j_2 \} \,\! \]

This process may be repeated for a third electron, then the fourth etc. until the total angular momentum has been found.

Figure \(\PageIndex{2}\): Vector model of total angular momentum: spin and orbital coupling (spin-1/2 particles).Image is used with permission (Public Domain; Maschen).