16.3. Total Angular Momentum

State function

\[| \Psi \rangle = | \psi \rangle \otimes | \sigma \rangle \quad \in \quad \mathcal{H} = \mathcal{H}_{space} \otimes \mathcal{H}_{spin} \ .\]

Total angular momentum operator

\[\hat{\mathbf{J}} = \hat{\mathbf{L}} + \hat{\mathbf{S}} = \hat{\mathbf{L}} \otimes \hat{\mathbf{1}}_{spin} + \hat{\mathbf{1}}_{space} \otimes \hat{\mathbf{S}}\]

Dot product of \(\hat{\mathbf{L}}\) and \(\hat{\mathbf{S}}\),

\[\hat{\mathbf{L}} \cdot \hat{\mathbf{S}} = L_x \otimes S_x + L_y \otimes S_y + L_z \otimes S_z \ .\]

Magnitude square of \(\hat{\mathbf{J}}\), \(\hat{J}^2\),

\[\begin{split}\begin{aligned} \hat{J}^2 & = ( \hat{\mathbf{L}} + \hat{\mathbf{S}} ) \cdot ( \hat{\mathbf{L}} + \hat{\mathbf{S}} ) = \\ & = \hat{L}^2 \otimes \hat{\mathbf{1}} + 2 \hat{\mathbf{L}} \cdot \hat{\mathbf{S}} + \hat{\mathbf{1}} \otimes \hat{S}^2 \ . \end{aligned}\end{split}\]

Orbital angular momentum.

\[\begin{split}\begin{aligned} \hat{L}^2 | \ell, m_l \rangle & = \hbar^2 \ell (\ell+1) | \ell, m_l \rangle && , \quad \ell \in \{ 0, 1, 2, \dots \} \\ \hat{L}_z | \ell, m_l \rangle & = \hbar m_l | \ell, m_l \rangle && , \quad m_l \in \{ - \ell, - \ell+1, \dots, \ell-1, \ell \} \end{aligned}\end{split}\]

Spin angular momentum.

\[\begin{split}\begin{aligned} \hat{S}^2 | s, m_s \rangle & = \hbar^2 s (s +1) | s, m_s \rangle && , \quad s \in \left\{ 0, \frac{1}{2}, 1, \dots \right\} \\ \hat{S}_z | s, m_s \rangle & = \hbar m_s | s, m_s \rangle && , \quad m_s \in \{ - s, - s+1, \dots, s-1, s \} \end{aligned}\end{split}\]

For an electron \(s = \frac{1}{2}\), so that \(m_s \in \left\{ - \frac{1}{2}, \frac{1}{2} \right\}\), and thus, without explicitly writing \(s\) in the eigenvectors - being that a given parameter -,

\[\begin{split}\begin{aligned} \hat{S}^2 | m_s \rangle & = \frac{3}{4}\hbar^2 | m_s \rangle && \\ \hat{S}_z | m_s \rangle & = \hbar m_s | m_s \rangle && , \quad m_s \in \left\{ -\frac{1}{2}, \frac{1}{2} \right\} \end{aligned}\end{split}\]

Total angular momentum. The \(z\)-component of the total angular momentum, and its magnitude squared are defined as

\[\begin{split}\begin{aligned} \hat{J}^2 | j, m_j \rangle & = \hbar^2 j (j +1) | j, m_j \rangle && , \quad j \in \left\{ 0, \frac{1}{2}, 1, \dots \right\} \\ \hat{J}_z | j, m_j \rangle & = \hbar m_j | j, m_j \rangle && , \quad m_j \in \{ - j, - j+1, \dots, j-1, j \} \end{aligned}\end{split}\]

16.3.1. Eigenvalue problem of the total angular momentum for an electron

Commutation of operators. The operators \(\hat{J}^2\), \(\hat{J}_z\), \(\hat{L}^2\), \(\hat{S}^2\) commute, and thus they share common eigenvectors

\[| j, m_j, \ell, s \rangle \ .\]

The operator \(\hat{J}_z\) commutes with \(\hat{L}^2\) and \(\hat{S}^2\), \(\left[ \hat{J}_z, \hat{L}^2 \right] = 0\), \(\left[ \hat{J}_z, \hat{S}^2 \right] = 0\).

