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

Note that we use the Einstein summation convention; in the final expression above the index \(k\) appears twice in the same term, which implies that we sum over it.

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

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 \(c \rightarrow \infty\).

\[\left[ \frac{\vec{p}^2}{2m_e} + \frac{e^2 \vec{A}^2}{2m_e} + \frac{e}{2m_e}(\mathbf{l} + 2 \mathbf{s}) \cdot \vec{B} + V \right] \psi^L = i \hbar \frac{\partial}{\partial t} \psi^L\]
  • 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 zero of energy to the non-relativistic limit one. This corresponds to the elimination of rest mass and is achieved by the substitution
\[V \rightarrow V - m_ec^2\]
  1. elimininate the small-component \(\psi^S\) from the upper component using the magnetic balance condition:
\[\psi^S \approx \frac{\vec{\sigma} \cdot \vec{\pi}}{2m_ec} \psi^L\]
  1. use \(\vec{A} \cdot \vec{p} = \vec{p} \cdot \vec{A} - (\vec{p} \cdot \vec{A})\)

    and remember that in Coulomb gauge we have

\[\vec{\nabla}\cdot\vec{A} = 0\]
  1. for a constant and homogeneous magnetic field \(\vec{B} = \vec{\nabla}\times\vec{A}\), we may write \(\vec{A} = \frac{1}{2}(\vec{B} \times \vec{r})\)
  • Problem 4: define the Bohr magneton and the gyromagnetic ratio g of the electron according to the Pauli Hamiltonian.

Literature and further reading

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