The course comprises two pen-and-paper exercises which are based on materials of the morning-session lectures of Day 1.

Dirac’s relation

A relation that is often exploited throughout the course, e.g. to achieve a separation of spin-dependent and independent terms in the Dirac equation (see lecture on Day 2), is Dirac’s relation, which can be written for two arbitrary vector operators \(\vec{u}\) and \(\vec{v}\) as:

\[(\vec{\sigma} \cdot \vec{u})(\vec{\sigma} \cdot \vec{v}) = \vec{u} \cdot \vec{v} I_{2} + i \vec{\sigma} \cdot (\vec{u} \times \vec{v})\]

where \(\vec{\sigma}\) are the Pauli spin matrices and \(I_{2}\) is a \(2 \times 2\) unit matrix. Note that \(\vec{u}\) and \(\vec{v}\) do not necessarily commute.

• Problem 1: Verify the relation expanding the left-hand side of Dirac’s relation. Hint: use the relations for the Pauli spin matrices:

\[\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z)\]\[\sigma_x^2 = \sigma_y^2 = \sigma_z^2 = I_{2}\]\[\sigma_i \sigma_j = \delta_{ij} I_{2} + i \sum_{k=1}^{3} \epsilon_{ijk} \sigma_k\]

• Problem 2: derive a final expression inserting for \(\vec{u} = \vec{v}\) the kinematical momentum operator \(\vec{\pi} = \vec{p} - \frac{q_e}{c}\vec{A}\) (where \(\vec{A}\) is an external electromagnetic vector potential). Hint:

\[[\pi_i,\pi_j] = \frac{q}{c} i \hbar \sum\limits_{k=1}^{3} \epsilon_{ijk} B_k\]

Two-component Pauli equation (0th order Pauli equation)

From the one-electron Dirac equation in external magnetic fields we may derive the Pauli equation (also known as 0th order Pauli equation) by considering the non-relativistic limit \(c \rightarrow \infty\).

\[\left[ \frac{\vec{p}^2}{2m_e} + \frac{q_e^2 \vec{A}^2}{2m_e c^2} - \frac{q_e}{2m_ec}(\bf{l} + 2 \bf{s}) \cdot \vec{B} + V \right] \Psi^L = i \hbar \frac{\partial}{\partial t} \Psi^L\]

• Problem 3: Derive the above equation starting from the one-electron Dirac equation in external magnetic fields written in two-spinor form:

\[\begin{split}i \hbar \frac{\partial}{\partial t} \left( \begin{array}{c} \Psi^L \\ \Psi^S \end{array} \right) = c \left( \begin{array}{c} (\vec{\sigma} \cdot \vec{\pi}) \Psi^S \\ (\vec{\sigma} \cdot \vec{\pi}) \Psi^L \end{array} \right) + m_ec^2 \left( \begin{array}{c} \Psi^L \\ -\Psi^S \end{array} \right) + V \left( \begin{array}{c} \Psi^L \\ \Psi^S \end{array} \right)\end{split}\]

The following hints may be useful:

1. shift the energy limit to the non-relativistic limit (E = 0 rather than E = \(m_ec^2\)):

\[V \rightarrow V - m_ec^2\]

2. elimininate the small-component \(\Psi^S\) from the upper component using the kinetic balance condition:

\[\Psi^S \approx \frac{\vec{\sigma} \cdot \vec{\pi}}{2m_ec^2} \Psi^L\]

3. use \(\vec{A} \cdot \vec{p} = \vec{p} \cdot \vec{A} - (\vec{p} \cdot \vec{A})\)
4. remember for the Coulomb gauge:

\[a div(\vec{A}) = 0\]

5. for a constant and homogeneous magnetic field \(\bf{B} = curl(\bf{A})\), we may write \(\bf{A} = \frac{1}{2}(\bf{B} \times \bf{r})\)

• Problem 4: define the Bohr magneton and the gyromagnetic ratio g of the electron (Lande factor) according to the Pauli Hamiltonian.

Literature and further reading

• 11. Dyall and K. Fœgri, Introduction to Relativistic Qunatum Chemistry, Chapter 4.
• 13. Reiher and M. Wolf, Relativistic Quantum Chemistry, Chapter 5.