Proof

\[\begin{split}\begin{aligned} \left[ \hat{J}_z, \hat{L}^2 \right] & = \hat{J}_z \hat{L}^2 - \hat{L}^2 \hat{J}_z = \\ & = \hat{L}_z \hat{L}^2 + \hat{S}_z \hat{L}^2 - \hat{L}^2 \hat{L}_z - \hat{L}^2 \hat{S}_z = \\ & = \underbrace{\left[ \hat{L}_z, \hat{L}^2 \right]}_{ = 0 } + \underbrace{\left[ \hat{S}_z, \hat{L}^2 \right]}_{ = 0} = 0 \ . \end{aligned}\end{split}\]

The last term holds because \(\hat{S}_z\) and \(\hat{L}^2\) acts on two different sub-spaces, or explicitly for any \(| \Psi \rangle = | \psi \rangle \otimes | \sigma \rangle\),

\[\begin{split}\begin{aligned} \hat{S}_z \hat{L}^2 | \Psi \rangle & = ( \hat{\mathbf{1}} \otimes \hat{S}_z ) ( \hat{L}^2 \otimes \hat{\mathbf{1}} ) | \psi \rangle \otimes | \sigma \rangle = \\ & = ( \hat{\mathbf{1}} \otimes \hat{S}_z ) ( \hat{L}^2 | \psi \rangle \otimes | \sigma \rangle ) = \\ & = \hat{L}^2 | \psi \rangle \otimes \hat{S}_z | \sigma \rangle \ , \end{aligned}\end{split}\]

and \(\hat{L}^2 \hat{S}_z | \Psi \rangle\) provides the same result for any \(| \Psi \rangle = | \psi \rangle \otimes | \sigma \rangle\).

The operator \(\hat{L}_z\) doesn’t commute with \(\hat{J}^2\), \(\left[ \hat{L}_z, \hat{J}^2 \right] = - 2 i \hbar \, \hat{\mathbf{z}} \cdot \hat{\mathbf{L}} \times \hat{\mathbf{S}}\).

Proof

\[\begin{split}\begin{aligned} \left[ \hat{L}_z, \hat{J}^2 \right] & = \hat{L}_z \hat{J}^2 - \hat{J}^2 \hat{L}_z = \\ & = \hat{L}_z \left( \hat{L}^2 + \hat{S}^2 + 2 \hat{\mathbf{L}} \cdot \hat{\mathbf{S}} \right) - \left( \hat{L}^2 + \hat{S}^2 + 2 \hat{\mathbf{L}} \cdot \hat{\mathbf{S}}\right) \hat{L}_z = \\ & = \underbrace{\left[ \hat{L}_z, \hat{L}^2 \right]}_{ = 0 } + \underbrace{\left[ \hat{L}_z, \hat{S}^2 \right]}_{ = 0} + 2 [\hat{L}_z, \hat{L}_a] \hat{S}_a = \\ & = 2 i \hbar \hat{L}_y \hat{S}_x - 2 i \hbar \hat{L}_x \hat{S}_y = \\ & = - 2 i \hbar \, \hat{\mathbf{z}} \cdot \hat{\mathbf{L}} \times \hat{\mathbf{S}} \ , \end{aligned}\end{split}\]

as \([\hat{L}_z, \hat{L}_x] = i \hbar \hat{L}_y\), \([\hat{L}_y, \hat{L}_z] = i \hbar \hat{L}_x\), \([\hat{L}_z, \hat{L}_z] = 0\),

\(z\)-component operators.

\[\begin{split}\begin{aligned} \hat{J}_z | \Psi_{m_\ell, m_s} \rangle & = ( \hat{L}_z + \hat{S}_z ) | m_\ell \rangle \otimes | m_s \rangle = \\ & = \hat{L}_z | m_\ell \rangle \otimes | m_s \rangle + \hat{S}_z | m_\ell \rangle \otimes | m_s \rangle = \\ & = \hbar m_\ell | m_\ell \rangle \otimes | m_s \rangle + \hbar m_s | m_\ell \rangle \otimes | m_s \rangle = \\ & = \hbar \left( m_\ell + m_s \right) | m_\ell \rangle \otimes | m_s \rangle = \\ & = \hbar \, m_j \, | m_\ell \rangle \otimes | m_s \rangle \ , \end{aligned}\end{split}\]

and thus the relation \(m_j = m_{\ell} + m_s\) for every pair of eigenvectors \(| m_{\ell} \rangle\) and \(| m_s \rangle\) of the operators \(\hat{L}_z\), and \(\hat{S}_z\). It also follows, that the extreme values of \(m_j\) are \(-j\) and \(j = \ell + s\).